Smooth K-Theory

SMOOTH K-THEOR Y by Ulric h Bunk e & Thomas Sc hic k De dic ate d to Je an-Michel Bismut on the o c c asion of his 60th birthday A bstr act . — In this paper we consider smooth extensions of cohomology theo- ries. In particular we construct an analytic multiplicat ive mo del of s mooth K -theory . W e further int ro duce the notion of a smo oth K -ori en tation of a prop er submersion p : W → B and deﬁne the asso ciated push-f orw ard ˆ p ! : ˆ K ( W ) → ˆ K ( B ). W e show that the push-forward has the exp ected prop erties as functoriali t y , compatibilit y with pull-back diagrams, pro jection formula and a bor dism formula. W e construct a multiplicative l ift of the Chern charac ter ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ), where ˆ H ( B, Q ) denote s the smooth exten sion of rational cohomology , and we show that ˆ c h induces a rational isomorphism. If p : W → B is a prop er submersion with a smo oth K -orienta tion, then we deﬁne a class A ( p ) ∈ ˆ H ev ( W, Q ) (see Lemma 6.17) and the modiﬁed push-forward ˆ p A ! := ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W , Q ) → ˆ H ( B, Q ) . One of our main r esults lifts the cohomological v ersi on of the Atiy ah-Singer index theorem to s mooth cohomology . It s tate s that ˆ p A ! ◦ ˆ c h = ˆ c h ◦ ˆ p ! . 2000 Mathematics Subje c t Classiﬁc ation . — 19L10, 58J28. Key wor ds and phr ases . — Deligne cohomology , smo oth K-theory , Chern c haracter, families of elliptic op erators, Atiy ah-Singer index theorem. Thomas Sc hick was funded by Couran t Researc h Cen ter G¨ ottingen “Higher order structures i n mathematics” via the German Initiativ e of Excellence. 2 ULRICH BUNKE & THOMAS SCHICK R´ esum´ e (K-theorie di ﬀeren tiable). — Nous considerons les extensions diﬀer- en tiables des theories de cohomology . En particulier, nous construisons un m od` ele analytique et av ec multiplication de la K-theorie diﬀerentiable. Nous introduisons le concept d’une K- or ien tation diﬀerentiable d’une submer sion propre p : W → B . Nous contruisons une application d’integration asso ci´ e ˆ p ! : ˆ K ( W ) → ˆ K ( B ); et nous demon trons les propri´ et´ es attendues comme functorialit´ e, compatibilit´ e av ec pull- bac k, formules de pro jection et de b ordism. Nous construisons une version diﬀerentiable du charact ` ere de Chern ˆ c h : ˆ K ( B ) → ˆ H ( B, Q ), o` u ˆ H ( B, Q ) est une extension diﬀerentiable de la cohomologie rationelle, et nous demontrons que ˆ c h induit un isomorphis me rationel. Si p : W → B est une submersion propre av ec une K -orienta tion diﬀerentiab le, nous deﬁnissons une classe A ( p ) ∈ ˆ H ev ( W, Q ) (compare Lemma 6.17) et une ap- plication d’inte gration mo diﬁ´ e ˆ p A ! := ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W , Q ) → ˆ H ( B, Q ) . Un de nos resultats principales est une version en cohomologie diﬀeren tiable du theor ` eme d’indice de Atiy ah-Singer. Cette version dits que ˆ p A ! ◦ ˆ c h = ˆ c h ◦ ˆ p ! . Con tent s 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. The main res ults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. A short introduction to smo oth cohomology theories . . . . . . . . 6 1.3. Related constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2. Deﬁnition of smo oth K-theory via cycles and relations . . . . . . . . . . 14 2.1. Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2. Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.3. Smo oth K -theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4. Natura l transformations and exact sequences . . . . . . . . . . . . . . . 22 2.5. Compa rison with the Hopkins-Singer theory and the ﬂat theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 6 3. Push-for w ar d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1. K -orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 9 3.2. Deﬁnition of the Push-forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 3.3. F u nctor ialit y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4. The cup pro duct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.1. Deﬁnition of the product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.2. Pr o jection for m ula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.3. Susp ension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5. Constructions of na tural smo oth K -theory classes . . . . . . . . . . . . . . . 5 1 5.1. Calcula tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.2. The smoo th K - theory class of a mapping torus . . . . . . . . . . . . . 52 5.3. The smo oth K -theor y class of a geometric family with kernel bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.4. A canonical ˆ K 1 -class on S 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.5. The pro duct of S 1 -v alued maps and line-bundles . . . . . . . . . . . . 56 5.6. A bi-inv ar ia n t ˆ K 1 - class on S U (2) . . . . . . . . . . . . . . . . . . . . . . . . . . 57 SMOOTH K-THEOR Y 3 5.7. Inv arian t classes o n homog eneous spac e s . . . . . . . . . . . . . . . . . . . . 59 5.8. Bo r dism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1 5.9. Z /k Z -inv arian ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.10. S pin c -b ordism inv a riants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.11. The e -inv ar ian t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6. The Cher n character a nd a smo oth Grothendieck-Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.1. Smo oth r a tional cohomo logy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.2. Cons tr uction o f the Cher n character . . . . . . . . . . . . . . . . . . . . . . . 70 6.3. The Chern ch ar acter is a rational is omorphism and mul tiplicative 76 6.4. Riemann Roch theor e m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 1. In tro duction 1.1. The m ain results. — 1.1.1 . — In this pap er we construct a mo del of a smo oth extension of the generalized cohomolog y theory K , co mplex K -theory . Historically , the concept of smo oth exten- sions of a co homology theory s ta rted with smo oth integral cohomo logy [ CS85 ], also called rea l Deligne cohomology , see [ B ry93 ]. A second, geometric mo del of smo oth in tegr al cohomo lo gy is given in [ CS85 ], where the smo oth integral cohomolo gy classes were called diﬀerent ial characters. One impor tan t motiv ation of its deﬁnition was that one can as s ocia te na tural diﬀeren tial c har acters to hermitean vector bundles with con- nection which reﬁne the Chern classes. The diﬀerential character in deg r ee tw o even classiﬁes hermitean line bundles with co nnection up to isomo r phism. The multiplica- tiv e s tructure of smo oth integral c o homology als o enco des cohomolog y op erations, s e e [ Gom ]. The ho lomorphic counterpart of the theory b ecame an imp ortant ingredient of arithmetic geometr y . 1.1.2 . — Mo tiv ated b y the pr oblem of setting up lagr angians for quan tum ﬁeld the- ories with diﬀeren tial form ﬁeld strength it was argued in [ FH00 ], [ F re00 ] that o ne may need smo oth extensions of other generalized cohomolo g y theories . The choice of the generalized coho mo logy theor y is here dictated by a charge qua n tization condi- tion, which ma thematically is reﬂected by a lattice in real cohomology . Let N b e a graded re a l vector space such that the ﬁeld strength lives in Ω d =0 ( B ) ⊗ N , the closed forms on the manifold B with co eﬃcients in N . Let L ( B ) ⊂ H ( B , N ) b e the lattice given by the c har ge quantization condition on B . Then one lo oks for a generalized cohomolog y theory h and a natural transformation c : h ( B ) → H ( B , N ) such that c ( h ( B )) = L ( B ). It was a rgued in [ FH00 ], [ F re00 ] that the ﬁelds of the theory 4 ULRICH BUNKE & THOMAS SCHICK should b e consider e d as cycles for a smo oth extension ˆ h of the pair ( h, c ). F or exam- ple, if N = R and the charge quantization leads to L ( B ) = im ( H ( B , Z ) → H ( B , R )), then the relev ant smo oth extension co uld b e the smo oth integral co ho mology theory of [ CS85 ]. In Subsection 1.2 we will introduce the notion of a smo oth extension in an axiomatic wa y . 1.1.3 . — [ F re00 ] pr opo ses in particular to consider smo oth extensions of co mplex and real versions of K -theory . In that pap er it was furthermore indicated how cycle mo dels of such smo oth extensions co uld lo ok like. The goal of the present pap er is to carry thro ug h this pr o gram in the case of complex K -theo ry . 1.1.4 . — In the remainder of the present subsectio n we des cribe, expa nding the a b- stract, our main results. The main ingredient is a constr uction of an analytic mo del of smo oth K -theory (1) using cycles and r elations. 1.1.5 . — O ur philosophy for the co nstruction of smo oth K -theory is that a vector bundle with connectio n or a family of Dirac op erators with some additional g eome- try s ho uld represe n t a smoo th K -theor y class tauto lo gically . In this wa y we follow the outline in [ F re00 ]. Our class of cycles is quite big . This makes the constr uc- tion of smo oth K -theory classes or tra ns fo rmations to smo oth K - theory easy , but it complicates the veriﬁcation that cer tain cycle lev el co nstructions o ut o f smo oth K - theory are well-deﬁned. The g reat adv an tage of o ur ch oice is that the constructions of the pro duct and the push- forward on the level of cy c les ar e of diﬀerential geometric nature. More precisely we us e the notion of a g eometric family which was introduced in [ Bun ] in order to s ubsume a ll g eometric data needed to deﬁne a Bism ut sup er- connection in one notion. A cycle of the smo oth K -theory ˆ K ( B ) o f a compact manifold B is a pair ( E , ρ ) of a geometric family E and a n element ρ ∈ Ω( B ) / i m ( d ), see Section 2. Therefor e, cycles are diﬀerential ge o metric o b jects. Secondar y sp ectral inv ar ian ts from lo cal index theory , namely η - fo r ms, en ter the deﬁnition of the relations (see Deﬁnition 2.10). The ﬁrs t main res ult is that our construction really yields a smo oth extension in the sense of Deﬁnition 1.1. 1.1.6 . — Our smo oth K -theor y ˆ K ( B ) is a contrav ar iant functor on the c a tegory of compact smo oth manifolds (poss ibly with boundar y) with v alues in the categor y of Z / 2 Z -graded rings. This multiplicativ e str uctur e is exp ected since K -theo ry is a mul- tiplicativ e generalize d co homology theory , and the Chern c har acter is m ultiplicative, to o. As said ab ov e, the co nstruction of the pro duct on the level of cycles (Deﬁni- tion 4.1) is of diﬀerential-geometric nature. Analysis en ters the veriﬁcation of w ell- deﬁnedness. The main re s ult is here that our construction produces a mult iplicative smo oth extension in the s ense of Deﬁnition 1.2. (1) or diﬀerent iable K - theory in the language of other authors SMOOTH K-THEOR Y 5 1.1.7 . — Let us co nsider a prop er submersion p : W → B with closed ﬁbres which has a top olog ica l K -o rien tation. Then we have a push-fo r w ar d p ! : K ( W ) → K ( B ), and it is an impor tan t part of the theory to extend this push-forward to the smo oth extension. F or this pur p ose one needs a smo oth reﬁnement of the notion of a K -orientation which we introduce in 3 .5. W e then deﬁne the as so c ia ted push-forward ˆ p ! : ˆ K ( W ) → ˆ K ( B ), aga in by a diﬀerential-geometric construction on the level o f cycles (17). W e show that the push-for w ar d has the e x pected prop erties: functorialit y , compatibility with pull-back diagra ms, pr o jection for m ula, b ordism form ula. 1.1.8 . — Let V = ( V , h V , ∇ V ) b e a hermitean vector bundle with co nnection. In [ CS85 ] a smo oth r e ﬁnement ˆ c h ( V ) ∈ ˆ H ( B , Q ) of the Cher n character was con- structed. In the present pap er we construct a lift o f the Cher n character c h : K ( B ) → H ( B , Q ) to a multiplicativ e natural transformation of smoo th co homology theories (see (30)) ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) such that ˆ c h ( V ) = ˆ c h ([ V , 0]), where V is the geometric family determined b y V . The Chern character induces a natural iso morphism of Z / 2 Z -g raded rings ˆ K ( B ) ⊗ Q ∼ → ˆ H ( B , Q ) (Prop osition 6.12 ). 1.1.9 . — If p : W → B is a prop er submer sion with a smo oth K - o rient atio n, then we deﬁne a class (see Lemma 6.17 ) A ( p ) ∈ ˆ H ev ( W , Q ) and the mo diﬁed push-forward ˆ p A ! := ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B , Q ) . Our index theorem 6.19 lifts the characteristic class version of the Atiy a h-Singer index theorem to smoo th cohomo logy . It states that the diagr am ˆ K ( W ) ˆ p !   ˆ c h / / ˆ H ( W, Q ) ˆ p A !   ˆ K ( B ) ˆ c h / / ˆ H ( B , Q ) commut es. 1.1.1 0. — In Subsection 1.2 w e pres e nt a shor t int ro duction to the theor y of smo oth extensions o f generalized co ho mology theories. In Subsection 1.3 we review in so me detail the literature abo ut v ariants of smo oth K -theory and as soc iated index theo- rems. In Section 2 we pres en t the cycle mo del of smo oth K -theory . The main result is the veriﬁcation that o ur co nstruction satisﬁes the a xioms g iv en b elow. Section 3 is devoted to the push-for w ar d. W e in tro duce the notion of a smo oth K -orientation, and we construct the push-forward on the cycle level. The main r esults are that the push-forward descends to smoo th K -theory , and the v eriﬁca tion of its functorial prop erties. In Sec tio n 4 we discuss the ring structure in smo oth K -theory and its 6 ULRICH BUNKE & THOMAS SCHICK compatibilit y with the push-forward. Section 5 presents a collection o f natura l con- structions of smo oth K -theor y classes. In Section 6 we construct the Chern character and prove the s mo oth index theorem. 1.2. A s hort in tro duction to smo oth cohomolo gy theories . — 1.2.1 . — The ﬁrst e x ample of a smo oth coho mology theo ry app eared under the name Cheeger-Simons diﬀerential characters in [ CS85 ]. Given a discrete subring R ⊂ R we hav e a functor (2) B 7→ ˆ H ( B , R ) fr o m smo oth ma nifolds to Z -graded r ings. It comes with natura l transformations 1. R : ˆ H ( B , R ) → Ω d =0 ( B ) (curv ature) 2. I : ˆ H ( B , R ) → H ( B , R ) (forget smo oth data) 3. a : Ω( B ) / im ( d ) → ˆ H ( B , R ) (action of for ms). Here Ω( B ) and Ω d =0 ( B ) denote the space of smo oth r e a l diﬀerent ial forms and its subspace of closed forms. The map a is of degree 1. F urthermor e, one has the following prop erties, all shown in [ CS85 ]. 1. The following diag ram commut es ˆ H ( B , R ) R   I / / H ( B , R ) R → R   Ω d =0 ( B ) dR / / H ( B , R ) , where dR is the de Rham homomorphism. 2. R and I are ring homo mo rphisms. 3. R ◦ a = d , 4. a ( ω ) ∪ x = a ( ω ∧ R ( x )), ∀ x ∈ ˆ H ( B , R ), ∀ ω ∈ Ω( B ) / im ( d ), 5. The s e quence H ( B , R ) → Ω( B ) / im ( d ) a → ˆ H ( B , R ) I → H ( B , R ) → 0 (1) is exact. 1.2.2 . — Cheeger - Simons diﬀerential c har acters a re the ﬁrst example of a mo re gen- eral structure which is describ ed for instance in the ﬁrst section of [ F re00 ]. In view of our constructions of examples for this structure in the ca s e of b ordism theories and K -theor y , and the presence of completely diﬀerent pictures like [ HS05 ] we think that an axio matic description o f smo o th cohomo logy theor ies is useful. Let N be a Z -graded vector space ov er R . W e co nsider a generalized cohomology theory h with a natura l transfo r mation of cohomology theor ies c : h ( B ) → H ( B , N ) . The natura l universal example is given by N := h ∗ ⊗ R , wher e c is the canonica l (2) In the literature, this group is sometimes denoted by ˆ H ( B, R / R ), p ossibly wi th a degree-shift by one. SMOOTH K-THEOR Y 7 transformation. Let Ω( B , N ) := Ω( B ) ⊗ R N . T o a pair ( h, c ) we as so ciate the notion o f a smo oth extensio n ˆ h . Note that manifolds in the pres en t pa p er may have b o undaries. Deﬁnition 1.1 . — A smo oth extension of the p air ( h, c ) is a functor B → ˆ h ( B ) fr om the c ate gory of c omp act smo oth manifolds to Z -gr ade d gr oups t o gether with natur al tr ansformations 1. R : ˆ h ( B ) → Ω d =0 ( B , N ) (curvatur e) 2. I : ˆ h ( B ) → h ( B ) (for get smo oth data) 3. a : Ω( B , N ) / im ( d ) → ˆ h ( B ) (action of forms) . These tr ansformations ar e r e quir e d to satisfy the fol lowing axioms: 1. The fol lowing diag r am c ommutes ˆ h ( B ) R   I / / h ( B ) c   Ω d =0 ( B , N ) dR / / H ( B , N ) . 2. R ◦ a = d . (2) 3. a is of de gr e e 1 . 4. The se quenc e h ( B ) c → Ω( B , N ) / im ( d ) a → ˆ h ( B ) I → h ( B ) → 0 . (3) is exact. The Cheeger- Simons smo oth cohomolog y B 7→ ˆ H ( B , R ) considere d in 1.2.1 is the smo oth extensio n of the pair ( H ( . . . , R ) , i ), where i : H ( B , R ) → H ( B , R ) is induced b y the inclusion R → R . The main ob ject of the present pap er, smo oth K - theo ry , is a smo oth extension of the pair ( K, c h R ), and we a c tually work with the ob vio us Z / 2 Z -graded version of these axioms. 1.2.3 . — If h is a multiplicativ e co homology theory , then one can consider a Z -graded ring N ov er R and a multiplicativ e transformation c : h ( B ) → H ( B , N ). In this case is makes sense to talk about a multiplicativ e smo oth extension ˆ h of ( h, c ). Deﬁnition 1.2 . — A smo oth exten s ion ˆ h of ( h, c ) is c al le d mult ipli c ative, if ˆ h to- gether with the t r ansformations R , I , a is a smo oth extension of ( h, c ) , and in additio n 1. ˆ h is a functor t o Z -gr ade d rings, 2. R and I ar e multiplic ative, 3. a ( ω ) ∪ x = a ( ω ∧ R ( x )) for x ∈ ˆ h ( B ) and ω ∈ Ω( B , N ) / im ( d ) . The smo oth extensio n ˆ H ( . . . , R ) of or dinary cohomology H ( . . . , R ) with co eﬃcient s in a subring R ⊂ R co nsidered in 1.2 .1 is multiplicativ e. The smo oth extension ˆ K of K -theor y which we construct in the pr esen t pap er is multiplic ative, to o. 8 ULRICH BUNKE & THOMAS SCHICK 1.2.4 . — Consider tw o pa ir s ( h i , c i ), i = 0 , 1 as in 1.2.2 and a transformation o f generalized cohomolog y theories u : h 0 → h 1 such that c 1 ◦ h = c 0 . Then we deﬁne the notion o f a natura l transformation of smo oth cohomolog y theories which reﬁnes u . Deﬁnition 1.3 . — A n atur al tr ansformation of smo oth extensions ˆ u : ˆ h 0 → ˆ h 1 which r eﬁn es u is a natur al t ra nsformation ˆ u : ˆ h 0 ( B ) → ˆ h 1 ( B ) such that the fol lowing diagr am c ommutes: Ω( B , N ) / im ( d ) a / / ˆ h 0 ( B ) R # # I / / ˆ u   h 0 ( B ) u   Ω d =0 ( B , N ) Ω( B , N ) / im ( d ) a / / ˆ h 1 ( B ) I / / R ; ; h 1 ( B ) Ω d =0 ( B , N ) . Our main example is the Chern c har acter ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) which reﬁnes the ordinar y Chern character ch : K ( B ) → H ( B , Q ). The Cher n c har- acter a nd its smo oth r eﬁnemen ts a re actually m ultiplicative. 1.2.5 . — O ne can show that t wo smo oth extensions of ( H ( . . . , R ) , i ) are c a nonically isomorphic (see [ SS ] and [ BS09 , Section 4 ]). Ther e is no uniqueness result for a r- bitrary pairs ( h, c ). Appropriate examples in the ca se o f K -theory a re pr esen ted in [ BS09 , Sectio n 6]. In order to ﬁx the uniqueness problem one has to require mor e conditions, which are a ll quite natura l. The pro jection pr 2 : S 1 × B → B has a canonical smo oth K -o r ien tation (see 4.3 .2 for details). Hence we have a push-forward ( ˆ pr 2 ) ! : ˆ K ( S 1 × B ) → ˆ K ( B ) (see Deﬁnition 3.18). This map plays the r ole of the susp ension for the smo oth e x tension. It is natural in B , a nd the following diagr am c o mm utes (see Prop osition 3.19 ) Ω( S 1 × B ) / im ( d ) R S 1 × B/ B   a / / ˆ K ( S 1 × B ) R # # ( ˆ p r 2 ) !   I / / K ( B ) ( pr 2 ) !   Ω( S 1 × B ) R S 1 × B/ B   Ω( B ) / im ( d ) a / / ˆ K ( B ) R ; ; I / / K ( B ) Ω( B ) . (4) SMOOTH K-THEOR Y 9 F urthermor e , it satisﬁes (see 4.6) ( ˆ pr 2 ) ! ◦ pr ∗ 2 = 0 . (5) W e hav e the following theorem, also discov er ed by Wiethaup. The or em 1.4 ( [ BS09 , Section 3, Section 4 ] ) . — Ther e is a unique (up to isomor- phism) smo oth extension of t he p air ( K, c h R ) for which in addition the push-forwar d along pr 2 : S 1 × B → B is deﬁne d, is natur al in B , satisﬁes (5), and is such that (4) c ommut es. If we r e qu ir e the isomorph ism to pr eserve ( ˆ pr 2 ) ! , then it is also unique. 1.2.6 . — The theory of [ HS05 ] g iv es the following g eneral existence res ult. The or em 1.5 ( [ HS05 ] ) . — F or every p air ( h, c ) of a gener alize d c ohomolo gy the ory and a natu r al tr ansformation h → H N ther e exists a smo oth ext ension ˆ h in the sense of Deﬁnition 1.1. A simila r general result ab out m ultiplicative extensions is not known. Besides smo oth extensions of o rdinary cohomolog y and K -theory we ha ve a collection of m ultiplicative extensions of b ordism theories, again by an an explicit cons tr uction in a cycle model. The details can be found in [ BSSW07 ] . 1.2.7 . — Let us now ass ume that ( h, c ) is multipli ca tive, and that ˆ h is a multip licative smo oth extension of the pair ( h, c ). Let p : W → B be a prop er submersio n with closed ﬁbres. An h -o rien tation of p is g iv en by a collection of compatible choices of h -Thom classes on repr esen tatives o f the stable nor mal bundle of p . Equiv a le ntly , we can ﬁx a Thom class on the vertical tang e n t bundle, and we will adopt this p oint of view in the pres en t pap er. If p is h - oriented, then w e ha ve a push-for ward p ! : h ( W ) → h ( B ) . It is a n inp ortant question for applications and ca lcula tions how o ne can lift the push-forward to the smo oth e x tensions. In the ca se of smo oth o rdinary cohomo logy with co eﬃcients in R it turns out that an or dina ry or ien tation of p suﬃces in order to deﬁne ˆ p ! : ˆ H ( W, R ) → ˆ H ( B , R ). This push-forward ha s b een considered e.g. in [ Bry93 ], [ DL05 ], [ K¨ o7 ]. W e r efer to 6.1.1 for mor e details. A push-forward for more general pa irs ( h, c ) has b een co nsidered in [ HS05 ] without a discussio n of functorial prop erties. 1.2.8 . — The philosophy in the present pap er is that the push-for w ar d in K -theory is r ealized a nalytically using families of ﬁbre-wise Dirac oper ators. Therefore, in the present pap er a smo oth K - o rient atio n is g iv en by a collection of geometric data which allows to deﬁne the push-forward on the level of cycles, whic h are given b y families of Dira c type op erator s. W e add a diﬀerential form to the data in or der to c apture the b ehaviour under defor mations. 10 ULRICH BUNKE & THOMAS SCHICK 1.2.9 . — W e hav e cycle mo dels of multip licative smo oth extensions of b ordism the- ories Ω G , wher e G in particular ca n b e S O, S pin, U, S pin c , see [ BSSW07 ]. In these examples the natura l transformation c is the g en us a sso ciated to a formal p ow er series φ ( x ) = 1 + a 1 x + . . . with co eﬃcients in some gr aded ring. These b ordism theories admit a theory of orientations a nd push- forward whic h is very similar to the case of K -theor y . Co nce r ning the pro duct and the in tegr ation b ordism theories turn out to be muc h simpler than or dina ry coho mology . Motiv ated by this fact, in a joint pr o ject with M. Kreck we develop a b ordism like version of the smo oth extension of integral cohomolog y based on the notion of o rient ifolds. W e also hav e an equiv aria nt version of the theory of the present pap er for ﬁnite groups which will b e present ed in a future publication. 1.3. Related cons tructions. — 1.3.1 . — Recall that [ HS05 ] pr ovides a top ological constructio n of smo oth K -theory . In this subsection we review the literature ab out a nalytic v a riant s of smo oth K -theor y and related index theorems. Note that w e will completely ignore the developmen t of holomorphic v aria n ts which a re more related to arithmetic questions tha n to top ology . This subsection will use the langua ge which is set up later in the pap er. It should b e read in detail o nly after obtaining s ome familiarity w ith the main deﬁnitions (though we tried to give suﬃciently ma ny forward references). 1.3.2 . — Let p : W → B be a prop er submersion with closed ﬁbres. T o give a K - orientation of p is equiv alent to g iv e a S pin c -structure on its vertical bundle T v p . The K -orientation o f p yields, by a s ta ble homotopy construction, a push-forward p ! : K ( W ) → K ( B ). Let ˆ A ( T v p ) denote the ˆ A -class of the v er tica l bundle, and let c 1 ( L 2 ) ∈ H 2 ( W , Z ) b e the cohomolog y class determined b y the S pi n c -structure (see 3.1.6). The ” index theor em fo r families” in the characteristic class version states that c h ( p ! ( x )) = Z W/B ˆ A ( T v p ) ∪ e 1 2 c 1 ( L 2 ) ∪ c h ( x ) , ∀ x ∈ K ( W ) . If one rea lizes the push-forward in an analy tic mo del, then this statement is indeed an index theorem for families o f Dirac op erator s. 1.3.3 . — The coﬁbr e of the map of sp ectra K → H R induced by the Cher n char- acter represents a g eneralized coho mo logy theory K R / Z , called R / Z - K -theor y . It is a mo dule theory over K - theory and therefor e also a dmits a push-for w ar d for K - oriented prop er submersions. This push-for w ar d is again deﬁned b y co nstructions in stable homotopy theory . An analytic/ge o metric mo del of R / Z - K -theor y was pro- po sed in [ Kar87 ], [ Kar97 ]. This led to the natura l question whether there is an analytic description o f the push-forward in R / Z - K -theo r y . This question was solved in [ Lot94 ]. The solution gives a top olog ical in terpre ta tion of ρ -inv ariants. F urthermor e , in [ Lot94 ] a Chern c hara cter fro m R / Z - K -theory to cohomology with R / Q -co eﬃcien ts has been constructed, and an index theorem ha s b een proved. SMOOTH K-THEOR Y 11 Let us now ex plain the relation of these constructio ns a nd res ults with the present pap er. In the present pap er we deﬁne the ﬂat theory ˆ K f lat ( B ) as the kernel of the cur- v ature R : ˆ K ( B ) → Ω d =0 ( B ). It turns out that ˆ K f lat ( B ) is isomorphic to K R / Z ( B ) up to a degree-shift by o ne (Prop osition 2.25 ). One can actually r epresent all clas s es of K 0 f lat ( B ) by pa irs ( E , ρ ), where E is a g eometric family with zero -dimensional ﬁbre (see 2.1.4). If one restricts to these sp ecial cycles , then our model o f K 0 f lat ( B ) a nd the mo del of K R / Z − 1 ( B ) of [ Lot94 ] coincide. By an inspec tio n of the constructions one can further chec k that the r estriction of our cycle level push-forward (17) to these particular ﬂat cycles is the same as the one in [ Lot94 ]. At a ﬁrst gla nce our push-forward o f ﬂat classes s eems to depend on a smo oth reﬁnement of the top ological K -orientation o f the map p , but it is in fact independent of these geometric choices as can b e s een using the homotopy inv ariance of the ﬂa t theory . The comparison with [ Lot94 ] shows that the restric tio n of our push-forward to ﬂat cla sses coincides with the homo top y theorists’ one. The restriction of our smo oth lift of the Chern character ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) (see Theorem 6.2) to the ﬂat theories exa c tly gives the Chern character of [ Lot94 ] ˆ c h : ˆ K f lat ( B ) → ˆ H f lat ( B , Q ) (using our no tation and the isomorphism of ˆ H ∗ f lat ( B ) ∼ = H ∗− 1 ( B , R / Q )). If we r estrict our index theorem 6 .19 to ﬂat classes, then it sp ecializes to ˆ c h ( ˆ p ! ( x )) = Z W/B ˆ A ( T v p ) ∪ e 1 2 c 1 ( L 2 ) ∪ ˆ c h ( x ) , ∀ x ∈ ˆ K ( W ) , and this is exa ctly the index theorem of [ Lot94 ]. In this sense the pre s en t pap er is a direct generalizatio n o f [ Lot94 ] fro m the ﬂat to the general ca se. 1.3.4 . — The analytic mo del o f R / Z - K -theo r y and the analytic cons truction of the push-forward in [ Lot94 ] ﬁts into a ser ies of co nstructions of ho motop y in v ariant func- tors with a push-forward which enco des secondary sp ectral in v ariants. Le t us mention the tw o examples in [ Lot00 ] which ar e based on ﬂat bundles or ﬂat bundles with dual- it y , re s pectively . T he sp ectral geo metric inv aria nts in these examples a re the ana lytic torsion forms of [ BL95 ] a nd the η -for ms introduced e.g . in [ BC9 0 a ]. The functor i- ality of the push-fow ards under comp ositions is discussed in [ Bun02 ] and [ BM04 ]. But these construction do not ﬁt (at least at the moment) into the w or ld of smo oth cohomolog y theory , a nd it is still an op en problem to in terpret the push-forward in topo logical terms. Let us also mention the pap er [ P e k93 ] devoted to smoo th lifts of Chern clas s es. 1.3.5 . — In [ Berb ], [ Bera ] se veral v ariants of functors der iv ed from K -theory are considered. In the fo llo wing we reca ll the names of these gro ups used in that reference and explain, if p ossible, their rela tio n with the pres e nt pap e r. 12 ULRICH BUNKE & THOMAS SCHICK 1. rela tive K -theory K r el : the cycles are triples ( V , ∇ V , f ) of Z / 2 Z -gr a ded ﬂat vector bundles and an o dd selfadjoint bundle automorphism f (whic h need not be para llel). 2. free m ultiplicative K -theory K ch (also called tra nsgressive in [ Be ra ]): it is essentially (3) a mo del of ˆ K 0 based on cycles of the for m ( E , ρ ), where E is a geometric family with zero-dimensiona l ﬁbre coming fro m a geometric vector bundle (see 2.1.4). 3. multiplicativ e K -theor y M K : it is the same mo del of K 0 f lat as in [ Lot94 ], see 1.3.3. 4. ﬂat K -theory K f lat : it is the Gr othendiec k gr o up of ﬂat vector bundles. Besides the deﬁnition of these g roups and the in vestigation of their int er r elation the main topic o f [ Be rb ], [ Bera ] is the constructio n o f push-forward op eratio ns . In the following we will only discuss multiplicativ e and trans g ressive K -theory since they are related to the presen t pap er. The diﬀerence to the constructio ns of [ Lot94 ] and the present pap er is that Ber tho mia u’s analytic push-forward (whic h we denote here by p B ! ) do es no t use the S pin c -Dirac op erator but the ﬁbre- wise de Rham co mplex. F rom the point of view of analysis the diﬀerence is essentially that the class ˆ A ( T v p ) ∪ e 1 2 c 1 ( L 2 ) or the cor r espo nding diﬀeren tial form ha s to b e replaced by the Euler class E ( T v p ) or the Euler form of the v er tical bundle. The adv antage of working with the de Rham c o mplex is that in order to deﬁne the push-forward p B ! one do es no t need a S pin c -structure. If there is one, then one can actually expr e ss p B ! in terms of ˆ p ! as p B ! ( x ) = ˆ p ! ( x ∪ s ∗ ) , where s ∗ ∈ K ( W ) is the class of the dua l o f the spinor bundle S c ( T v p ), or the ˆ K ( W )- class represented b y the geometric version of this bundle in the case of transg r essive K -theor y , resp ectiv ely . The p oint here is that the Dirac op erator induced by the de Rham complex is the S p in c -Dirac o pera tor twisted by S c ( T v p ) ∗ . As said a bove, the homotopy theor ists’ p ! is the push-forward as so ciated to a K -or ien tation of p . In co n tras t, the homotopy theorists’ version o f p B ! is the Gottlieb- Beck er transfer. The motiv a tion of [ Berb ] , [ Be ra ] to deﬁne the push-forward with the de Rham complex is that it is compatible with the push-forward for ﬂat K -theory . The push- forward of a ﬂat vector bundle is ex pr essed in terms of ﬁbre-wise cohomolog y which forms ag ain a ﬂat vector bundle on the base. This additional structure also plays a crucial r ole in [ Lot00 ], [ BL95 ], [ Bun02 ], a nd [ BM04 ]. If o ne interprets the push- forward using the S pin c -calculus, then the ﬂat co nnection is lost. Let us men tion that (3) The connections are not assumed to b e hermitean and the corresp onding diﬀerential f orms hav e complex co eﬃcien ts. SMOOTH K-THEOR Y 13 the ﬁrst circulated v ersio n of the present pap er predates the pape r s [ Berb ] , [ Bera ] which actually adapt s o me of our ideas. 1.3.6 . — The topics of [ Bis05 ] ar e t wo index theorems inv olving ˆ H ( B , Q )-v alued characteristic classes. Her e we only review the ﬁrs t one, since the seco nd is r elated to ﬂat vector bundles. (Compare a lso [ MZ04 ] for a “ ﬂat version”). Let us formulate the res ult of [ Bi s 05 ] in the language of the present pap er. Let p : W → B b e a pr oper submersion with closed ﬁbr es with a ﬁbre-wise spin - structure ov er a compact base B . The spin structure induces a S pin c -structure, and w e c ho ose a r epresent a tive of a smooth K -o rien tation o := ( g T v p , T h p, ˜ ∇ , 0), where ˜ ∇ is indced from the Lev i- Civita connection on T v p (see 3.1 .9 for deta ils ). Let V = ( V , h V , ∇ V ) be a ge o metric vector bundle ov er W with asso ciated geometric family V (compare 2 .1.4). Then we ca n for m the g eometric family E := p ! V (see 3.7 ) ov er B . The family of Dirac op erator s D ( E ) acts on sections of a bundle of Hilbert spa c e s H ( E ) → B . The geometric structures of the K -orientation o and V induce a con- nection ∇ H ( E ) (it is the connection part of the Bismut sup erconnection [ BGV04 , Prop. 10.15 ] as so c ia ted to this situation). W e a s sume that the family of Dirac o p- erators of D ( E ) has a k er nel bundle K := ker( D ( E )). This bundle has an induced metric h K . The pr o jection of ∇ H ( E ) to K g iv es a hermitean co nnection ∇ K . W e th us get a geo metr ic bundle K := ( K, h K , ∇ K ), and an asso ciated geometric family K (see 5.3.1). The index theo r em in [ Bis0 5 ] calculates the smo oth Cher n character ˆ c h ( K ) ∈ ˆ H ( B , Q ) of [ CS85 ] and s tates: ˆ c h ( K ) = ˆ p ! ( ˆ ˆ A ( T v p ) ∪ ˆ c h ( V )) + a ( η B C ( E )) , where we refer to (33) and 5.3.3 for notation. Note that this theorem could also b e derived from o ur index Theorem 6 .19. By Corollar y 5 .5, (17) , our sp ecial choice o f o , and Theorem 6.19 (the marked step) we hav e ˆ c h ( K ) − a ( η B C ( E )) = ˆ c h [ K , η B C ( E )] = ˆ c h [ E , 0] = ˆ c h ([ p ! V , 0]) = ˆ c h ( p ! ([ V , 0])) ! = ˆ p K ! ( ˆ c h ( V )) = p ! ( ˆ ˆ A ( T v p ) ∪ ˆ c h ( V )) . A cknow le dgement: We thank Moritz Wiethaup for explaining to us his insights and r esu lt . We further thank Mike H opkins and Dan F re e d for their inter est in t his work and many helpful r emarks. We thank the r efer e e for many helpf ul c omments which le ad to c onsider able impr ovements of the exp osition. 14 ULRICH BUNKE & THOMAS SCHICK 2. Deﬁnition of smo o th K-theory via cycles and relations 2.1. Cycles. — 2.1.1 . — One goal of the pr e s en t pap er is to constr uct a multiplicativ e smo oth ex- tension of the pair ( K, c h R ) of the multiplicativ e g eneralized cohomolog y theory K , complex K -theory , and the co mposition c h R : K c h → H Q → H R o f the Chern c har ac- ter with the natural map fro m ordinary co ho mology with rational to r eal co eﬃcien ts induced b y the inclusion Q → R . In this section we deﬁne the s moo th K -theo r y group ˆ K ( B ) of a smo oth compact manifold, p ossibly with b oundary , and construct the natural tra nsformations R, I , a . The main r esult of the present section is that our co nstruction really yields a smo oth e x tension in the sense of Deﬁnition 1.1. Wi discuss the m ultiplicative structure in Section 4 . Our res triction to compact manifolds with b o undary is due to the fac t that w e work with absolute K -g roups. One co uld in fact mo dify the co nstructions in order to pro duce compactly supp orted smoo th K -theory or rela tiv e smo oth K -theory . But in the pr esen t pa per, for simplicity , we will not discuss rela tiv e s mo oth cohomo logy theories. 2.1.2 . — W e deﬁne the smo oth K -theory ˆ K ( B ) a s the group completion o f a quotien t of a semigr oup of isomorphism classes o f cyc les by an equiv alence relatio n. W e start with the description of the cyc les. Deﬁnition 2.1 . — L et B b e a c omp act manifold , p ossibly with b oundary. A cycle for a smo oth K -the ory class over B is a p air ( E , ρ ) , wher e E is a ge ometric family, and ρ ∈ Ω( B ) / im ( d ) is a class of diﬀer ent ial forms. 2.1.3 . — The notio n of a geometric family has been in tro duced in [ Bun ] in order to hav e a short name for the da ta needed to deﬁne a Bismut s up er-co nnection [ BGV04 , Prop. 10 .15]. F o r the conv enience of the reader we ar e going to explain this notion in so me detail. Deﬁnition 2.2 . — A ge ometric family over B c onsists of the fol lowing data: 1. a pr op er submersion with close d ﬁbr es π : E → B , 2. a vertic al Riemannian metric g T v π , i.e. a met ric on the vertic al bund le T v π ⊂ T E , deﬁne d as T v π := ker( d π : T E → π ∗ T B ) . 3. a horizontal distribution T h π , i.e. a bu nd le T h π ⊆ T E such that T h π ⊕ T v π = T E . 4. a family of Dir ac bund les V → E , 5. an orientation of T v π . Here, a family of Dirac bundles consists of 1. a hermitean vector bundle with connection ( V , ∇ V , h V ) on E , 2. a Cliﬀord mult iplication c : T v π ⊗ V → V , 3. on the comp onents wher e dim( T v π ) has even dimension a Z / 2 Z -grading z . SMOOTH K-THEOR Y 15 W e require that the restrictions of the family Dirac bundles to the ﬁbres E b := π − 1 ( b ), b ∈ B , give Dirac bundles in the usual sense (see [ Bun , Def. 3.1]): 1. The vertical metric induces the Riemannian structure on E b , 2. The Cliﬀord mu ltiplication turns V | E b in to a Cliﬀord module (see [ BGV04 , Def.3.32]) which is g raded if dim( E b ) is ev en. 3. The r estriction of the co nnection ∇ V to E b is a Cliﬀord connection (see [ BGV04 , Def.3.39]). A ge o metric fa mily is ca lled even or odd, if dim( T v π ) is even-dimensional o r o dd- dimensional, res p ectively . 2.1.4 . — Here is a s imple example of a g e ometric fa mily with zero - dimensional ﬁbres. Let V → B be a complex Z / 2 Z -graded vect or bundle. Assume that V comes with a hermitean metric h V and a hermitean connectio n ∇ V which are co mpatible with the Z / 2 Z -grading . The geometric bundle ( V , h V , ∇ V ) will usually b e denoted by V . W e consider the submersion π := id B : B → B . In this c a se the vertical bundle is the zero -dimensional bundle which has a canonical vertical Riemannian metr ic g T v π := 0 , and for the horizontal bundle w e must take T h π := T B . F urthermore , there is a canonical or ien tation of p . The geometric bundle V can natur a lly be in terpr eted as a family of Dirac bundles on B → B . In this w ay V gives r ise to a geometric family ov er B which w e will usually denote by V . 2.1.5 . — In o r der to deﬁne a representativ e of the negative of the smo oth K -theor y class r epresented b y a cycle ( E , ρ ) we introduce the notion of the opp osite geometric family . Deﬁnition 2.3 . — The opp osite E op of a ge ometric family E is obtaine d by r evers- ing the signs of the Cliﬀor d mult ipli c ation and the gr ading (in the even c ase) of the underlying family of Cliﬀor d bun d les, and of the orientation of the vertic al bund le. 2.1.6 . — Our smo oth K -theor y gr oups will b e Z / 2 Z -graded. On the le vel of cycles the gr ading is reﬂected by the notions of even a nd o dd cycles. Deﬁnition 2.4 . — A cycle ( E , ρ ) is c al le d even (or o dd, r esp.), if E is even (or o dd, r esp.) and ρ ∈ Ω odd ( B ) / im ( d ) ( or ρ ∈ Ω ev ( B ) / im ( d ) , r esp.). 2.1.7 . — Let E and E ′ be tw o geo metr ic families o ver B . An isomorphism E ∼ → E ′ consists o f the following data: V   F / / V ′   E π   @ @ @ @ @ @ @ f / / E ′ π ′ ~ ~ } } } } } } } B 16 ULRICH BUNKE & THOMAS SCHICK where 1. f is a diﬀeomorphism over B , 2. F is a bundle isomor phism over f , 3. f preserves the horizontal distribution, the vertical metric and the o rien tation. 4. F preserves the connection, Cliﬀord m ultiplication and the grading. Deﬁnition 2.5 . — Two cycles ( E , ρ ) and ( E ′ , ρ ′ ) ar e c al le d isomorphic if E and E ′ ar e isomorphic and ρ = ρ ′ . We let G ∗ ( B ) denote the set of isomorp hism classes of cycles over B of p arity ∗ ∈ { e v , odd } . 2.1.8 . — Given tw o geometric families E a nd E ′ we can form their sum E ⊔ B E ′ ov er B . The underlying prop er submersion with closed ﬁbres o f the sum is π ⊔ π ′ : E ⊔ E ′ → B . The r emaining structures of E ⊔ B E ′ are induced in the obvious way . Deﬁnition 2.6 . — The sum of t wo cycles ( E , ρ ) and ( E ′ , ρ ′ ) is deﬁne d by ( E , ρ ) + ( E ′ , ρ ′ ) := ( E ⊔ B E ′ , ρ + ρ ′ ) . The sum of cycles induces o n G ∗ ( B ) the s tr ucture of a graded ab e lia n semigroup. The ident ity element of G ∗ ( B ) is the cycle 0 := ( ∅ , 0 ), where ∅ is the empty geo metric family . 2.2. Relations. — 2.2.1 . — In this subsection w e in tro duce an equiv a lence rela tion ∼ on G ∗ ( B ). W e show that it is compatible with the semigr oup structure so that we get a semigroup G ∗ ( B ) / ∼ . W e then deﬁne the smo oth K -theor y ˆ K ∗ ( B ) as the group completion of this quo tient. In or der to deﬁne ∼ we ﬁrst intro duce a simpler relation ”paired” whic h has a nice lo cal index-theoretic meaning. The relation ∼ will b e the eq uiv alence relation generated by ”pair e d” . 2.2.2 . — The main ingredients of our deﬁnition of ”paired” are the notions of a taming of a geometric family E introduced in [ Bun , Def. 4.4], and the η -form o f a tamed family [ Bun , Def. 4.1 6]. In this par agraph we shortly review the notio n of a taming. F or the deﬁnition of eta-forms we refer to [ Bun , Sec. 4 .4]. In the pr esen t pap er w e will use η -forms as a black box with a few imp ortant prop erties whic h we explicitly state at the a ppropriate places b elow. If E is a geometric family over B , then we can for m a family o f Hilb ert spaces ( H b ) b ∈ B , where H b := L 2 ( E b , V | E b ). If E is even, then this family is in addition Z / 2 Z - graded. The geometric family E g iv es rise to a family o f Dirac o pera tors ( D ( E b )) b ∈ B , where D ( E b ) is a n unbounded selfadjoin t oper ator on H b , whic h is o dd in the even case. A pre-taming of E is a family ( Q b ) b ∈ B of selfadjoint op e r ators Q b ∈ B ( H b ) given b y a smo oth ﬁbrewise in tegr a l kernel Q ∈ C ∞ ( E × B E , V ⊠ V ∗ ). In the even case we SMOOTH K-THEOR Y 17 assume in addition tha t Q b is o dd, i.e. tha t it anticomm utes with the grading z . The pre-taming is called a taming if D ( E b ) + Q b is inv ertible for all b ∈ B . The family of Dirac op erato rs ( D ( E b )) b ∈ B has a K - theoretic index which w e deno te b y index ( E ) ∈ K ( B ) . If the geometric family E admits a taming, then the a sso ciated fa mily of Dirac op er- ators op erator s admits a n inv ertible compact perturba tion, and hence ind ex ( E ) = 0. Vice versa, if ind ex ( E ) = 0 and the even part is empty or has a co mponent with dim( T v π ) > 0, then by [ Bun , Lemma. 4.6] the geo metr ic family admits a taming. If the ev en pa rt of E has zero- dimensional ﬁbres, then the existence of a ta ming may r equire some stabilization. This means that w e m ust add a geometric family V ⊔ B V op (see 2.1.4 and Deﬁnition 2.3), wher e V is the bundle B × C n → B for suﬃcient ly large n . 2.2.3 . — Deﬁnition 2.7 . — A ge ometric family E to gether with a taming wil l b e denote d by E t and c al le d a tame d ge ometric family. Let E t be a taming of the geometric family E by the family ( Q b ) b ∈ B . Deﬁnition 2.8 . — The opp osite tame d family E op t is given by the taming ( − Q b ) b ∈ B of E op . 2.2.4 . — The lo cal index for m Ω( E ) ∈ Ω( B ) is a diﬀerential for m canonica lly as so- ciated to a g eometric family . F or a detailed deﬁnition we refer to [ Bun , Def..4.8], but we c an br ieﬂy formulate its constr uctio n as follows. The vertical metric T v π and the ho rizontal distribution T h π together induce a c o nnection ∇ T v π on T v π (see 3.1.3 for more details). Lo cally on E we can assume that T v π has a spin s tructure. W e let S ( T v π ) b e the asso ciated s pinor bundle. Then we c a n write the family o f Dirac bundles V as V = S ⊗ W fo r a twisting bundle ( W, h W , ∇ W , z W ) with metric, metric connection, a nd Z / 2 Z -gra ding which is determined uniquely up to isomo rphism. The form ˆ A ( ∇ T v π ) ∧ c h ( ∇ W ) ∈ Ω( E ) is g lobally deﬁned, and we g et the lo cal index form b y applying the in tegra tion over the ﬁbre R E /B : Ω( E ) → Ω( B ): Ω( E ) := Z E /B ˆ A ( ∇ T v π ) ∧ ch ( ∇ W ) . The lo cal index form is clo sed and represents a co homology class [Ω( E )] ∈ H dR ( B ). W e let c h dR : K ( B ) → H dR ( B ) be the co mposition c h dR : K ( B ) c h → H ( B ; Q ) can → H dR ( B ) . The characteristic class version of the index theorem for families is 18 ULRICH BUNKE & THOMAS SCHICK The or em 2.9 ( [ AS71 ] ) . — c h dR ( index ( E )) = [Ω( E )] . A pro of using metho ds of lo cal index theory has b een given by [ Bis 85 ]. F or a presentation of the pro of we refer to [ BGV04 ]. An alternative pro of ca n b e obtained from [ Bun , Thm.4.18] by sp ecializing to the case of a family of closed manifolds. 2.2.5 . — If a geometric family E admits a taming E t (see Deﬁnition 2.7), then w e hav e in dex ( E ) = 0 . In pa rticular, the lo cal index form Ω( E ) is exact. The imp ortant feature o f lo cal index theory in this case is that it provides a n explicit form whose bo undary is Ω ( E ) (see equation (6) b elow). Let E t be a tamed geometric family over B . In [ Bun , Def. 4.16] we have deﬁned the η -for m η ( E t ) ∈ Ω( B ). By [ Bun , Theor em 4.13]) it satisﬁes dη ( E t ) = Ω( E ) . (6) The ﬁr st construction o f η -forms has b een given in [ BC90a ], [ BC90b ], [ BC91 ] under the a ssumption that k er( D ( E b )) v anishes or has constan t dimension. The v ar iant which we use her e has a lso b een co nsidered in [ Lot94 ], [ MP97b ], [ MP97a ]. Since the analytic details of the deﬁnition o f the η - fo r m η ( E t ) ar e quite co mplicated we will not re p eat them her e but refer to [ Bun , Def. 4.16]. F or most of the present pap er we can use the construction of the η -for m as a bla c k box refering to [ Bun ] for details of the co nstruction and the pro o fs of prop erties. E xceptions ar e arg umen ts in volving adiabatic limits for which we use [ BM04 ] as the reference. 2.2.6 . — Now we can introduce the relations ”paired” and ∼ . Deﬁnition 2.10 . — We c al l two cycles ( E , ρ ) and ( E ′ , ρ ′ ) p air e d if t her e exists a taming ( E ⊔ B E ′ op ) t such that ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ) . We let ∼ denote the e quivalenc e r elation gener ate d by the r elation ”p air e d”. L emma 2.11 . — The r elation ”p air e d” is symmetric and r eﬂexive. Pr o of . — In or der to show that ”pa ired” is reﬂexiv e and s y mmetric we are going employ the relation [ Bun , Lemma 4 .12] η ( E op t ) = − η ( E t ) . (7) Let E be a geometric family ov er B , and let H b denote the Hilbe r t space of sections of the Dirac bundle a long the ﬁbre ov er b ∈ B . The family E ⊔ B E op has an inv olution τ which ﬂips the compo nen ts, the signs of the Cliﬀord multipli ca tio ns , the gra ding and the orientations. W e use the sa me symbol τ in or der to denote the action of τ on the Hilbert space of sections of the Dirac bundle of E b ⊔ B E op b . The latter ca n b e SMOOTH K-THEOR Y 19 iden tiﬁed with H b ⊕ H op b , a nd in this picture τ =  0 1 1 0  . Note that τ anticomm utes with D b := D ( E b ⊔ B E op b ) =  D ( E b ) 0 0 − D ( E b )  . W e choose an even, compa c tly supp orted s moo th function χ : R → [0 , ∞ ) such that χ (0) = 1 and form Q b := τ χ ( D b ) . This op erator also anticomm utes with D b , and ( D b + Q b ) 2 = D 2 b + χ 2 ( D b ) is p ositive and therefore inv e rtible for all b ∈ B . The family ( Q b ) b ∈ B th us deﬁnes a taming ( E ⊔ B E op ) t . The inv olution σ :=  0 i − i 0  on the Hilbert space H b ⊕ H op b is induced b y a n isomo rphism ( E ⊔ B E op ) t ∼ = ( E ⊔ B E op ) op t . Because of the relatio n (7) we have η (( E ⊔ B E op ) t ) = 0. It follows that ( E , ρ ) is paired with ( E , ρ ). Assume now tha t ( E , ρ ) is paired with ( E ′ , ρ ′ ) via the taming ( E ⊔ B E ′ op ) t so that ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ). Then ( E ⊔ B E ′ op ) op t is a taming o f E ′ ⊔ B E op such that ρ ′ − ρ = η (( E ⊔ B E ′ op ) op t ), again by (7 ). It follows that ( E ′ , ρ ′ ) is pa ired with ( E , ρ ). L emma 2.12 . — The r elations ”p air e d” and ∼ ar e c omp atible with the semigr oup structur e on G ∗ ( B ) . Pr o of . — In fact, if ( E i , ρ i ) ar e paired with ( E ′ i , ρ ′ i ) via tamings ( E i ⊔ B E ′ op i ) t for i = 0 , 1, then ( E 0 , ρ 0 ) + ( E ′ 0 , ρ ′ 0 ) is paired with ( E 1 , ρ 1 ) + ( E ′ 1 , ρ ′ 1 ) via the taming ( E 0 ⊔ B E 1 ⊔ B ( E ′ 0 ⊔ B E ′ 1 ) op ) t := ( E 0 ⊔ B E ′ op 0 ) t ⊔ B ( E 1 ⊔ B E ′ op 1 ) t . In this calculation we use the additivity o f the η -for m [ Bun , Le mma 4.12] η ( E t ⊔ B F t ) = η ( E t ) + η ( F t ) . The co mpatibilt y of ∼ with the sum follo ws from the compatibilit y of ”pair e d” . W e ge t an induced semigroup structure on G ∗ ( B ) / ∼ . L emma 2.13 . — If ( E 0 , ρ 0 ) ∼ ( E 2 , ρ 2 ) , then ther e ex ists a cycle ( E ′ , ρ ′ ) such that ( E 0 , ρ 0 ) + ( E ′ , ρ ′ ) is p air e d with ( E 2 , ρ 2 ) + ( E ′ , ρ ′ ) . 20 ULRICH BUNKE & THOMAS SCHICK Pr o of . — Let ( E 0 , ρ 0 ) b e paired w ith ( E 1 , ρ 1 ) via a taming ( E 0 ⊔ B E op 1 ) t , and ( E 1 , ρ 1 ) b e paired with ( E 2 , ρ 2 ) v ia ( E 1 ⊔ B E op 2 ) t . Then ( E 0 , ρ 0 ) + ( E 1 , ρ 1 ) is paired with ( E 2 , ρ 2 ) + ( E 1 , ρ 1 ) via the taming (( E 0 ⊔ B E 1 ) ⊔ B ( E 2 ⊔ B E 1 ) op ) t := ( E 0 ⊔ B E op 1 ) t ⊔ B ( E 1 ⊔ B E op 2 ) t . If ( E 0 , ρ 0 ) ∼ ( E 2 , ρ 2 ), then there is a chain ( E 1 ,α , ρ 1 ,α ), α = 1 , . . . , r with ( E 1 , 1 , ρ 1 , 1 ) = ( E 0 , ρ 0 ), ( E 1 ,r , ρ 1 ,r ) = ( E 2 , ρ 2 ), suc h that ( E 1 ,α , ρ 1 ,α ) is pa ired with ( E 1 ,α +1 , ρ 1 ,α +1 ). The asser tion of the Lemma follows from an ( r − 1 )-fold a pplication o f the argument ab ov e . 2.3. Smo oth K -theory. — 2.3.1 . — In this subsection we deﬁne the con trav aria n t functor B → ˆ K ( B ) from compact smoo th ma nifolds to Z / 2 Z -graded a belian g roups. Recall the deﬁnition 2.6 of the semig r oup of isomor phism cla sses of cycles. By Lemma 2.12 we can form the semigroup G ∗ ( B ) / ∼ . Deﬁnition 2.14 . — We deﬁne the sm o oth K - the ory ˆ K ∗ ( B ) of B to b e the gr oup c ompletion of the ab elian semigr oup G ∗ ( B ) / ∼ . If ( E , ρ ) is a cycle, then let [ E , ρ ] ∈ ˆ K ∗ ( B ) denote the corr espo nding class in smo oth K -theor y . W e now collect some simple facts which a re helpful for computations in ˆ K ( B ) on the level of cycles. L emma 2.15 . — We have [ E , ρ ] + [ E op , − ρ ] = 0 . Pr o of . — W e show that ( E , ρ ) + ( E op , − ρ ) = ( E ⊔ B E op , 0) is paired with 0 = ( ∅ , 0 ). In fact, this relation is given by the taming (( E ⊔ B E op ) ⊔ B ∅ op ) t = ( E ⊔ E op ) t in tro duced in the pro of of Le mma 2.1 1 with η (( E ⊔ B E op ) t ) = 0. L emma 2.16 . — Every element of ˆ K ∗ ( B ) c an b e r epr esente d in the form [ E , ρ ] . Pr o of . — An elemen t of ˆ K ∗ ( B ) can be re pr esent ed b y a diﬀere nce [ E 0 , ρ 0 ] − [ E 1 , ρ 1 ]. Using Lemma 2.15 w e get [ E 0 , ρ 0 ] − [ E 1 , ρ 1 ] = [ E 0 , ρ 0 ] + [ E op 1 , − ρ 1 ] = [ E 0 ⊔ B E op 1 , ρ 0 − ρ 1 ]. L emma 2.17 . — If [ E 0 , ρ 0 ] = [ E 1 , ρ 1 ] , then ther e exists a cycle ( E ′ , ρ ′ ) such that ( E 0 , ρ 0 ) + ( E ′ , ρ ′ ) is p air e d with ( E 1 , ρ 1 ) + ( E ′ , ρ ′ ) . Pr o of . — The rela tion [ E 0 , ρ 0 ] = [ E 1 , ρ 1 ] implies that there ex ists a cycle ( ˜ E , ˜ ρ ) such that ( E 0 , ρ 0 ) + ( ˜ E , ρ ) ∼ ( E 1 , ρ 1 ) + ( ˜ E , ˜ ρ ). The asser tion now follows from Lemma 2.13. SMOOTH K-THEOR Y 21 2.3.2 . — In this parag raph we extend B 7→ ˆ K ∗ ( B ) to a contrav aria n t functor fro m smo oth manifolds to Z / 2 Z -graded gr oups. Let f : B 1 → B 2 be a smo oth map. Then we hav e to deﬁne a map f ∗ : ˆ K ∗ ( B 2 ) → ˆ K ( B 1 ). W e will ﬁrst deﬁne a ma p of abe lia n semigroups f ∗ : G ∗ ( B 2 ) → G ∗ ( B 1 ), and then we show that it passes to ˆ K . If E is a g eometric family over B 2 , then we c an deﬁne a n induced geometric family f ∗ E over B 1 . The underlying s ubmer sion a nd vector bundle of f ∗ E a re g iv en by the cartesian diagr am f ∗ V   / / V   f ∗ E f ∗ π   F / / E π   B 1 f / / B 2 . The metric g T v f ∗ π and the orientation of T v f ∗ π are deﬁned such that dF : T v f ∗ π → F ∗ T v π is a n isometry and o rient ation preserv ing . The ho r izont al distribution T h f ∗ π is given by the condition that dF ( T h f ∗ π ) ⊆ F ∗ T h π . Finally , the Dirac bundle structure of f ∗ V is induced from the Dira c bundle structure on V in the usua l w ay . F or b 2 ∈ B 2 let H b 2 be the Hilb ert space of sections of V along the ﬁbre E b 2 . If b 1 ∈ B 1 satisﬁes f ( b 1 ) = b 2 , then we can ident ify the Hilb ert space of sections of f ∗ V along the ﬁbre f ∗ E b 1 canonically with H b 2 . If ( Q b 2 ) b 2 ∈ B 2 deﬁnes a taming E t of E , then the family ( Q f ( b 1 ) ) b 1 ∈ B is a taming f ∗ E t of f ∗ E . W e hav e the following r elation of η -forms: η ( f ∗ E t ) = f ∗ η ( E t ) . (8) In o rder to see this note the following facts. The g e o metric family E giv es rise to a bundle of Hilbert spaces H ( E ) → B 2 with ﬁbres H ( E ) b 2 = H b 2 , using the nota tion in tro duced ab ov e. W e hav e a na tural isomorphism H ( f ∗ E ) ∼ = f ∗ H ( E ). The ge o metry of E together with the taming induces a family of supe r -connections A s ( E t ) on H parametrized by s ∈ (0 , ∞ ) (see [ Bun , 4 .4.4] fo r explicit fo rm ulas). By cons truction we hav e f ∗ A s ( E t ) = A s ( f ∗ E t ). The η -form η ( E t ) is deﬁned a s an int egr al o f the trace of a family of o p era tors on H ( E ) (with diﬀerential form co eﬃcients) build from ∂ s A s ( E t ) and A s ( E ) 2 [ Bun , Deﬁnition 4.16 ]. Equation (8) no w follows from f ∗ ∂ s A s ( E t ) = ∂ s A s ( f ∗ E t ) and f ∗ A s ( E ) 2 = A s ( f ∗ E t ) 2 . If ( E , ρ ) ∈ G ( B 2 ), then we deﬁne f ∗ ( E , ρ ) := ( f ∗ E , f ∗ ρ ) ∈ G ( B 2 ). The pull-back preserves the disjoin t union and opp osites of g eometric families. In particular , f ∗ is a semigroup homomorphism. Assume now that ( E , ρ ) is paired with ( E ′ , ρ ′ ) via the taming ( E ⊔ B 2 E ′ op ) t . Then we can pull back the ta ming as well a nd get a taming f ∗ ( E ⊔ B 2 E ′ op ) t of f ∗ E ⊔ B 1 f ∗ E ′ op . Equation (8) now implies that f ∗ ( E , ρ ) is paired with f ∗ ( E ′ , ρ ′ ) via the taming f ∗ ( E ⊔ B 2 E ′ op ) t . 22 ULRICH BUNKE & THOMAS SCHICK Hence, the pull-back f ∗ passes to G ∗ ( B ) / ∼ , a nd being a semigroup homomo r- phism, it induces a map of group completions f ∗ : ˆ K ∗ ( B 2 ) → ˆ K ∗ ( B 1 ) . Evidently , ( id B ) ∗ = ˆ id ˆ K ∗ ( B ) . Le t f ′ : B 0 → B 1 be another smo oth map. If E is a geometric family ov er B 2 , then ( f ◦ f ′ ) ∗ E is isomorphic to f ′∗ f ∗ E . This obser v ation implies that f ′∗ f ∗ = ( f ◦ f ′ ) ∗ : ˆ K ∗ ( B 2 ) → ˆ K ( B 0 ) . This ﬁnishes the co nstruction of the contra v ariant functor ˆ K ∗ on the lev el of mor- phisms. 2.4. Natural transformations and exact sequences. — 2.4.1 . — In this s ubse c tion we in tro duce the transformations R, I , a , a nd we sho w that they turn the functor ˆ K in to a smo oth extensio n of ( K, c h R ) in the sense of Deﬁnition 1.1 . 2.4.2 . — W e ﬁrst deﬁne the natural transformation I : ˆ K ( B ) → K ( B ) b y I [ E , ρ ] := i ndex ( E ) . W e mu st s ho w that I is well-deﬁned. Consider ˜ I : G ( B ) → K ( B ) deﬁned b y ˜ I ( E , ρ ) := index ( E ). If ( E , ρ ) is paired with ( E ′ , ρ ′ ), then the existence o f a taming ( E ⊔ B E ′ op ) t implies that index ( E ) = index ( E ′ ). The rela tion index ( E ⊔ B E ′ ) = inde x ( E ) + index ( E ′ ) (9) together with Lemma 2.13 now implies that ˜ I descends to G ( B ) / ∼ . The additivit y (9) and the deﬁnition of ˆ K ( B ) as the g roup co mpletion of G ( B ) / ∼ implies that ˜ I further descends to the homomorphism I : ˆ K ( B ) → K ( B ). The r elation ind ex ( f ∗ E ) = f ∗ index ( E ) shows that I is a natural transforma tio n of functors from smo oth ma nifolds to Z / 2 Z -graded ab elian gr oups. 2.4.3 . — L emma 2.18 . — F or every c omp act manifold B , the tr ansformation I : ˆ K ( B ) → K ( B ) is surje ctive. Pr o of . — W e dis c us s even and odd deg r ees s epera tely . In the even case, a K-theory class ξ ∈ K ( B ) is represented by a Z / 2 Z -gra ded vector bundle V on B . Simply choose a hermitean metric and a connection o n V . W e obtain a resulting geo metric family V on B , with underlying submers ion id : B → B (i.e. 0 -dimensional ﬁbre s ) as in 2 .1 .4, and clea rly I ( V ) = ind ex ( V ) = [ V ] = ξ ∈ K 0 ( B ). F or o dd degrees, the statement is prov ed in [ Bun , 3.1.6.7]. SMOOTH K-THEOR Y 23 2.4.4 . — W e co nsider the functor B 7→ Ω ∗ ( B ) / im ( d ), ∗ ∈ { ev , odd } as a functor fro m manifolds to Z / 2 Z -graded ab elian groups. W e constr uct a par it y-reversing natural transformation a : Ω ∗ ( B ) / im ( d ) → ˆ K ∗ ( B ) b y a ( ρ ) := [ ∅ , − ρ ] . 2.4.5 . — Let Ω ∗ d =0 ( B ) b e the group of clos ed fo r ms of parity ∗ on B . Aga in we consider B 7→ Ω ∗ d =0 ( B ) as a functor fr o m smoo th ma nifolds to Z / 2 Z -gra ded ab elian groups. W e deﬁne a natural transfor mation R : ˆ K ( B ) → Ω d =0 ( B ) b y R ([ E , ρ ]) = Ω( E ) − dρ . Again we must show that R is well-deﬁned. W e will use the relatio n (6) o f the η -form and the loc a l index form, and the obvious prop erties of lo ca l index forms Ω( E ⊔ B E ′ ) = Ω( E ) + Ω( E ′ ) , Ω( E op ) = − Ω( E ) . W e star t with ˜ R : G ( B ) → Ω( B ) , ˜ R ( E , ρ ) := Ω( E ) − dρ . Since Ω( E ) is closed, ˜ R ( E , ρ ) is clo s ed. If ( E , ρ ) is paired with ( E ′ , ρ ′ ) via the taming ( E ⊔ B E ′ op ) t , then ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ). It follo ws R ( E , ρ ) = Ω( E ) − dρ = Ω( E ) − dρ ′ − dη (( E ⊔ B E ′ op ) t ) = Ω( E ) − dρ ′ − Ω( E ) − Ω( E ′ op ) = Ω( E ′ ) − dρ ′ = R ( E ′ , ρ ′ ) . Since ˜ R is additiv e it descends to G ( B ) / ∼ and ﬁnally to the map R : ˆ K ( B ) → Ω d =0 ( B ). It fo llows fro m Ω( f ∗ E ) = f ∗ Ω( E ) that R is a natural transformation. 2.4.6 . — The natural tra ns fo rmations satisfy the following relations: L emma 2.19 . — 1. R ◦ a = d 2. ch dR ◦ I = [ . . . ] ◦ R . Pr o of . — The ﬁrst relation is a n immediate co nsequence of the deﬁnition o f R a nd a . The second relation is the lo cal index theorem 2 .9. 24 ULRICH BUNKE & THOMAS SCHICK 2.4.7 . — Via the embedding H dR ( B ) ⊆ Ω( B ) / im ( d ), the Chern character c h dR : K ( B ) → H dR ( B ) can b e considered as a natural tr ansformation c h dR : K ( B ) → Ω( B ) / i m ( d ) . Pr op osition 2.20 . — The fol lowing se quenc e is exact: K ( B ) c h dR → Ω( B ) / i m ( d ) a → ˆ K ( B ) I → K ( B ) → 0 . W e give the pro of in the following couple of subsection. 2.4.8 . — W e start with the surjectivit y of I : ˆ K ( B ) → K ( B ). The main po in t is the fact that every element x ∈ K ( B ) can b e realized as the index of a family of Dirac oper ators by Lemma 2 .18. So let x ∈ K ( B ) and E b e a geometric family with index ( E ) = x . Then w e have I ([ E , 0]) = x . 2.4.9 . — Next we show exa c tness at ˆ K ( B ). F o r ρ ∈ Ω( B ) / im ( d ) we hav e I ◦ a ( ρ ) = I ([ ∅ , − ρ ]) = index ( ∅ ) = 0, hence I ◦ a = 0. Cons ider a class [ E , ρ ] ∈ ˆ K ( B ) which satisﬁes I ([ E , ρ ]) = 0. W e ca n assume that the ﬁbres of the underlying submersion ar e not zero-dimensiona l. Indeed, if necessa ry , we can replace E by E ⊔ B ( ˜ E ⊔ B ˜ E op ) for s ome even family with nonzero-dimensiona l ﬁbres without ch ang ing the smo oth K -theory class by Lemma 2.15. Since inde x ( E ) = 0 this family admits a taming E t (2.2.2). Therefore, ( E , ρ ) is pair ed with ( ∅ , ρ − η ( E t )). It follows that [ E , ρ ] = a ( η ( E t ) − ρ ). 2.4.1 0. — In order to pr epare the pr oo f of exactness at Ω( B ) / im ( d ) in 2 .4.11 we need some facts ab out the classiﬁca tion of taming s of a geometr ic family E . The main idea is to measure the diﬀerence b et ween tamings of E using a lo cal index theorem for E × [0 , 1] (compar e [ Bun , Cor. 2.2.19]). Let us assume that the underlying submersion π : E → B decompo ses as E = E ev ⊔ B E odd such that the r estriction of π to the even a nd o dd parts is surjective with nonzer o- a nd even -dimensio nal and o dd-dimensional ﬁbres, a nd which is suc h that the Cliﬀor d bundle is nowhere zero - dimensional. If in dex ( E ) = 0, then there exis ts a taming E t (see 2 .2.2). Assume that E t ′ is a seco nd taming. Both tamings to gether induce a b oundary taming o f the family with b oundary ( E × [0 , 1]) bt . In [ Bun ] we hav e discussed in detail g eometric families with boundar ies a nd the op eration of taking a bo undary of a geo metric family with b oundary . In the present case E × [0 , 1] has t wo boundar y faces lab eled b y the endpo in ts { 0 , 1 } of the in terv al. W e hav e ∂ 0 ( E × [0 , 1 ]) ∼ = E a nd ∂ 1 ( E × [0 , 1]) ∼ = E op . A b oundary taming ( E × [0 , 1]) bt is given by tamings of ∂ i ( E × [0 , 1]) fo r i = 0 , 1 (see [ Bun , Def. 2.1.4 8]). W e use E t at E × { 0 } a nd E op t ′ at E × { 1 } . The b oundary tamed family has an index index (( E × [0 , 1]) bt ) ∈ K ( B ) whic h is the obstructio n against extending the boundar y taming to a taming [ Bun , Lemma 2.2.6]. The c onstruction of the lo cal index form extends to geometric families with bo undaries. Because of the geometric pro duct structure o f E × [0 , 1] we hav e Ω( E × [0 , 1 ]) = 0. The index theorem for b oundary tamed families [ Bun , Theor em 2.2.1 8] gives c h dR ◦ index (( E × [0 , 1]) bt ) = [ η ( E t ) − η ( E t ′ )] . SMOOTH K-THEOR Y 25 On the other ha nd, g iv en x ∈ K ( B ) and E t , since we hav e c hosen our family E suﬃcient ly big, there exists a taming E t ′ such that index (( E × [0 , 1]) bt ) = x . T o prove this, we argue as follows. Giv en tamings E t and E t ′ we obtain a family D ( E t , E t ′ ) of per tur b ed Dirac op erato rs ov er B × R whic h r estricts to D ( E t ) on B × { β } for β < 0 , and to D ( E t ′ ) for β ≥ 1, and which interpolates these families for β ∈ [0 , 1]. Since the restriction o f D ( E t , E t ′ ) is inv e r tible o utside of a compac t subset of B × R (note that B is co mpact) it gives rise to a class [ E t , E t ′ ] ∈ K K ( C , C ( B ) ⊗ C 0 ( R )). The Dirac o per a tor on R provides a clas s [ ∂ ] ∈ K K ( C 0 ( R ) , C ), a nd one chec ks —using the method of connections as in [ Bun95 , pro of o f Prop osition 2.1 1 ] or directly working with the unbounded picture [ BJ83 ]— that D ( E × [0 , 1]) bt represents the K asparov pro duct [ E t , E t ′ ] ⊗ C 0 ( R ) [ ∂ ] ∈ K K ( C , C ( B )) . The ma p K c ( B × R ) ∼ → K K ( C , C ( B ) ⊗ C 0 ( R )) ·⊗ C 0 ( R ) [ ∂ ] → K K ( C , C ( B )) ∼ → K ( B ) is by [ Kas81 , Paragraph 5, Theorem 7] the inv erse of the s us p ension isomor phism, so in particula r s urjectiv e. It r emains to see that one ca n exhaust K K ( C , C ( B ) ⊗ C 0 ( R )) with clas s es o f the form [ E t , E t ′ ] by v ar ying the taming E t ′ . W e s k etch an argument in the even-dimensional case. The o dd-dimensional cas e is similar. F or a sepa rable inﬁnite-dimensional Hilb ert s pace H let GL 1 ( H ) ⊂ GL ( H ) be the group of inv ertible o pera tors o f the form 1 + K with K ∈ K ( H ) compact. The space GL 1 ( H ) has the homotopy type of the classifying space for K 1 . The bundle o f Hilb ert spaces H ( E ) + → B gives rise to a (canonica lly trivial, up to ho- motopy) bundle o f gr oups GL 1 ( H ( E ) + ) → B by taking GL 1 ( . . . ) ﬁbr ewise (it is here where we use that the family is suﬃciently big so that H ( E ) + is inﬁnite-dimensional). Let Γ( GL 1 ( H ( E ) + )) b e the top olog ica l group of sections. Then we hav e an isomo r - phism π 0 Γ( GL 1 ( H ( E ) + )) ∼ = K 1 ( B ). Let x ∈ K 1 ( B ) be represented by a s ection s ∈ Γ( GL 1 ( H ( E ) + )). W e can a pproximate s − 1 b y a s moo th family of smo othing op erators. Therefor e we ca n assume that s − 1 is given by a s moo th ﬁbrewise in tegr al kernel (a pretaming in the la ng uage of [ Bun ]) (4) . There is a bijection b etw e en ta mings E t ′ and sections s ∈ Γ( GL 1 ( H ( E ) + )) of this t yp e which maps E t ′ to s := D + ( E t ) − 1 D + ( E t ′ ). The ma p which asso ciates the K K - class [ E t , E t ′ ] to the sec tio n s is just one realiza tion o f the susp ension isomorphism K 1 ( B ) → K 0 c ( B × R ) (using the Kasparov picture of the latter g roup). In pa rticular we see that all classes in K 0 c ( B × R ) arise as [ E t , E t ′ ] for v ar io us ta mings E t ′ . 2.4.1 1. — W e now show exactness a t Ω( B ) / im ( d ). Let x ∈ K ( B ). Then we have a ◦ c h dR ( x ) = [ ∅ , − c h dR ( x )]. W e choos e a geometric family E as in 2.4.10 and set ˜ E := E ⊔ B E op . In the pro of o f Lemma 2.11 we hav e constructed a taming ˜ E t such that η ( ˜ E t ) = 0. Using the discussion 2.4.10 we choo se a second taming ˜ E t ′ such that (4) Alternativ ely one can dir ectly produce s uc h a section using the setup describ ed in [ MR0 7 ]. 26 ULRICH BUNKE & THOMAS SCHICK index (( ˜ E × [0 , 1 ]) bt ) = − x , hence η ( ˜ E t ′ ) = c h dR ( x ). By the taming ˜ E t ′ we see that the cycle ( ˜ E , 0) pairs with ( ∅ , − c h dR ( x )). On the other hand, via ˜ E t the cycle ( ˜ E , 0) pairs with 0. It follows that ( ∅ , − c h dR ( x )) ∼ 0 a nd hence a ◦ c h dR = 0 . Let now ρ ∈ Ω( B ) / im ( d ) b e such that a ( ρ ) = [ ∅ , − ρ ] = 0 . Then b y Lemma 2.17 there exists a cycle ( ˆ E , ˆ ρ ) such that ( ˆ E , ˆ ρ − ρ ) pairs with ( ˆ E , ˆ ρ ). Therefore there ex is ts a taming E t ′ of E := ˆ E ⊔ B ˆ E op such that η ( E t ′ ) = − ρ . Let E t be the taming with v a nishing η -for m constructed in the pro of of Lemma 2.11. The tw o tamings induce a b oundary taming ( E × [0 , 1 ]) bt such that ch dR ◦ inde x (( E × [0 , 1 ]) bt ) = − η ( E t ′ ) = ρ . This shows that ρ is in the image o f c h dR . ✷ 2.4.1 2. — W e now improve Lemma 2.13. This result will be very helpful in verifying well-deﬁnedness o f maps o ut of s moo th K -theory , e.g. the smo oth Cher n c har acter. L emma 2.21 . — If [ E 0 , ρ 0 ] = [ E 1 , ρ 1 ] and at le ast one of these families has a higher- dimensional c omp onent, then ( E 0 , ρ 0 ) is p air e d with ( E 1 , ρ 1 ) . Pr o of . — By Lemma 2.13 there exists [ E ′ , ρ ′ ] such that ( E 0 , ρ 0 ) + ( E ′ , ρ ′ ) is paired with ( E 1 , ρ 1 ) + ( E ′ , ρ ′ ) by a taming ( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t . W e hav e ρ 1 − ρ 0 = η (( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t ) . Since in dex ( E 0 ) = inde x ( E 1 ) there exists a taming ( E 0 ⊔ B E op 1 ) t . F urthermore , there exists a taming ( E ′ ⊔ B E ′ op ) t with v a nishing η -inv aria nt (see the pro of o f Lemma 2 .11). These t wo tamings combine to a taming ( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t ′ . There ex is ts ξ ∈ K ( B ) s uch that c h dR ( ξ ) = η (( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t ) − η (( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t ′ ) = η (( E 0 ⊔ B E ′ ⊔ B ( E 1 ⊔ B E ′ ) op ) t ) − η (( E 0 ⊔ B E op 1 ) t ) . W e can now adjust (using 2.4.10 ) the taming ( E 0 ⊔ B E op 1 ) t such that we ca n choos e ξ = 0. It follows that ρ 1 − ρ 0 = η (( E 0 ⊔ B E op 1 ) t ). 2.5. Comparison with the Hopkins-Singe r theory and the ﬂat theory. — 2.5.1 . — An imp ortant consequence o f the axioms 1.1 for a smo oth generalized co- homology theory is the homotopy formula. Let ˆ h be a smo oth extension of a pair ( h, c ). Let x ∈ ˆ h ([0 , 1 ] × B ), and let i k : B → { k } × B ⊂ [0 , 1] × B , k = 0 , 1, b e the inclusions. L emma 2.22 . — i ∗ 1 ( x ) − i ∗ 0 ( x ) = a Z [0 , 1] × B /B R ( x ) ! . SMOOTH K-THEOR Y 27 Pr o of . — Let p : [0 , 1] × B → B denote the pro jection. If x = p ∗ y , then on the o ne hand the left-hand side of the equation is zer o. O n the other hand, R ( x ) = p ∗ R ( y ) so that R [0 , 1] × B /B R ( x ) = 0 , to o. Since p is a homo top y equiv alence there ex ists ¯ y ∈ h ( B ) such that I ( x ) = p ∗ ( ¯ y ). Because of the surjectivity of I we ca n c ho ose y ∈ ˆ h ( B ) such that I ( y ) = ¯ y . It follows that I ( x − p ∗ y ) = 0. By the exactness of (3) there e x ists a form ω ∈ Ω( I × B ) / im ( d ) such that x − p ∗ y = a ( ω ). By Stokes’ theorem we ha ve the equa lit y i ∗ 1 ω − i ∗ 0 ω = R [0 , 1] × B /B dω in Ω( B ) / i m ( d ). By (2) we hav e dω = R ( a ( ω )). It fo llows that Z [0 , 1] × B /B dω = Z [0 , 1] × B /B R ( a ( ω )) = Z [0 , 1] × B /B R ( x − p ∗ y ) = Z [0 , 1] × B /B R ( x ) . This implies i ∗ 1 x − i ∗ 0 x = i ∗ 1 a ( ω ) − i ∗ 0 a ( ω ) = a i ∗ 1 ω − i ∗ 0 ω ) = a ( Z [0 , 1] × B /B R ( x ) ! . 2.5.2 . — Let ˆ h be a s mo oth e x tension o f a pa ir ( h, c ). W e use the notation introduced in 1 .2 .2. Deﬁnition 2.23 . — The asso ciate d ﬂ at functor is deﬁne d by B 7→ ˆ h f lat ( B ) := ker { R : ˆ h ( B ) → Ω d =0 ( B , N ) } . Recall that a functor F fro m smo oth manifolds is homotopy inv ar iant , if for the t wo embeddings i k : B → { k } × B → [0 , 1] × B , k = 0 , 1, we have F ( i 0 ) = F ( i 1 ). As a consequence of the homotopy formula Le mma 2.22 the functor ˆ h f lat is homotopy in v ariant. In int eres ting cases it is part o f a genera lized c o homology theor y . The map c : h → H N gives rise to a co ﬁbre sequence in the s table homotopy categ ory h c → H N → h N , R / Z which deﬁnes a sp ectrum h N , R / Z . Pr op osition 2.24 . — If ˆ h is the Hopkins-Singer extension of ( h, c ) , then we have a natur al isomorph ism ˆ h f lat ( B ) ∼ = h N , R / Z ( B )[ − 1] . In the special case that N = h ∗ ⊗ Z R this is [ HS05 , (4.57)]. 28 ULRICH BUNKE & THOMAS SCHICK 2.5.3 . — In the case of K -theory and the Chern character c h R : K → H ( K ∗ ⊗ Z R ) one usually writes K R / Z := h K ∗ ⊗ Z R , R / Z . The functor B 7→ K R / Z ( B ) is called R / Z - K - theory . Since R / Z is an injectiv e a belian group we hav e a univ ersa l co eﬃcient formula K R / Z ∗ ( B ) ∼ = Hom ( K ∗ ( B ) , R / Z ) , (10) where K ∗ ( B ) denotes the K -homology of B . A geometric in terpr etation of R / Z - K -theor y w as ﬁr st pr opo sed in [ Kar87 ], [ Kar97 ]. I n these reference it w as called m ultiplicative K -theory . The analytic construction of the push-forward has been given in [ Lot94 ]. 2.5.4 . — Pr op osition 2.25 . — Ther e is a natur al isomorphi sm of functors ˆ K f lat ( B ) ∼ = K R / Z ( B )[ − 1] . Pr o of . — In the following (the par agraphs 2.5.5 , 2.5.6 ) w e sketch tw o conceptually very diﬀerent arg uments. F o r details we refer to [ BS09 , Section 5, Section 7]. 2.5.5 . — In the ﬁrst step one extends ˆ K f lat to a reduced coho mology theory on smo oth manifolds. The reduced gro up of a p ointed manifold is deﬁned as the kernel of the restriction to the p oin t. The missing structure is a susp ension iso mo rphism. It is induced b y the map ˆ K ( B ) → ˆ K ( S 1 × B ) giv en by x 7→ pr ∗ 1 x S 1 ∪ pr ∗ 2 x , where x S 1 ∈ ˆ K 1 ( S 1 ) is deﬁned in Deﬁnition 5.6, and the ∪ -pr oduct is deﬁned b elow in 4.1. The in verse is induced b y the push-forward ( ˆ pr 2 ) ! : ˆ K ( S 1 × B ) → ˆ K ( B ) along pr 2 : S 1 × B → B int ro duced b elow in 3.18. Finally one v eriﬁes the exactness of mapping cone sequences. In order to identif y the r esulting reduced co homology theory with R / Z - K -theory one constructs a pair ing b et ween ˆ K f lat and K -homology , using an analytic mo del as in [ Lot94 ]. This pa iring, in view of the universal coe ﬃcien t formula (10) gives a map of cohomolog y theor ies ˆ K f lat ( B ) → K R / Z ( B )[ − 1] which is an iso morphism by comparison of co e ﬃcien ts. 2.5.6 . — The second argument is ba sed on the comparison with the Hopkins-Singer theory . W e let B 7→ ˆ K H S ( B ) denote the version of the smo oth K -theor y functor deﬁned b y Hopkins- Singer [ HS05 ]. In [ BS09 , Section 5 ] we show that there is a unique natura l isomorphism ˆ K ev ∼ → ˆ K ev H S . In view of 2 .2 4 we get the isomor phism ˆ K ev f lat ( B ) ∼ → ˆ K ev H S,f lat ( B ) ∼ → K R / Z ev [ − 1]( B ) . In [ BS09 ] we furthermore show that using the integration for ˆ K a nd the suspe ns io n isomorphism for K R / Z this isomorphism extends to the o dd parts. SMOOTH K-THEOR Y 29 2.5.7 . — Many of the interesting examples given in Section 5 can b e understo o d (at least to a larg e extend) alr eady a t this stage. W e r e commend to loo k them up no w, if one is less interseted in structural questio ns . This should a ls o serve as a motiv ation for the constructions in Sections 3 and 4. 3. Push-forw ard 3.1. K -orienta tio n. — 3.1.1 . — The groups S pin ( n ) a nd S pin c ( n ) ﬁt in to exact sequences 1 − − − − → Z / 2 Z − − − − → S pin ( n ) − − − − → S O ( n ) − − − − → 1   y   y   y id 1 − − − − → U (1) i − − − − → S pin c ( n ) π − − − − → S O ( n ) − − − − → 1 1 → Z / 2 Z → S p in c ( n ) ( λ,π ) → U (1) × S O ( n ) → 1 such that λ ◦ i : U (1) → U (1) is a do uble cov ering. Let P → B b e a n S O ( n )-pr incipal bundle. W e let S pin c ( n ) act on P via the pro jection π . Deﬁnition 3.1 . — A S pin c -r e duction of P is a diagr am Q   ? ? ? ? ? ? ? f / / P           B , wher e Q → B is a S pin c ( n ) -princip al bund le and f is S pin c ( n ) -e quivariant. 3.1.2 . — Let p : W → B be a prop er submersion with v ertical bundle T v p . W e assume that T v p is orie nted. A choice o f a vertical metric g T v p gives a n S O -reduction S O ( T v p ) o f the frame bundle F r ( T v p ), the bundle of oriented orthonorma l frames. Usually one calls a map b etw e e n manifolds K -oriented if its stable normal bundle is equipped with a K -theor y Thom class. It is a well-kno wn fact [ ABS64 ] that this is equiv alent to the choice of a S pi n c -structure on the stable no rmal bundle. Finally , isomorphism classes of choices of S pi n c -structures on T v p and the stable norma l bundle of p are in bijective corresp ondence. So for the purp ose of the pre s en t paper we ado pt the following deﬁnition. Deﬁnition 3.2 . — A top olo gic al K -orientation of p is a S pin c -r e duction of S O ( T v p ) . In the present pap er we prefer to work with S pin c -structures on the vertical bundle since it directly gives r ise to a family of Dirac op erators a lo ng the ﬁbres . The g oal of this sectio n is to intro duce the notion o f smo oth K -orientation whic h reﬁnes a g iv en topo logical K - orientation. 30 ULRICH BUNKE & THOMAS SCHICK 3.1.3 . — In or der to deﬁne s uch a family of Dirac op erators we must choose additional geometric data. If we choo se a horizontal distribution T h p , then we get a connection ∇ T v p which r estricts to the Levi-Civita connection a long the ﬁbres. Its construction go es as follows. First one ch o oses a metric g T B on B . It induces a horizo n tal metr ic g T h p via the iso morphism dp | T h p : T h p ∼ → p ∗ T B . W e get a metric g T v p ⊕ g T h p on T W ∼ = T v p ⊕ T h p which gives rise to a Levi- Civita c o nnection. Its pr o jection to T v p is ∇ T v p . Finally one checks that this connection is indep enden t of the choice o f g T B . 3.1.4 . — The connection ∇ T v p can b e co nsidered as an S O ( n )-principal bundle con- nection on the frame bundle S O ( T v p ). In order to deﬁne a family of Dira c o pera tors, or be tter, the Bismut sup er-connection we m ust cho o se a S pin c -reduction ˜ ∇ of ∇ T v p , i.e. a connection o n the S pi n c -principal bundle Q which reduces to ∇ T v p . If we think of the connections ∇ T v p and ˜ ∇ in terms of horizontal distributions T h S O ( T v p ) and T h Q , then w e say that ˜ ∇ r e duces to ∇ T v p if dπ ( T h Q ) = π ∗ ( T h S O ( T v p )). 3.1.5 . — The S pin c -reduction of Fr ( T v p ) deter mines a s pino r bundle S c ( T v p ), a nd the choice of ˜ ∇ turns S c ( T v p ) into a family of Dirac bundles. In this wa y the choices of the S pin c -structure a nd ( g T v p , T h p, ˜ ∇ ) turn p : W → B in to a geometric family W . 3.1.6 . — Lo cally on W we can choo s e a S pin - structure on T v p with asso ciated spinor bundle S ( T v p ). Then we can write S c ( T v p ) = S ( T v p ) ⊗ L for a her mitean line bundle L with connectio n. The s pin structure is given by a S p i n -reduction q : R → S O ( T v p ) (similar to 3.1) which can a c tua lly be considered as a subbundle of Q . Since q is a double covering and thus has discrete ﬁbres, the connection ∇ T v p (in contrast to the S pin c -case) has a unique lift to a S p in ( n )-connection on R . The spinor bundle S ( T v p ) is asso ciated to R and ha s an induced connection. In view of the relations of the g roups 3.1.1 the squa re o f the lo cally deﬁned line bundle L is the globally deﬁned bundle L 2 → W ass o cia ted to the S pi n c -bundle Q via the representation λ : S pin c ( n ) → U (1). The connection ˜ ∇ thus induces a connection on ∇ L 2 , a nd hence a co nnection on the lo cally deﬁned s q uare ro ot L . Note that vice versa, ∇ L 2 and ∇ T v p determine ˜ ∇ uniquely . 3.1.7 . — W e in tro duce the form c 1 ( ˜ ∇ ) := 1 4 π i R L 2 (11) which w ould be the Cher n form of the bundle L in c a se o f a globa l S pin -s tructure. Let R ∇ T v p ∈ Ω 2 ( W , End ( T v p )) denote the curv ature o f ∇ T v p . The closed form ˆ A ( ∇ T v p ) := det 1 / 2   R ∇ T v p 4 π sinh  R ∇ T v p 4 π    represents the ˆ A -class of T v p . SMOOTH K-THEOR Y 31 Deﬁnition 3.3 . — The r elevant diﬀer ential form for lo c al index the ory in t he S pin c - c ase is ˆ A c ( ˜ ∇ ) := ˆ A ( ∇ T v p ) ∧ e c 1 ( ˜ ∇ ) . If we consider p : W → B with the geometry ( g T v p , T h p, ˜ ∇ ) and the Dirac bundle S c ( T v p ) as a geometric family W o ver B , then b y compa rison with the des cription 2.2.4 of the lo cal index form Ω( W ) w e s e e that Z W/B ˆ A c ( ˜ ∇ ) = Ω( W ) . 3.1.8 . — The dep endence of the form ˆ A c ( ˜ ∇ ) o n the data is describ ed in terms of the transgres s ion form. Let ( g T v p i , T h i p, ˜ ∇ i ), i = 0 , 1 , b e tw o choices of geometric data. Then we ca n choose geometric data ( g T v p , T h p, ˜ ∇ ) on p = id [0 , 1] × p : [0 , 1] × W → [0 , 1 ] × B (with the induced S pin c -structure on T v p ) which restr icts to ( g T v p i , T h i p, ˜ ∇ i ) on { i } × B . The class ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) := Z [0 , 1] × W /W ˆ A c ( ˜ ∇ ) ∈ Ω( W ) / im ( d ) is indep enden t of the extension and satisﬁes d ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) = ˆ A c ( ˜ ∇ 1 ) − ˆ A c ( ˜ ∇ 0 ) . (12) Deﬁnition 3.4 . — The form ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) is c al le d the tr ansgr ession form. Note that w e ha ve the identit y ˜ ˆ A c ( ˜ ∇ 2 , ˜ ∇ 1 ) + ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) = ˜ ˆ A c ( ˜ ∇ 2 , ˜ ∇ 0 ) . (13) As a consequence we get the identit ies ˜ ˆ A c ( ˜ ∇ , ˜ ∇ ) = 0 , ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) = − ˆ A c ( ˜ ∇ 0 , ˜ ∇ 1 ) . (14) 3.1.9 . — W e can now introduce the notion of a smo oth K - orientation of a pro p er submersion p : W → B . W e ﬁx an underlying top olog ical K -o rien tation of p (see Deﬁnition 3.2) whic h is given by a S p in c -reduction of S O ( T v p ). In order to ma k e this prec is e we must choos e a n orientation and a metric on T v p . W e co nsider the set O o f tuples ( g T v p , T h p, ˜ ∇ , σ ) where the ﬁrst three entries hav e the same mea ning as ab ov e (see 3.1.3 ), a nd σ ∈ Ω odd ( W ) / im ( d ). W e int ro duce a relation o 0 ∼ o 1 on O : Two tuples ( g T v p i , T h i p, ˜ ∇ i , σ i ), i = 0 , 1 are related if and only if σ 1 − σ 0 = ˜ ˆ A ( ˜ ∇ 1 , ˜ ∇ 0 ). W e claim that ∼ is an eq uiv alence rela tion. In fact, symmetry and reﬂexiv ity follow from (14 ), while tra nsitivit y is a co ns equence of (13). Deﬁnition 3.5 . — The set of smo oth K -orientations which r eﬁne a ﬁx e d underlying top olo gic al K - orientation of p : W → B is t he set of e quivalenc e classes O / ∼ . 32 ULRICH BUNKE & THOMAS SCHICK 3.1.1 0. — Note that Ω odd ( W ) / im ( d ) acts on the set of smo oth K - o rient atio ns . If α ∈ Ω odd ( W ) / im ( d ) and ( g T v p , T h p, ˜ ∇ , σ ) repr esen ts a smo oth K -orientation, then the transla te of this orientation by α is repr esen ted by ( g T v p , T h p, ˜ ∇ , σ + α ). As a consequence of (13) we get: Cor ol lary 3.6 . — The set of smo oth K -orientations r eﬁn ing a ﬁxe d underlying top o- lo gic al K -orientation is a torsor over Ω odd ( W ) / im ( d ) . 3.1.1 1. — If o = ( g T v p , T h p, ˜ ∇ , σ ) ∈ O represents a smo oth K -o rien tation, then we will wr ite ˆ A c ( o ) := ˆ A c ( ˜ ∇ ) , σ ( o ) := σ . 3.2. Deﬁnition of the Push-forw ard. — 3.2.1 . — W e consider a prop er submersion p : W → B with a choice o f a top ologica l K -or ien tation. Assume that p has clo sed ﬁbres. Let o = ( g T v p , T h p, ˜ ∇ , σ ) represent a smo oth K -orientation which reﬁnes the given topo logical one. T o every geometric family E over W we want to asso ciate a g eometric family p ! E ov er B . Let π : E → W denote the underlying pro per submersion with closed ﬁbres of E which comes with the geometric da ta g T v π , T h π and the family of Dirac bundles ( V , h V , ∇ V ). The underlying prope r submersion with closed ﬁbres of p ! E is q := p ◦ π : E → B . The hor izon tal bundle o f π admits a decomp osition T h π ∼ = π ∗ T v p ⊕ π ∗ T h p , where the isomorphism is induced by dπ . W e deﬁne T h q ⊆ T h π such that dπ : T h q ∼ = π ∗ T h p . F urthermor e we hav e a n identiﬁcation T v q = T v π ⊕ π ∗ T v p . Using this decomp osition we deﬁne the vertical metric g T v q := g T v π ⊕ π ∗ g T v p . The o rient ations o f T v π a nd T v p induce an o rien tation of T v q . Finally w e must construct the Dirac bundle p ! V → E . Lo cally on W w e choose a S pin -structure o n T v p and let S ( T v p ) b e the spinor bundle. Then we can write S c ( T v p ) = S ( T v p ) ⊗ L for a hermitean line bundle with connection. Lo cally o n E we can choose a S pin -structure on T v π with spinor bundle S ( T v π ). Then we ca n write V = S ( T v π ) ⊗ Z , where Z is the twisting bundle of V , a hermitean vector bundle with connection ( Z / 2 Z -gra ded in the e ven cas e). The lo cal s pin structures on T v π and π ∗ T v p induce a lo cal S pin -str ucture o n T v q = T v π ⊕ π ∗ T v p . Therefore lo cally w e can deﬁne the family o f Dirac bundles p ! V := S ( T v q ) ⊗ π ∗ L ⊗ Z . It is easy to see that this bundle is well-deﬁned independent of the choices of lo cal S pin - structures a nd therefore is a globally deﬁned family of Dirac bundles. Deﬁnition 3.7 . — L et p ! E denote the ge ometric family given by q : E → B and p ! V → E with the ge ometric struct ur es deﬁne d ab ove. It immediately follo ws from the deﬁnitions, that p ! ( E op ) ∼ = ( p ! E ) op . SMOOTH K-THEOR Y 33 3.2.2 . — Let p : W → B be a prop er submersion with a smo oth K - orientation rep- resented b y o . In 3.2.1 we hav e constructed for each geo metric family E over W a push-forward p ! E . Now we introduce a parameter λ ∈ (0 , ∞ ) int o this co ns tr uction. Deﬁnition 3.8 . — F or λ ∈ (0 , ∞ ) we deﬁne the ge ometric family p λ ! E as in 3.2.1 with the only diﬀer enc e that the metric on T v q = T v π ⊕ π ∗ T v p is given by g T v q λ = λ 2 g T v π ⊕ π ∗ g T v p . More sp eciﬁcally , we use scaling inv ar iance of the spinor bundle to canonically iden tify the Dirac bundle for the metric g λ lo cally with p ! V := S ( T v q ) ⊗ π ∗ L ⊗ Z (for g 1 ). This uses the description of S ( T v p ) in terms of tensor pro ducts of S ( T v π ) and π ∗ S ( T v p ) (compare [ Bun , Section 2.1.2]) and the scaling inv ariance of S ( T v π ). How e ver, with this ident iﬁcatio n the Cliﬀord multiplication by v ectors in T v q = T v π ⊕ π ∗ T v p is resca led on the summand T v π by λ . The connection is s ligh tly mo re complicated, but conv er ges for λ → 0 to so me kind of sum co nnection. The family of geometr ic families p λ ! E is called the adia batic defo r mation of p ! E . There is a na tural wa y to deﬁne a geometric family F on (0 , ∞ ) × B such that its restriction to { λ } × B is p λ ! E . In fact, we deﬁne F := ( id (0 , ∞ ) × p ) ! ((0 , ∞ ) × E ) with the ex c e ptio n that w e ta ke the appropr iate vertical metric. No te again that the underlying bundle can b e cano nically identiﬁed with (0 , ∞ ) × p ! V . In the following, we work with this ident iﬁcatio ns throughout. Although the vertical metrics of F and p λ ! E co llapse as λ → 0 the induced connec- tions and the curv a ture tensors on the vertical bundle T v q converge and simplify in this limit. This fact is heavily used in lo cal index theory , and we refer to [ BGV04 , Sec 10.2 ] for details. In particular, the int egr al ˜ Ω( λ, E ) := Z (0 ,λ ) × B /B Ω( F ) (15) conv erges, and we have lim λ → 0 Ω( p λ ! E ) = Z W/B ˆ A c ( o ) ∧ Ω( E ) , Ω( p λ ! E ) − Z W/B ˆ A c ( o ) ∧ Ω( E ) = d ˜ Ω( λ, E ) . (16) 3.2.3 . — Let p : W → B b e a pr oper submers ion with closed ﬁbres with a smo oth K - orientation represented b y o . W e now start with the construction of the push-forward p ! : ˆ K ( W ) → ˆ K ( B ). F or λ ∈ (0 , ∞ ) and a cycle ( E , ρ ) we deﬁne ˆ p λ ! ( E , ρ ) := [ p λ ! E , Z W/B ˆ A c ( o ) ∧ ρ + ˜ Ω( λ, E ) + Z W/B σ ( o ) ∧ R ([ E , ρ ])] ∈ ˆ K ( B ) . (17) Since ˆ A c ( o ) and R ([ E , ρ ]) are closed, the maps Ω( W ) / im ( d ) ∋ ρ 7→ Z W/B ˆ A c ( o ) ∧ ρ ∈ Ω( B ) / im ( d ) , 34 ULRICH BUNKE & THOMAS SCHICK Ω( W ) / im ( d ) ∋ σ ( o ) 7→ Z W/B σ ( o ) ∧ R ([ E , ρ ]) ∈ Ω( B ) / im ( d ) are well-deﬁned. It immediately follows fro m the deﬁnition that ˆ p λ ! : G ( W ) → ˆ K ( B ) is a homomorphism of semigroups. 3.2.4 . — The homomorphism ˆ p λ ! : G ( W ) → ˆ K ( B ) commutes with pull-bac k. More precisely , let f : B ′ → B be a smo oth map. Then we deﬁne the submersion p ′ : W ′ → B ′ b y the car tes ia n diagram W ′ p ′   F / / W p   B ′ f / / B . The diﬀerential dF : T W ′ → F ∗ T W induces an isomo rphism dF : T v W ′ ∼ → F ∗ T v W . Therefore the metric, the orientation, and the S pin c -structure of T v p induce by pull- back corresp onding structures on T v p ′ . W e deﬁne the horizo n tal distribution T h p ′ such that dF ( T h p ′ ) ⊆ F ∗ T h p . Fina lly we se t σ ′ := F ∗ σ . The representativ e of a smo oth K -o rien tation given by these s tr uctures will b e denoted b y o ′ := f ∗ o . An inspec tio n of the deﬁnitions sho ws: L emma 3.9 . — The pul l-b ack of r epr esent atives of smo oth K -orientations pr eserves e quivalenc e and henc e induc es a pul l-b ack of smo oth K -orientations. Recall from 3.1.5 that the representativ es o and o ′ of the smo oth K -orientations enhance p and p ′ to geometr ic families W a nd W ′ . W e hav e f ∗ W ∼ = W ′ . Note that we have F ∗ ˆ A c ( o ) = ˆ A c ( o ′ ). If E is a geometric family ov er W , then an inspec tio n o f the deﬁnitions shows that f ∗ p ! ( E ) ∼ = p ′ ! ( F ∗ E ). The following lemma now follows immediately from the deﬁnitions L emma 3.10 . — We have f ∗ ◦ ˆ p λ ! = ˆ p ′ λ ! ◦ F ∗ : G ( W ) → ˆ K ( B ′ ) . 3.2.5 . — L emma 3.11 . — The class ˆ p λ ! ( E , ρ ) do es not dep end on λ ∈ (0 , ∞ ) . Pr o of . — Consider λ 0 < λ 1 . Note that ˆ p λ 1 ! ( E , ρ ) − ˆ p λ 0 ! ( E , ρ ) = [ p λ 1 ! E , ˜ Ω( λ 1 , E )] − [ p λ 0 ! E , ˜ Ω( λ 0 , E )] . Consider the inclusion i λ : B → { λ } × B ⊂ [ λ 0 , λ 1 ] × B and let F b e the family ov er [ λ 0 , λ 1 ] × B as in 3.2 .2 such that p λ ! E = i ∗ λ F . W e apply the homotopy formula L e mma 2.22 to x = [ F , 0 ]: i ∗ λ 1 ( x ) − i ∗ λ 0 ( x ) = a Z [ λ 0 ,λ 1 ] × B /B R ( x ) ! = a Z [ λ 0 ,λ 1 ] × B /B Ω( F ) ! = a  ˜ Ω( λ 1 , E ) − ˜ Ω( λ 0 , E )  , SMOOTH K-THEOR Y 35 where the last e q ualit y follo ws directly fro m the deﬁnition of ˜ Ω. This eq ualit y is equiv alent to [ p λ 1 ! E , ˜ Ω( λ 1 , E )] = [ p λ 0 ! E , ˜ Ω( λ 0 , E )] . In view of this Lemma we can omit the sup erscript λ and wr ite ˆ p ! ( E , ρ ) for ˆ p λ ! ( E , ρ ). 3.2.6 . — Let E b e a geometric family ov er W whic h admits a taming E t . Reca ll that the taming is g iven b y a family of smo othing oper ators ( Q w ) w ∈ W . W e have identiﬁed the Dirac bundle of p λ ! E with the Dirac bundle of p 1 ! E in a natural way in 3.2.2. The λ -dependence of the Dirac op e r ator takes the form D ( p λ ! E ) = λ − 1 D ( E ) + ( D H + R ( λ )) , where D H is the ho rizontal Dirac oper ator, and R ( λ ) is o f zer o order and remains bo unded as λ → 0 . W e now replace D ( E ) b y the inv ertible op erator D ( E ) + Q . Then for small λ > 0 the op erator λ − 1 ( D ( E ) + Q ) + ( D H + R ( λ )) is inv ertible. T o see this, we consider its s quare which has the structure λ − 2 ( D ( E ) + Q ) 2 + λ − 1 { D ( E ) + Q, ( D H + R ( λ )) } + ( D H + R ( λ )) 2 . The anticomm utator { D ( E ) , D H + R ( λ ) } is a ﬁrst-order vertical op erator which is thus dominated by a multiple of the p ositive seco nd order ( D ( E ) + Q ) 2 . The remaining parts of the a n ticommutator ar e zero-o rder and therefore also dominated by multiples of ( D ( E ) + Q ) 2 . The la st summand is a square of a selfadjoint op erator and hence non-negative. The family of op erato rs along the ﬁbr es o f p ! E induced by Q is not a taming since it is not given by a family of integral ope r ators a long the ﬁbres of p ! E → B . In order to unders tand its structure note the following. F or b ∈ B the ﬁbr e of ( p ! E ) b is the total space of the bundle E | W b → W b . The int egr al kernel Q induces a family of smo othing oper ators on the bundle of Hilb ert spaces H ( E | W b ) → W b . Using the natural identiﬁcation H ( p ! E ) b ∼ = L 2 ( W , S ( T v p ) ⊗ H ( E | W b )) we get the induced oper ator on H ( p ! E ) b . W e will call a family o f op erators with this structure a generalized taming. Now r ecall that the η -form η ( F t ) of a tamed o r generalize d tamed family F t is build from a family of sup erconnections A s ( F t ) parametrized b y s ∈ (0 , ∞ ) (see [ Bun , 2.2 .4.3]). F or 0 < s < 1 the family co incides with the usua l res caled Bis mut supe r connection a nd is indep endent of the taming. Therefo re the taming do es not aﬀect the ana lysis of ∂ s A s ( F t ) e − A s ( F t ) 2 for s → 0. In the interv a l s ∈ [1 , 2] the family A s ( F t ) smo othly connects with the family of sup erconnections given by A s ( F t ) = sD ( F t ) + ter ms with higher form degree 36 ULRICH BUNKE & THOMAS SCHICK for s ≥ 2. In or der to deﬁne the η -form η ( F t ) the main p oints are: 1. F o r small s the family A s ( F t ) behav es lik e the Bism ut sup erconnection. The formula (6) dη ( F t ) = Ω( F ) only depends on the b ehavior o f A s ( F t ) for small s . Therefore this formula contin ues to hold for generalized tamings. 2. ∂ s A s ( F t ) e − A s ( F t ) 2 is given by a family o f integral op e r ators with smo oth integral kernel. This holds true for tamed families as well as for familes which are tamed in the genera lized sense explained ab ov e. A pro of can be based on Duhamel’s principle. 3. The in tegr al kernel of ∂ s A s ( F t ) e − A s ( F t ) 2 together with all deriv atives v anishes exp o nen tially as s → ∞ . This follows by s pectra l estimates from the in vertibilit y and selfadjoint ness of D ( F t ). Now the in vertibilit y of D ( F t ) is exactly the desired eﬀect of a ta ming or generalized taming. Coming back to our iterated ﬁbre bundle we see that we ca n use the generalized taming for suﬃcientl y small λ > 0 like a taming in order to deﬁne an η -form which we will denote by η ( p λ ! E t ). T o b e precise this eta form is asso ciated to the family o f op erators A s ( p λ ! E ) + χ ( sλ − 1 ) sλ − 1 Q , s ∈ (0 , ∞ ) , where χ v anishes near zero and is equal to 1 on [1 , ∞ ). This means that we switch on the taming at time s ∼ λ , and we rescale it in the same wa y as the vertical par t of the Dirac op erator . W e ca n control the b ehaviour o f η ( p λ ! E t ) in the a dia batic limit λ → 0. The or em 3.12 . — lim λ → 0 η ( p λ ! E t ) = Z W/B ˆ A c ( o ) ∧ η ( E t ) . Pr o of . — T o write out a formal pro of o f this theorem seems to o long for the pre s en t pap er, without giving fundamental new insights. Instead we point out the following references. Adiaba tic limits o f η -forms of t wisted s ignature o pera tors w ere studied in [ BM04 , Section 5]. The same metho ds apply in the pr esen t case. The L -form in [ BM04 , Section 5] is the lo cal index form of the signa ture oper ator. In the present case it m ust b e replaced b y the form ˆ A c ( o ), the loca l index form of the S pin c -Dirac op erator. The absence of small eige nv alues simpliﬁes matters considerably . Since the geometric family p λ ! E admits a generalized taming it follows that index ( p λ ! E ) = 0. Hence we ca n also c ho ose a taming ( p λ ! E ) t . The latter choice together with the generalized taming induce a genera lized bounda r y taming of the family p λ ! E × [0 , 1] ov er B . The index theor em [ Bun , Theorem 2.2.1 8] can b e extended to genera lized b oundary tamed families (by copying the pro of ) and gives: SMOOTH K-THEOR Y 37 L emma 3.13 . — The diﬀer enc e of η -forms η (( p λ ! E ) t ) − η ( p λ ! E t ) is close d. Its de Rha m c ohomolo gy class satisﬁes [ η (( p λ ! E ) t ) − η ( p λ ! E t )] ∈ c h dR ( K ( B )) . 3.2.7 . — W e no w show that ˆ p ! : G ( W ) → ˆ K ( B ) passes thro ugh the equiv alence rela- tion ∼ . Since ˆ p ! is additive it suﬃces by Lemma 2.13 to show the following assertion. L emma 3.14 . — If ( E , ρ ) is p air e d with ( ˜ E , ˜ ρ ) , then ˆ p ! ( E , ρ ) = ˆ p ! ( ˜ E , ˜ ρ ) . Pr o of . — Let ( E ⊔ W ˜ E op ) t be the taming which induces the relation b etw een the t wo cycles, i.e. ρ − ˜ ρ = η  ( E ⊔ W ˜ E op ) t  . In view of the discuss ion in 3.2.6 we can choose a taming p λ ! ( E ⊔ ˜ E op ) t . [ p λ ! E , 0] − [ p λ ! ˜ E , 0] = [ p λ ! ( E ⊔ W ˜ E op ) , 0] = a  η  p λ ! ( E ⊔ W ˜ E op ) t  . By Prop osition 2.20 and Lemma 3.13 we ca n replac e the taming by the genera lize d taming and still g et [ p λ ! E , 0] − [ p λ ! ˜ E , 0] = a  η  p λ ! ( E ⊔ W ˜ E op ) t  . F or suﬃciently small λ > 0 w e thus get ˆ p ! ( E , ρ ) − ˆ p ! ( ˜ E , ˜ ρ ) = a  η  p λ ! ( E ⊔ W ˜ E op ) t  − Z W/B ˆ A c ( o ) ∧ ( ρ − ˜ ρ ) + ˜ Ω( λ, E ) − ˜ Ω( λ, ˜ E )) W e now go to the limit λ → 0 and use Theorem 3 .12 in o rder to get ˆ p ! ( E , ρ ) − ˆ p ! ( ˜ E , ˜ ρ ) = a Z W/B ˆ A c ( o ) ∧ η  ( E ⊔ W ˜ E op ) t  ! = − Z W/B ˆ A c ( o ) ∧ ( ρ − ˜ ρ ) = 0 W e let ˆ p ! : ˆ K ( W ) → ˆ K ( B ) denote the ma p induced by the construction (17 ). Though not indicated in the no ta - tion until now this map may dep end o n the choice of the re pr esen tative of the smo oth K -or ien tation o (later in Lemma 3 .17 we see that it only dep ends on the s mo oth K -or ien tation). 38 ULRICH BUNKE & THOMAS SCHICK 3.2.8 . — Let p : W → B b e a pro per submersion with closed ﬁbres with a smo oth K -or ien tation represented b y o . W e now hav e constructed a ho momorphism ˆ p ! : ˆ K ( W ) → ˆ K ( B ) . In the pre s en t para graph w e study the compatibilty of this construction with the curv ature map R : ˆ K → Ω d =0 . Deﬁnition 3.15 . — We deﬁne the int e gr ation of forms p o ! : Ω( W ) → Ω( B ) by p o ! ( ω ) = Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ ω Since ˆ A c ( o ) − dσ ( o ) is c lo sed we a lso hav e a factorization p o ! : Ω( W ) / im ( d ) → Ω( B ) / im ( d ) . L emma 3.16 . — F or x ∈ ˆ K ( W ) we have R ( ˆ p ! ( x )) = p o ! ( R ( x )) . Pr o of . — Let x = ( E , ρ ). W e insert the deﬁnitions, R ( x ) = Ω( E ) − dρ , and (16) in the mar k ed step. R ( ˆ p ! ( x )) = Ω( p λ ! E ) − d ( Z W/B ˆ A c ( o ) ∧ ρ + ˜ Ω( λ, E ) + Z W/B σ ( o ) ∧ R ( x )) ! = Ω( p λ ! E ) − Z W/B ˆ A c ( o ) ∧ dρ + Z W/B ˆ A c ( o ) ∧ Ω( E ) − Ω( p λ ! E ) − Z W/B dσ ( o ) ∧ R ( x ) = Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ R ( x ) = p o ! ( R ( x )) 3.2.9 . — O ur c onstructions of the homomor phisms ˆ p ! : ˆ K ( W ) → ˆ K ( B ) , p o ! : Ω( W ) → Ω( B ) in volve an explicit c hoice of a repres e ntative o = ( g T v p , T h p, ˜ ∇ , σ ) of the smo oth K -or ien tation lifting the g iv en topo logical K -o rien tation of p . In this paragr aph we show: L emma 3.17 . — The homomorphisms ˆ p ! : ˆ K ( W ) → ˆ K ( B ) and p o ! : Ω( W ) → Ω( B ) only dep end on the smo oth K -orientation r epr esente d by o . Pr o of . — Let o k := ( g T v p k , T h k p, ˜ ∇ k , σ k ), k = 0 , 1 b e tw o representativ es of a smo oth K -or ien tation. Then we hav e σ 1 − σ 0 = ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ). F o r the momen t w e indicate SMOOTH K-THEOR Y 39 b y a sup erscr ipt ˆ p k ! which r e pr esen tative of the smo oth K -o rien tation is used in the deﬁnition. Let ω ∈ Ω( W ). Then using (12) we g et p o 1 ! ( ω ) − p o 0 ! ( ω ) = Z W/B ( ˆ A c ( o 1 ) − ˆ A c ( o 0 ) − d ( σ 1 − σ 0 )) ∧ ω = Z W/B ( ˆ A c ( ˜ ∇ 1 ) − ˆ A c ( ˜ ∇ 0 ) − d ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 )) ∧ ω = 0 . W e now consider the pro jection p : [0 , 1] × W → [0 , 1] × B with the induced top ologica l K -or ien tation. It can b e r e ﬁned to a smo oth K -orientation o which restricts to o k at { k } × B . Let q : [0 , 1] × W → W b e the pro jection and x ∈ ˆ K ( W ). F urthermor e let i k : B → { k } × B → [0 , 1] × B b e the em b eddings. The following c hain of equalities follows from the homotopy formula Lemma 2.22 , the curv ature formula Lemma 3.16 , Stokes’ theor em and the deﬁnition of ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ), and ﬁnally fro m the fact that o 0 ∼ o 1 . ˆ p 1 ! ( x ) − ˆ p 0 ! ( x ) = i ∗ 1 ˆ p ! q ∗ ( x ) − i ∗ 0 ˆ p ! q ∗ ( x ) = a Z [0 , 1] × B /B R ( ˆ p ! q ∗ x ) ! = a Z [0 , 1] × B /B p o ! R ( q ∗ ( x )) ! = a Z [0 , 1] × B /B p o ! q ∗ ( R ( x )) ! = a Z [0 , 1] × B /B Z [0 , 1] × W / [0 , 1] × B ( ˆ A c ( o ) − dσ ( o )) ∧ q ∗ R ( x ) ! = a Z W/B [ Z [0 , 1] × W /W ( ˆ A c ( o ) − dσ ( o ))] ∧ R ( x ) ! = a Z W/B [ ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) − ( σ ( o 1 ) − σ ( o 0 ))] ∧ R ( x ) ! = 0 . 3.2.1 0. — L e t p : W → B be a pr oper submersion with closed ﬁbres with a top o logical K -or ien tation. W e c ho ose a smo oth K -o rient ation which reﬁnes the top olog ic a l K - orientation. In this case we say that p is smo othly K -oriented. Deﬁnition 3.18 . — We deﬁne the push-forwar d ˆ p ! : ˆ K ( W ) → ˆ K ( B ) to b e t he map induc e d by (17) for some choic e of a r epr esentative of the smo oth K -orientation 40 ULRICH BUNKE & THOMAS SCHICK W e als o hav e well-deﬁned maps p o ! : Ω( W ) → Ω( B ) , p o ! : Ω( W ) / im ( d ) → Ω( B ) / im ( d ) given b y integration of forms along the ﬁbres. Let us state the res ult ab out the compatibilit y of ˆ p ! with the structure maps of smo oth K -theory as follows. Pr op osition 3.19 . — The fol lowing diagr ams c ommute: K ( W ) c h dR − − − − → Ω( W ) / im ( d ) a − − − − → ˆ K ( W ) I − − − − → K ( W )   y p !   y p o !   y ˆ p !   y p ! K ( B ) c h dR − − − − → Ω( B ) / im ( d ) a − − − − → ˆ K ( B ) I − − − − → K ( B ) (18) ˆ K ( W ) R − − − − → Ω d =0 ( W )   y ˆ p !   y p o ! ˆ K ( B ) R − − − − → Ω d =0 ( B ) (19) Pr o of . — The maps b etw een the top ological K -groups are the usual push-forward maps deﬁned by the K -or ien tation of p . The other tw o are deﬁned a b ov e. The square (19) commutes by Lemma 3.16 . The r ight squar e of (18) comm utes be c a use we have the well-known fact from index theory index ( p ! ( E )) = p ! ( index ( E )) . Let ω ∈ Ω( W ) / im ( d ). Then we hav e ˆ p ! ( a ( ω )) = [ ∅ , Z W/B σ ( o ) ∧ dω − Z W/B ˆ A c ( o ) ∧ ω ] = [ ∅ , − Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ ω ] = a ( p ! ( ω )) . This shows that the middle square in (18) commutes. Finally , the commu tativity of the left sq uare in (18) is a consequence of the Cher n character version of the family index theor em c h dR ( p ! ( x )) = Z W/B ˆ A c ( T v p ) ∧ c h dR ( x ) , x ∈ K ( W ) . If f : B ′ → B is a smo oth map then w e co nsider the cartesian diagra m W ′ F − − − − → W   y p ′   y p B ′ f − − − − → B . W e equip p ′ with the induced smo oth K -or ien tation (see 3 .2.4). SMOOTH K-THEOR Y 41 L emma 3.20 . — The fol lowing diagr am c ommutes: ˆ K ( W ) F ∗ − − − − → ˆ K ( W ′ )   y p !   y p ′ ! ˆ K ( B ) f ∗ − − − − → ˆ K ( B ′ ) . Pr o of . — This follo ws from Lemma 3 .10. 3.3. F unctorialit y. — 3.3.1 . — W e now discuss the functoriality o f the push-for w ar d with resp ect to iter- ated ﬁbr e bundles. Let p : W → B b e as b efore to gether with a re pr esent ative of a smo oth K -orie ntation o p = ( g T v p , T h p, ˜ ∇ p , σ ( o p )). Let r : B → A be another prop er submersion with close d ﬁbres with a top ological K -o rient ation whic h is reﬁned b y a smo oth K -o rient atio n r e present ed by o r := ( g T v r , T h r , ˜ ∇ r , σ ( o r )). W e can consider the geometric family W := ( W → B , g T v p , T h p, S c ( T v p )) and apply the construction 3 .2 .2 in order to deﬁne the geometric family r λ ! ( W ) ov e r A . The underlying submersion o f the family is q := r ◦ p : W → A . Its vertical bundle has a metric g T v q λ and a hor iz o n tal distribution T h q . The topo logical S pin c -structures of T v p a nd T v r induce a top ological S pi n c -structure on T v q = T v p ⊕ p ∗ T v r . The family of Cliﬀord bundles o f p ! W is the spinor bundle asso ciated to this S pin c -structure. In order to understand how the connection ˜ ∇ λ q behaves as λ → 0 we c ho ose lo cal spin structures on T v p and T v r . Then we write S c ( T v p ) ∼ = S ( T v p ) ⊗ L p and S c ( T v r ) ∼ = S ( T v r ) ⊗ L r for one- dimensional twisting bundles with connection L p , L r . The tw o lo cal spin structures induce a lo ca l spin structure on T v q ∼ = T v p ⊕ p ∗ T v r . W e g et S c ( T v q ) ∼ = S ( T v q ) ⊗ L q with L q := L p ⊗ p ∗ L r . The co nnection ∇ λ,T v q q conv erges as λ → 0 . Mo reov er, the twist ing connection on L q do es not depend on λ a t all. Since ∇ λ,T v q q and ∇ L q determine ˜ ∇ λ q (see 3.1.5 ) we conclude that the connection ˜ ∇ λ q conv erges as λ → 0 . W e in tro duce the following notation for this adiaba tic limit: ˜ ∇ adia := l im λ → 0 ˜ ∇ λ q . 3.3.2 . — W e k eep the situation descr ibed in 3.3.1 . Deﬁnition 3.21 . — We deﬁne the c omp osite o λ q := o r ◦ λ o p of the re pr esentatives of smo oth K -orientations of p and r by o λ q := ( g T v q λ , T h q , ˜ ∇ λ q , σ ( o λ q )) , wher e σ ( o λ q ) := σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ λ q ) − dσ ( o p ) ∧ p ∗ σ ( o r ) . L emma 3.22 . — This c omp osition of r epr esentatives of smo oth ˆ K -orientations pr e- serves e quivalenc e and induc es a wel l-deﬁne d c omp osition of smo oth K -orientations which is indep endent of λ . 42 ULRICH BUNKE & THOMAS SCHICK Pr o of . — W e ﬁrst sho w that o λ q is indep endent of λ . In view of 3.1.9 for λ 0 < λ 1 we m ust show that σ ( o λ 1 q ) − σ ( o λ 0 q ) = ˜ ˆ A c ( ˜ ∇ λ 1 q , ˜ ∇ λ 0 q ). In fact, inserting the deﬁnitions and using (13) and (14) we hav e σ ( o λ 1 q ) − σ ( o λ 0 q ) = − ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ λ 1 q ) + ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ λ 0 q ) = ˜ ˆ A c ( ˜ ∇ λ 1 q , ˜ ∇ λ 0 q ) . Let us now take another representativ e o ′ p . The following equalities hold in the limit λ → 0. σ ( o q ) − σ ( o ′ q ) = ( σ ( o p ) − σ ( o ′ p )) ∧ p ∗ ˆ A c ( o r ) + ( ˆ A c ( o p ) − ˆ A c ( o ′ p )) ∧ p ∗ σ ( o r ) − d ( σ ( o p ) − σ ( o ′ p )) ∧ p ∗ σ ( o r ) = ˜ ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p ) ∧ p ∗ ˆ A c ( o r ) + ( ˆ A c ( ˜ ∇ p ) − ˆ A c ( ˜ ∇ ′ p ) − d ˜ ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p )) ∧ p ∗ σ ( o r ) = ˜ ˆ A c ( ˜ ∇ adia q , ˜ ∇ ′ adia q ) The la st equality uses (12) and that in the adiaba tic limit ˆ A c ( ˜ ∇ adia q ) = ˆ A c ( ˜ ∇ p ) ∧ p ∗ ˆ A c ( ∇ r ) , (20) which implies a cor r espo nding for m ula for the adiabatic limit o f transgressio ns, ˜ ˆ A c ( ˜ ∇ adia q , ˜ ∇ ′ adia q ) = ˜ ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p ) ∧ p ∗ ˆ A c ( ∇ r ) . Next we consider the eﬀect of changing the r epresent ative o r to the equiv a len t one o ′ r . W e compute in the adiabatic limit σ ( o q ) − σ ( o ′ q ) = σ ( o p ) ∧ ( p ∗ ˆ A c ( o r ) − p ∗ ˆ A c ( o ′ r )) + ( ˆ A c ( o p ) − dσ ( o p )) ∧ p ∗ ( σ ( o r ) − σ ( o ′ r )) = σ ( o p ) ∧ dp ∗ ˜ ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) + ( ˆ A c ( o p ) − dσ ( o p )) ∧ p ∗ ˜ ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) = ˆ A c ( o p ) ∧ p ∗ ˜ ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) = ˜ ˆ A c ( ˜ ∇ adia q , ˜ ∇ ′ adia q ) . In the last equality we have use d again (20) and the corresp onding equality ˜ ˆ A c ( ˜ ∇ adia q , ˜ ∇ ′ adia q ) = ˆ A c ( o p ) ∧ p ∗ ˜ ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) . 3.3.3 . — W e consider the co mposition of pro per K -oriented submersions W q 9 9 p / / B r / / A with r epresentativ es of smo oth K -orientations o p of p a nd o r of r . W e let o q := o r ◦ o p be the co mposition. These choices deﬁne push-forwards ˆ p ! , ˆ r ! and ˆ q ! in smo oth K - theory . The or em 3.23 . — We have the e quality of homomorphisms ˆ K ( W ) → ˆ K ( A ) ˆ q ! = ˆ r ! ◦ ˆ p ! . SMOOTH K-THEOR Y 43 Pr o of . — W e ca lculate the push-for w ar ds and the comp osition of the K -o rien tations using the parameter λ = 1 (though w e do not indicate this in the notation). W e take a class [ E , ρ ] ∈ ˆ K ( W ). The following equality holds since λ = 1: q ! E = r ! ( p ! E ) . So we m ust show that Z W/ A ˆ A c ( o q ) ∧ ρ + ˜ Ω( q , 1 , E ) + Z W/ A σ ( o q ) ∧ R ([ E , ρ ]) (21) ≡ Z B / A ˆ A c ( o r ) ∧ " Z W/B ˆ A c ( o p ) ∧ ρ + ˜ Ω( p, 1 , E ) + Z W/B σ ( o p ) ∧ R ([ E , ρ ]) # + ˜ Ω( r , 1 , p ! E ) + Z B / A σ ( o r ) ∧ R ( p ! [ E , ρ ]) . where ≡ means equality mo dulo i m ( d ) + ch dR ( K ( A )). The fo rm Ω ( q , 1 , E ) is given by (15). Since in the present para graph we consider these transgress io n for ms for v ar io us bundles we hav e included the pro jection q as an argument. By Pr o pos ition 3 .1 9 we hav e R ( ˆ p ! [ E , ρ ]) = Z W/B ( ˆ A c ( o p ) − dσ ( o p )) ∧ R ([ E , ρ ]) . Next we observe that ˜ Ω( q , 1 , E ) ≡ ˜ Ω( r , 1 , p ! E ) + Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z B / A ˆ A c ( o r ) ∧ ˜ Ω( p, 1 , E ) , (22) (where ≡ mea ns equality up to im ( d )). T o see this we consider the tw o- pa rameter family r λ ! ◦ p µ ! ( E ), λ, µ > 0, o f g eometric families. Ther e is a natur a l geometric fam- ily F ov er (0 , 1 ] 2 × A which res tricts to r λ ! ◦ p µ ! ( E ) on { ( λ, µ ) } × A (see 3.2.2 for the one-pa rameter ca s e). Note that the lo cal index form Ω( F ) extends by contin u- it y to [0 , 1] 2 × A . If P : [0 , 1] ֒ → [0 , 1] 2 is a path, then one can for m the in tegra l R P × A/ A Ω( F | P × A ), the trans gression of the lo cal index form of r λ ! ◦ p µ ! ( E ) a lo ng the path P . The following squar e indicates four paths in the ( λ, µ )-plane. The a rrows are lab eled by the ev aluations of Ω( F ) (which follow from the a diabatic limit formula 44 ULRICH BUNKE & THOMAS SCHICK 16), and their integrals, the co rresp onding transgr e s sion forms: (0 , 1) ˜ Ω( r , 1 , p ! E ) Ω( r λ ! ◦ p ! ( E )) / / (1 , 1) (0 , 0) R B/A ˆ A c ( o r ) ∧ Ω( p µ ! E ) R B/A ˆ A c ( o r ) ∧ ˜ Ω( p, 1 , E ) O O R W/A ˆ A c ( o r ◦ λ o p ) ∧ Ω( E ) R W/A ˜ ˆ A c ( ˜ ∇ q , ˜ ∇ adia ) ∧ Ω( E ) / / (1 , 0) ˜ Ω( q , 1 , E ) Ω( r ! ◦ p µ ! ( E )) O O . Note the equality r ! ◦ p µ ! ( E ) = q µ ! ( E ) whic h is relev ant for the rig h t vertical path. Also note that for the low er horizontal path that , as µ → 0, the ﬁbres of E are scaled to zero, wher e a s the ﬁbr es o f p are scaled by λ . The latter is exactly the eﬀect of the scaled comp osition o r ◦ λ o p of orientations deﬁned in 3.3 .1, explaining its appea rence in the ab ov e formula. The equa tion (22) follows since the transgressio n is additive under comp osition of paths, and since the transgr ession along a closed con tra ctible path g iv es an exact form. W e now insert Deﬁnition 3.21 of σ ( o q ) in order to get Z W/ A σ ( o q ) ∧ R ([ E , ρ ]) = Z W/ A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) − ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) i ∧ R ([ E , ρ ]) = Z W/ A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o r ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ dρ = Z W/ A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/ A  ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) − ˆ A c ( o q )  ∧ ρ (23) W e insert (23) a nd (22) in to the left-hand s ide of (2 1). SMOOTH K-THEOR Y 45 Z W/ A ˆ A c ( o q ) ∧ ρ + ˜ Ω( q , 1 , E ) + Z W/ A σ ( o q ) ∧ R ([ E , ρ ]) ≡ Z W/ A ˆ A c ( o q ) ∧ ρ + ˜ Ω( r , 1 , p ! E ) + Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z B / A ˆ A c ( o r ) ∧ ˜ Ω( p, 1 , E ) + Z W/ A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/ A ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/ A  ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) − ˆ A c ( o q )  ∧ ρ = ˜ Ω( r , 1 , p ! E ) + Z B / A ˆ A c ( o r ) ∧ ˜ Ω( p, 1 , E ) + Z W/ A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) + Z W/ A ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) ∧ ρ . An insp ection shows that this is ex actly the r ight -hand side of (21). 4. The cup pro duct 4.1. Deﬁnition of the pro duct. — 4.1.1 . — In this section w e deﬁne a nd study the cup pr oduct ∪ : ˆ K ( B ) ⊗ ˆ K ( B ) → ˆ K ( B ) . It turns smo oth K -theory into a functor on manifolds with v a lues in Z / 2 Z -gr aded rings and into a m ultiplicative extensio n of the pair ( K, c h R ) in the sense o f Deﬁnition 1.2. 4.1.2 . — Let E a nd F b e geometric fa milies over B . The formula for the pro duct in volves the pr oduct E × B F of geometr ic fa milies ov er B . The detailed description of the pro duct is easy to guess, but let us employ the following trick in order to give an alterna tiv e deﬁnition. Let p : F → B b e the prop er submersion with closed ﬁbres underlying F . Let us fo r the moment assume that the vertical metric, the horizontal distribution, and the orientation of p are co mplemen ted by a top ological S pin c -structure together with a S pin c -connection ˜ ∇ as in 3.2 .1. The Dirac bundle V of F has the form V ∼ = W ⊗ S c ( T v p ) for a twisti ng bundle W with a hermitean metric and unitary connection (and Z / 2 Z -grading in the even ca s e), which is uniquely determined up to isomor phism. Let p ∗ E ⊗ W denote the geometric family which is o btained from p ∗ E by twisting its Dirac bundle with δ ∗ W , where δ : E × B F → F denotes the under lying prop er 46 ULRICH BUNKE & THOMAS SCHICK submersion with closed ﬁbres of p ∗ E . Then we have E × B F ∼ = p ! ( p ∗ E ⊗ W ) . This description may help to unders tand the meaning of the adiabatic deformation which blows up F , which in this no ta tion is giv en by p λ ! ( p ∗ E ⊗ W ). In the description of the pr oduct of geometric families w e could interc hang e the roles of E and F . If the vertical bundle of E does not hav e a globa l S pin c -structure, then it ha s at least a lo cal one. In this case the description ab ov e aga in gives a complete description of the loca l geometry of E × B F . 4.1.3 . — W e now pro ceed to the deﬁnition of the pro duct in terms o f cycles. In order to write down the formula we a ssume that the cycles ( E , ρ ) and ( F , θ ) a r e homogeneous o f degree e and f , re s pectively . Deﬁnition 4.1 . — We deﬁne ( E , ρ ) ∪ ( F , θ ) := [ E × B F , ( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ θ ] . Pr op osition 4.2 . — The pr o duct is wel l-deﬁne d. It turns B 7→ ˆ K ( B ) into a functor fr om smo oth manifolds to unital gr ade d-c ommutative rings. Pr o of . — W e ﬁrst show that this pro duct is bilinear and compatible with the equiv a- lence rela tion ∼ (2.10). The pro duct is o b viously biadditiv e and natura l with r espec t to pull-backs along maps B ′ → B . W e now show that the pro duct preserves the equiv alence relation in the ﬁrs t argument. Assume that E admits a taming E t . Then we have ( E , ρ ) ∼ ( ∅ , ρ − η ( E t )). Using the latter representativ e we get ( ∅ , ρ − η ( E t )) ∪ ( F , θ ) = [ ∅ , ( ρ − η ( E t )) ∧ Ω( F ) − ( − 1) e dρ ∧ θ + ( − 1) e dη ( E t ) ∧ θ ] = [ ∅ , ρ ∧ Ω( F ) + ( − 1) e Ω( E ) ∧ θ − ( − 1) e dρ ∧ θ − η ( E t ) ∧ Ω( F )] . On the other hand, similar to in 3.2.6, the taming E t induces a genera lized taming ( E × B F ) t . Using L e mma 3.13 and argueing a s in the pro of o f Lemma 3.14 we get [ E × B F , ( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ σ ] = [ ∅ , ( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ σ − η (( E × B F ) t )] . It suﬃces to show that η ( E t ) ∧ Ω( F ) − η (( E × B F ) t ) ∈ im ( c h dR ) . (24) W e will actually show that this diﬀerence is exact. W e ﬁrst co nsider the a dia batic limit in which we blow up the metric of F . W e g et from Theor em 3.12 lim adia η (( E × B F ) t ) = η ( E t ) ∧ Ω( F ) . (25) In o rder to see this we use that E × B F ∼ = p ! ( p ∗ E ⊗ W ) (see 4.1.2 ), where p : F → B and W → F is the twisting bundle of this family . The taming E t induces a taming SMOOTH K-THEOR Y 47 p ∗ E t , a nd hence a taming ( p ∗ E ⊗ W ) t . It follows fro m standard prop erties of the induced sup erconnection o n a tensor pro duct bundle (alternatively one can use the sp e c ia l cas e of Theorem 3 .12 where the second ﬁbration has ze r o-dimensional ﬁbres) that η ( p ∗ E ⊗ W ) t = p ∗ η ( E t ) ∧ c h ( ∇ W ). F rom Theorem 3.12 w e get ( ˜ ∇ is asso ciated to p ) lim adia η (( E × B F ) t ) = lim λ → 0 η ( p λ ! ( p ∗ E ⊗ W ) t ) = η ( E t ) ∧ Z F /B ˆ A c ( ˜ ∇ ) ∧ ch ( ∇ W ) ! = η ( E t ) ∧ Ω( F ) As in 3.2.2 we now let G t be the ta med family over (0 , ∞ ) × B with underlying pro jection r : (0 , ∞ ) × E × B F → (0 , ∞ ) × B which restricts to p λ ! ( p ∗ E ⊗ W ) t on { λ } × B . Then we hav e dη ( G t ) = Ω( G ). Using the formulas for ∇ T v r given in [ BGV04 , Pr op. 10.2] w e observe that i ∂ H λ R ∇ T v r = 0 , where ∂ H λ is a horizontal lift of ∂ λ . This implies that i ∂ λ dη ( G t ) = i ∂ λ Ω( G ) = 0. W e get η ( p λ ! ( p ∗ E ⊗ W ) t ) − η ( p 1 ! ( p ∗ E ⊗ W ) t ) = d Z [ λ, 1] × B /B η ( G t ) . The ex a ctness o f the diﬀerence (24) now follows by taking the limit λ → 0 a nd the fact that the rang e of d is closed since l im λ → 0 η ( p λ ! ( p ∗ E ⊗ W ) t ) = η ( E t ) ∧ Ω( F ) by (25) and η ( p 1 ! ( p ∗ E ⊗ W ) t ) = η (( E × B F ) t ) by construction. In order to avoid re p eating this a rgument for the second ar gumen t we show that the pro duct is g raded commutativ e. Note that E × B F ∼ = F × B E except if b oth families are o dd, in whic h ca se E × B F ∼ = ( F × B E ) op [ E , ρ ] ∪ [ F , θ ] = [ E × B F , ( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ θ ] = [( − 1) ef F × B E , ( − 1) e + e ( f − 1) θ ∧ Ω( E ) + ( − 1) f ( e − 1) Ω( F ) ∧ ρ − ρ ∧ dθ ] = [( − 1) ef F × B E , ( − 1) ef θ ∧ Ω( E ) + ( − 1 ) ef ( − 1) f Ω( F ) ∧ ρ − ( − 1) ef ( − 1) f dθ ∧ ρ ] = ( − 1) ef [ F , θ ] ∪ [ E , ρ ] . 4.1.4 . — W e no w ha ve a well-deﬁned Z / 2 Z -g raded commutativ e pro duct ∪ : ˆ K ( B ) ⊗ ˆ K ( B ) → ˆ K ( B ) . W e show next that it is a sso ciative. Fir s t of all o bserve that the ﬁbr e pro duct of geometric families is asso ciative. Le t e, f , g b e the parities o f the homogeneo us cla sses 48 ULRICH BUNKE & THOMAS SCHICK [ E , ρ ], [ F , θ ], a nd [ G , κ ]. ([ E , ρ ] ∪ [ F , θ ]) ∪ [ G , κ ] = [ E × B F , ( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1 ) e dρ ∧ θ ] ∪ [ G , κ ] = [ E × B F × B G , (( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ θ ) ∧ Ω( G ) +( − 1) e + f Ω( E × B F ) ∧ κ − ( − 1) e + f d (( − 1) e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − 1) e dρ ∧ θ ) ∧ κ ] = [ E × B F × B G , ( − 1) e Ω( E ) ∧ θ ∧ Ω( G ) + ρ ∧ Ω( F ) ∧ Ω( G ) − ( − 1) e dρ ∧ θ ∧ Ω( G ) + ( − 1 ) e + f Ω( E ) ∧ Ω( F ) ∧ κ − ( − 1) e + f Ω( E ) ∧ dθ ∧ κ − ( − 1) e + f dρ ∧ Ω( F ) ∧ κ + ( − 1) e + f dρ ∧ dθ ∧ κ ] On the other hand [ E , ρ ] × ([ F , θ ] × [ G , κ ]) = [ E , ρ ] × [ F × B G , ( − 1) f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − 1) f dθ ∧ κ ] = [ E × B ∧F × B G , ( − 1) e Ω( E ) ∧ (( − 1) f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − 1 ) f dθ ∧ κ ) + ρ ∧ Ω( F × B G ) − ( − 1) e dρ ∧ (( − 1) f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − 1 ) f dθ ∧ κ )] = [ E × B F × B G , ( − 1) e + f Ω( E ) ∧ Ω( F ) ∧ κ + ( − 1) e Ω( E ) ∧ θ ∧ Ω( G ) − ( − 1) e + f Ω( E ) ∧ dθ ∧ κ + ρ ∧ Ω( F ) ∧ Ω( G ) − ( − 1) e + f dρ ∧ Ω( F ) ∧ κ − ( − 1) e dρ ∧ θ ∧ Ω( G ) + ( − 1) e + f dρ ∧ dθ ∧ κ ] By an inspection we see that the tw o rig h t-hand s ides agree. 4.1.5 . — Let us observe that the unit 1 ∈ ˆ K ( B ) is simply given by ( B × C , 0), i.e. the trivial 0-dimensio nal family with ﬁbr e the graded vector space C concent ra ted in even degree, and with curv atur e form 1. The deﬁnition shows that this is actually a unit on the level of cycles. This ﬁnishes the pro of of Pro pos ition 4.2 . 4.1.6 . — In this par agraph we study the compatibilit y of the cup pro duct in smo oth K -theor y with the cup pro duct in top ologica l K-theory and the w edge pro duct of diﬀerent ial forms. L emma 4.3 . — F or x, y ∈ ˆ K ( B ) we have R ( x ∪ y ) = R ( x ) ∧ R ( y ) , I ( x ∪ y ) = I ( x ) ∪ I ( y ) . F u rthermor e, for α ∈ Ω( B ) / im ( d ) we have a ( α ) ∪ x = a ( α ∧ R ( x )) . Pr o of . — Straight forward ca lcula tion using the deﬁnitions. Cor ol lary 4.4 . — With the ∪ -pr o duct smo oth K -the ory ˆ K is a multiplic ative exten- sion of the p air ( K , c h R ) . 4.2. Pro jection formula. — SMOOTH K-THEOR Y 49 4.2.1 . — Let p : W → B b e a pro per submersion with closed ﬁbres with a smo oth K -or ien tation r epresented by o . In this ca se we hav e a well-deﬁned push-forward ˆ p ! : ˆ K ( W ) → ˆ K ( B ). The explicit formula in ter ms of cycles is (17). The pro jection formula s tates the co mpatibilit y of the push- forward with the ∪ -pro duct. Pr op osition 4.5 . — L et x ∈ ˆ K ( W ) and y ∈ ˆ K ( B ) . The n ˆ p ! ( p ∗ y ∪ x ) = y ∪ ˆ p ! ( x ) . Pr o of . — Let x = [ F , σ ] a nd y = [ E , ρ ]. By a n insp ection of the constructions we observe that the pr o jection for m ula ho lds true on the level of geometric families p ! ( p ∗ E × W F ) ∼ = E × B p ! F . This implies Ω( p λ ! ( p ∗ E × W F )) = Ω( E ) ∧ Ω( p λ ! ( F )) . Consequently we hav e ˜ Ω( λ, p ∗ E × W F ) = ( − 1) e Ω( E ) ∧ ˜ Ω( λ, F ). Inse r ting the deﬁni- tions of the pro duct a nd the push-forward we get up to exact forms ˆ p ! ( p ∗ y ∪ x ) = ˆ p ! ([ p ∗ E × W F , ( − 1) e p ∗ Ω( E ) ∧ σ + p ∗ ρ ∧ Ω( F ) − ( − 1) e p ∗ dρ ∧ σ ]) = [ p ! ( p ∗ E × W F ) , Z W/B ˆ A c ( o ) ∧ [( − 1) e p ∗ Ω( E ) ∧ σ + p ∗ ρ ∧ Ω( F ) − ( − 1) e p ∗ dρ ∧ σ ] + Z W/B σ ( o ) ∧ R ( p ∗ y ∪ x ) + ˜ Ω(1 , p ∗ E × W F )] = [ E × B p ! F , ρ ∧ Z W/B ˆ A c ( o ) ∧ Ω( F ) + ( − 1) e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ +( − 1) e Ω( E ) ∧ ˜ Ω(1 , F ) − ρ ∧ Z W/B ˆ A c ( o ) ∧ dσ + ( − 1) e R ( y ) ∧ Z W/B σ ( o ) ∧ R ( x )] . (26) 50 ULRICH BUNKE & THOMAS SCHICK Up to exact for ms we have ρ ∧ Z W/B ˆ A c ( o ) ∧ Ω( F ) + ( − 1 ) e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ +( − 1) e Ω( E ) ∧ ˜ Ω(1 , F ) − ρ ∧ Z W/B ˆ A c ( o ) ∧ dσ + ( − 1) e R ( y ) ∧ Z W/B σ ( o ) ∧ R ( x ) = ( − 1) e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜ Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ Z W/B ˆ A c ( o ) ∧ (Ω( F ) − dσ )) − ( − 1 ) e dρ ∧ Z W/B σ ( o ) ∧ R ( x ) = ( − 1) e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜ Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ R ( x ) = ( − 1) e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜ Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ R ( ˆ p ! x ) . Thu s the for m component of (26) is exactly the one needed for the pr oduct y ∪ p ! ( x ). 4.3. Susp ension. — 4.3.1 . — W e consider the pro jection pr 2 : S 1 × B → B . The goa l of this subsection is to verify the relation ( ˆ pr 2 ) ! ◦ pr ∗ 2 = 0 which is an imp ortant ing redien t in the uniqueness res ult Theorem 1.4. 4.3.2 . — The pro jection pr 2 ﬁts into the ca rtesian diagram S 1 × B pr 1 / / pr 2   S 1 p   B r / / ∗ . W e choose the metric g T S 1 of unit volume and the b ounding spin s tructure on T S 1 . This s pin s tructure induces a S pi n c structure on T S 1 together with the connection ˜ ∇ . In this way we get a repr esen tative o of a smo oth K -orientation of p . By pull-back we get the representativ e r ∗ o of a smo oth K -o r ien tation of pr 2 which is used to deﬁne ( ˆ pr 2 ) ! . SMOOTH K-THEOR Y 51 4.3.3 . — Using the pro jection formul a P rop osition 4.5 we get for x ∈ ˆ K ( B ) ( ˆ pr 2 ) ! ( pr ∗ 2 ( x )) = ( ˆ pr 2 ) ! ( pr ∗ 2 ( x ) ∪ 1) = x ∪ ( ˆ pr 2 ) ! 1 . Using the c o mpatibilit y of the push-forward with cartesian diagrams Lemma 3.20 we get ( ˆ pr 2 ) ! 1 = ( ˆ pr 2 ) ! ( pr ∗ 1 (1)) = r ∗ ˆ p ! (1) . W e let S 1 denote the geo metric family over ∗ given by p : S 1 → ∗ with the g eometry describ ed ab ove. Since S 1 has the b ounding S p i n -structure the Dirac o p era to r is in vertible and has a symmetric sp ectrum. The family S 1 therefore has a canonical taming S 1 t b y the zero s moo thing op erator, and we have η ( S 1 t ) = 0. This implies ˆ p ! (1) = [ S 1 , 0] = [ ∅ , η ( S 1 t )] = [ ∅ , 0] = 0 . Cor ol lary 4.6 . — We have ( ˆ pr 2 ) ! ◦ pr ∗ 2 = 0 . 5. Constructions o f natural smo oth K -theory classe s 5.1. Calculations. — 5.1.1 . — L emma 5.1 . — We have ˆ K ∗ ( ∗ ) ∼ =  Z ∗ = 0 R / Z ∗ = 1 . Pr o of . — W e use the e x act sequence g iv en by Pr opo sition 2 .20. The ass ertion follows from the obvious identities ˆ K 0 ( ∗ ) ∼ = K 0 ( ∗ ) ∼ = Z , ˆ K 1 ( ∗ ) ∼ = Ω ev ( ∗ ) / c h dR ( K 0 ( ∗ )) ∼ = R / Z . 5.1.2 . — L emma 5.2 . — Ther e ar e exact se quenc es 0 → R / Z → ˆ K 0 ( S 1 ) → Z → 0 0 → C ∞ ( S 1 ) / Z → ˆ K 1 ( S 1 ) → Z → 0 . Pr o of . — These asser tions a gain follow from Pro p osition 2.20 and the identiﬁcations K 0 ( S 1 ) ∼ = Z , K 1 ( S 1 ) ∼ = Z , Ω ev ( S 1 ) / c h dR ( K 0 ( S 1 )) ∼ = C ∞ ( S 1 ) / Z . 52 ULRICH BUNKE & THOMAS SCHICK 5.1.3 . — Let V := ( V , h V , ∇ V , z ) b e a geometr ic Z / 2 Z -graded bundle ov er S 1 such that dim( V + ) = dim ( V − ). Let V denote the corresp onding geometric family . By Lemma 5.2 the class [ V , 0] ∈ ˆ K 0 ( S 1 ) satisﬁes I ([ V , 0]) = 0 and hence corres p onds to an element of R / Z . This element is calcula ted in the following lemma. Let φ ± ∈ U ( n ) /conj deno te the holonomies o f V ± (w ell deﬁned mo dulo conjugation in the group U ( n )). L emma 5.3 . — We have [ V , 0] = a  1 2 π i log det ( φ + ) det ( φ − )  . Pr o of . — W e consider the map q : S 1 → ∗ with the canonical K -o rien tation 4.3.2. By Pr o pos ition 3 .1 9 we hav e a commutativ e dia gram R / Z ∼ − − − − → Ω 1 ( S 1 ) / ( im ( d ) + im ( c h dR )) a − − − − → ˆ K 1 ( S 1 )   y =   y q o !   y ˆ q ! R / Z ∼ − − − − → Ω 0 ( ∗ ) / im ( c h dR ) a − − − − → ˆ K 0 ( ∗ ) . In order to deter mine [ V , 0] it therefor e suﬃces to calculate ˆ q ! ([ V , 0]). Now observe that q : S 1 → ∗ is the bo unda r y of p : D 2 → ∗ . Since the underlying topolog ical K -or ien tation of q is g iven by the b ounding S pin -structure w e can choose a smo oth K -or ien tation of p with pro duct s tructure which r e stricts to the smo oth K -orientation of q . The bundle V is top ologica lly trivial. Therefor e we can ﬁnd a g eometric bundle W = ( W , h W , ∇ W , z ), a gain with pro duct str uctur e , on D 2 which restricts to V o n the boundar y . Let W denote the corr e s ponding geometric family ov er D 2 . Later we prov e the bordism formula P rop osition 5.18. It gives ˆ q ! ([ V , 0]) = [ ∅ , p ! R ([ W , 0])] = − a Z D 2 / ∗ Ω 2 ( W ) ! . Note that Ω 2 ( W ) = ch 2 ( ∇ W ) = c h 2 ( ∇ det ( W + ) ) − c h 2 ( ∇ det ( W − ) ) = − 1 2 π i  R ∇ det W + − R det ∇ W −  . The holonomy det ( φ ± ) ∈ U (1) of det ( V ± ) is equal to the integral of the curv ature of det W ± : log det ( φ ± ) = Z D 2 R ∇ det ( W ± ) . It follows that ˆ q ! ([ V , 0]) = a  1 2 π i log det ( φ + ) det ( φ − )  . 5.2. The s mo oth K -theory class of a mapping torus. — SMOOTH K-THEOR Y 53 5.2.1 . — Le t E b e a g eometric family ov er a po int and consider an automorphism φ of E . Then we can for m the mapping torus T ( E , φ ) := ( R × E ) / Z , where n ∈ Z acts on R by x 7→ x + n , and by φ n on E . The pro duct R × E is a Z -equiv a riant ge o metric family ov er R (the pull-back of E b y the pro jection R → ∗ ). The geometr ic structures descend to the quo tient and turn the mapping torus T ( E , φ ) into a geo metr ic family ov er S 1 = R / Z . In the present subsection we study the class [ T ( E , φ ) , 0] ∈ ˆ K ( S 1 ) . In the following we will a ssume that the parity of E is even, and that index ( E ) = 0. 5.2.2 . — Let dim : K 0 ( S 1 ) → Z b e the dimension homomor phism, whic h in this case is an isomo rphism. Since dim I ([ T ( E , φ ) , 0]) = dim( inde x ( E )) = 0 we have in fact [ T ( E , φ ) , 0 ] ∈ R / Z ⊂ ˆ K 0 ( S 1 ), where we consider R / Z as a subgroup o f ˆ K 0 ( S 1 ) according to Lemma 5 .2. Let V := ker( D ( E )). This gra ded vector space is pre s erved by the action of φ . W e use the same symbol in order to denote the induced action o n V . W e form the zer o-dimensional family V := ( R × V ) / Z ov er S 1 . This bundle is isomorphic to the k ernel bundle of T ( E , φ ). The bundle of Hilbert s paces of the family T ( E , φ ) ⊔ S 1 V op has a ca no nical subbundle of the for m V ⊕ V op . W e choo se the taming ( T ( E , φ ) ⊔ S 1 V op ) t which is induced by the iso morphism  0 1 1 0  on this subbundle. Note tha t [ T ( E , φ ) , 0] = [ V , η (( T ( E , φ ) ⊔ S 1 V op ) t )]. Since the pull- back of ( T ( E , φ ) ⊔ S 1 V op ) t under R → R / Z is isomorphic to a tamed family pulled back under R → ∗ we see that the o ne- form η (( T ( E , φ ) ⊔ S 1 V op ) t ) = 0. 5.2.3 . — Th us it r emains to ev aluate [ T ( E , φ ) , 0] = [ V , 0 ] ∈ R / Z . By Lemma 5.3 this n umber can b e expre s sed in terms of the holono my of the determinant bundle det ( V ). Let φ ± ∈ A ut ( V ± ) b e the induced transformations. Pr op osition 5.4 . — We have [ T ( E , φ ) , 0] = [ 1 2 π i log( det φ + det φ − )] R / Z . In p articular, if D ( E ) is invertible, then [ T ( E , φ ) , 0] = 0 . 5.3. The smo oth K -theory class of a geom etric family with k ernel bundl e. — 5.3.1 . — Let E b e an even-dimensional g eometric family ov e r the bas e B . By ( D b ) b ∈ B we deno te the asso ciated family of Dirac op erators on the family of Hilber t spaces ( H b ) b ∈ B . The geometry o f E induces a connection ∇ H on this family (the connec- tion part of the Bismut sup erconnection [ BGV04 , Prop. 10.1 5]). W e assume that dim(k er( D b )) is co nstan t. In this cas e we can form a vector bundle K := ker( D ). The pro jection o f ∇ H to K gives a connection ∇ K . Hence we get a geometric bundle K := ( K, h K , ∇ K ) and an a sso ciated g eometric family K (see 2.1.4). 54 ULRICH BUNKE & THOMAS SCHICK 5.3.2 . — The sum E ⊔ B K op has a natural taming ( E ⊔ B K op ) t which is given by  0 u u ∗ 0  ∈ End ( H b ⊕ K op b ) , where u : K b → H b is the embedding. W e th us hav e the following equa lit y in ˆ K ( B ): [ E , 0] = [ K , η (( E ⊔ B K op ) t )] . 5.3.3 . — Under the sta nding assumption that dim(ker( D b )) is co nstan t w e also hav e the η -form of Bismut-Cheeger η B C ( E ) ∈ Ω( B ) (see [ BC91 ], [ BC90b ], [ BC90a ]) . Since other authors us e η B C ( E ), in the following tw o paragr aphs we s ha ll analyse the relation b etw een this and η (( E ⊔ B K op ) t ). W e form the geo metric family [0 , 1] × ( E ⊔ B K op ) ov er B . The taming ( E ⊔ B K op ) t induces a b oundary ta ming at { 0 } × ( E ⊔ B K op ). In index theo r y the b oundary taming is used to constr uct a p erturbation of the Dirac o per a tor which is inv ertible at −∞ of ( −∞ , 1] × ( E ⊔ B K op ) (see [ Bun ] for details). On the other side { 1 } × ( E ⊔ B K op ) we consider APS-b oundar y co nditions. W e thus get a family o f p erurb e d Dirac op erators on ( −∞ , 1] × ( E ⊔ B K op ). The L 2 -b oundary condition at {−∞} × ( E ⊔ B K op ) and the APS-b oundary condition a t { 1 } × ( E ⊔ B K op ) together imply the F redholm pro perty (whic h c a n b e chec ked lo ca lly for the v ario us b oundary co mponents or ends). In this wa y the family of Dirac op erators o n [0 , 1] × ( E ⊔ B K op ) gives rise to a family of F redholm oper a tors. W e will denote this s tructure by ([0 , 1 ] × ( E ⊔ B K op )) bt,AP S . The Cher n ch ar acter of its index i ndex (([0 , 1] × ( E ⊔ B K op )) bt,AP S ) ∈ K ( B ) can be calculated using the metho ds of lo ca l index theory . 5.3.4 . — Using 2.4 .10 w e can choo se a p ossibly diﬀerent taming ( E ⊔ B K op ) t ′ such that the co rresp onding index index (([0 , 1 ] × ( E ⊔ B K op )) bt ′ ,AP S ) ∈ K ( B ) v anishes. In this case we can extend the b oundary taming to a taming index (([0 , 1 ] × ( E ⊔ B K op )) t ′ ,AP S ). W e se t up the metho d of local index theory as us ual b y forming the family of rescaled Bis m ut s uperco nnections A s := A s (([0 , 1] × ( E ⊔ B K op )) t ′ ,AP S ) whic h take the tamings and b oundar y tamings in to account as explained in [ Bun , 2.2.4.3], see also 3.2 .6. In vertibilit y of D (([0 , 1] × ( E ⊔ B K op )) t ′ ,AP S ) ensures exp onential v anishing of the in tegra l kernel o f e − A 2 s for s → ∞ . The usual transgre s sion integral expresses the lo ca l index form Ω([0 , 1] × ( E ⊔ B K op )) as a sum of contributions of the b oundary comp onen ts o r ends (see [ Bun , pro of of Lemma 2 .2.15 ]). These co n tributions can b e calculated sepa r ately for each part. Because o f the pro duct structure we have Ω([0 , 1 ] × ( E ⊔ B K op )) = 0. The con- tribution o f the b oundary { 1 } × ( E ⊔ B K op ) is given b y the pro of o f the APS-index theorem of [ BC91 ], [ BC90b ], [ BC90a ], a nd it is eq ual to η B C ( E ⊔ B K op ) = η B C ( E ). The second equalit y holds true, since the Dirac o pera tor for K op is tr ivial. The con- tribution of the b oundary { 0 } × ( E ⊔ B K op ) is calculated in the pro of of [ Bun , Lemma 2.2.15] and equal to − η (( E ⊔ B K op ) t ′ ). Ther efore we hav e η B C ( E ) = η (( E ⊔ B K op ) t ′ ) (note that we calculate mo dulo exact for ms). W e now use 2.4.10 and a rela tiv e index SMOOTH K-THEOR Y 55 theorem (compar e (28)) in order to see that η (( E ⊔ B K op ) t ′ ) − η (( E ⊔ B K op ) t ) = c h dR ( index (([0 , 1 ] × ( E ⊔ B K op )) bt,AP S )) ∈ c h dR ( K ( B )) . Using Pro pos ition 2 .2 0 we get: Cor ol lary 5.5 . — We have [ E , 0] = [ K , η B C ( E )] . 5.3.5 . — Let p : W → B b e a pro per submersion with closed ﬁbres with a smo oth K -or ien tation r e pr esen ted by o . Let V b e a geometr ic vector bundle over W , and let V denote the a sso ciated geo metric family . Then we can form the geometric family E := p ! V (see Deﬁnition 3.7). Assume that the kernel of the family of Dirac op era to rs ( D ( E b )) b ∈ B has cons tan t dimension, forming thus the kernel bundle K . Since V has zero-dimensiona l ﬁbres we have ˜ Ω(1 , V ) = 0. F ro m (17) we get ˆ p ! [ V , ρ ] = [ p ! V , Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )] = [ E , Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )] = [ K , η B C ( E ) + Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )] . 5.4. A canonical ˆ K 1 -class on S 1 . — 5.4.1 . — W e construct in a natura l w ay an element x S 1 ∈ ˆ K 1 ( S 1 ) coming from the Poincar ´ e bundle ov er S 1 × S 1 . Let us identify S 1 ∼ = R / Z . W e co nsider the complex line bundle L := ( R × R / Z × C ) / Z ov er R / Z × R / Z , wher e the Z -action is g iv en b y n ( s, t, z ) = ( s + n, t, exp( − 2 π int ) z ). On R × R / Z × C → R × R / Z w e hav e the Z -equiv aria n t connectio n ∇ := d + 2 π i sdt with curv a ture R ∇ = 2 π ids ∧ dt . This connection descends to a co nnection ∇ L on L . The unitary line bundle with con- nection L := ( L, h L , ∇ L ) gives a geometric family L over R / Z × R / Z . It represents v := [ L , 0] ∈ ˆ K 0 ( R / Z × R / Z ). Note that R ( v ) = 1 + ds ∧ dt . W e now consider the pro jection p : R / Z × R / Z → R / Z on the second factor . This ﬁbre bundle has a natural smo oth ˆ K -or ien tation ( g T v p , T h p, ˜ ∇ , 0). The vertical metric and the horizon- tal distribution come from the metric o f S 1 and the pro duct structure. More over, T v p is trivialized by the S 1 -action. Hence it has a preferred orientation. W e take the b ounding S pin -structure on the ﬁbres which induces the S pin c -structure and the connection ˜ ∇ . Deﬁnition 5.6 . — We deﬁne x S 1 := ˆ p ! v ∈ ˆ K 1 ( S 1 ) . 5.4.2 . — W e hav e R ( x S 1 ) = dt . Let t ∈ S 1 . Then we compute t ∗ x S 1 ∈ ˆ K 1 ( ∗ ) ∼ = R / Z (iden tiﬁcation again as in Lemma 5 .2 ). No te that 0 ∗ x S 1 is r epresented by the trivia l line bundle over S 1 . Since we cho ose the b ounding s pin structure, the corresp onding Dirac op erator is inv ertible. Its sp ectrum is sy mmetric and its η -inv ariant v a nishes 56 ULRICH BUNKE & THOMAS SCHICK (compare 4.3.3). Therefore we hav e 0 ∗ x S 1 = 0. It now follows by the homo top y formula (o r by a n explicit co mputation of η -inv ariants), that t ∗ x S 1 = − t . (27) 5.4.3 . — Le t f : B → S 1 be given. Then w e deﬁne Deﬁnition 5.7 . — < f > := f ∗ x S 1 ∈ ˆ K 1 ( B ) . Assume now that we hav e tw o s uc h maps f , g : B → S 1 . As an interesting illustra - tion we characterize < f > ∪ < g > ∈ ˆ K 0 ( B ) . It suﬃces to consider the universal ex ample B = T 2 = S 1 × S 1 . W e consider the pro jections pr i : S 1 × S 1 → S 1 , i = 1 , 2. Let x := ˆ pr ∗ 1 x S 1 and y := ˆ pr ∗ 2 x S 1 . Then we m ust compute x ∪ y ∈ ˆ K 0 ( T 2 ). W e iden tify T 2 = R / Z × R / Z with co ordina tes s, t . First note that R ( x ∪ y ) = R ( x ) ∪ R ( y ) = ds ∧ dt . Thus the class x ∪ y − v + 1 is ﬂat, i.e. x ∪ y − v + 1 ∈ K 0 f lat ( T 2 ) . In fact, since K 0 ( T 2 ) is torsion- fr e e , we hav e K 0 f lat ( T 2 ) ∼ = H odd ( T 2 ) / im ( c h dR ) = R 2 / Z 2 . In order to determine this elemen t w e m ust compute its holonomies a long the cir cles S 1 × 0 a nd 0 × S 1 . The ho lonomy of v along these circles is trivial. Since 0 ∗ x = 0 and 0 ∗ y = 0 we see that x × y also has tr iv ial holonomies a long these cir cles. Therefore we conclude Pr op osition 5.8 . — x ∪ y = v − 1 W e can now solve our original problem. The tw o maps f , g induce a map f × g : B → T 2 . Cor ol lary 5.9 . — We have < f > ∪ < g > = ( f × g ) ∗ v − 1 . 5.5. The pro duct o f S 1 -v alued maps and li ne -bundles. — 5.5.1 . — Let f : B → S 1 be a smo oth map and L := ( L, ∇ L , h L ) b e a hermitean line bundle with connection over B . It giv es rise to a geometric family L (se e 2.1.4 ). W e consider the smo oth K -theory classes < f > and < L > := [ L , 0 ] − 1. It is again in teres ting to determine the class < f > ∪ < L > ∈ ˆ K 1 ( B ) . An explicit answer is only known in sp ecial cases. First we compute the curv a tur e : R ( < f > ∪ < L > ) = R ( < f > ) ∧ R ( < L > ) = d f ∧ ( e c 1 ( ∇ L ) − 1 ) , where d f := f ∗ dt a nd c 1 ( ∇ L ) := − 1 2 π i R ∇ L . SMOOTH K-THEOR Y 57 5.5.2 . — Note that the degree-one co mponent of the o dd form R ( < f > ∪ < L > ) v anishes. Let now q : Σ → B be a smo oth map from an or ien ted closed surface. Then R ( q ∗ ( < f > ∪ < L > )) = q ∗ R (( < f > ∪ < L > )) = 0 . Therefore q ∗ ( < f > ∪ < L > ) ∈ ˆ K 1 f lat (Σ) ∼ = H ev (Σ , R ) / im ( c h ) ∼ = R / Z ⊕ R / Z , where the ﬁrst comp onent corresp onds to H 0 (Σ , R ) a nd the second to H 2 (Σ , R ). In order to ev aluate the ﬁrst comp onent we restrict to a p oint. Since the restriction o f < L > to a p oin t v anishes, the ﬁrst co mp onent of q ∗ ( < f > ∪ < L > ) v a nishes. Therefore it remains to determine the second comp onent. 5.5.3 . — Let us assume that q ∗ L is triv ial. W e choose a trivialization. Then we can deﬁne the transg ression Chern form ˜ c 1 ( ∇ q ∗ L , ∇ tr iv ) ∈ Ω 1 (Σ) such that d ˜ c 1 ( ∇ q ∗ L , ∇ tr iv ) = q ∗ c 1 ( ∇ L ). By the homotopy for mula we hav e q ∗ < L > = [ ∅ , − ˜ c 1 ( ∇ q ∗ L , ∇ tr iv )] . In this sp ecial case w e can compute q ∗ ( < f > ∪ < L > ) = q ∗ < f > ∪ q ∗ < L > = < q ∗ f > ∪ q ∗ < L > = [ ∅ , q ∗ d f ∧ ˜ c 1 ( ∇ q ∗ L , ∇ tr iv )] . W e see that the second compo nen t is  Z Σ q ∗ d f ∧ ˜ c 1 ( ∇ q ∗ L , ∇ tr iv )  R / Z . W e do not know a go od answer in the gener al case where q ∗ L is non-trivial. 5.6. A bi -in v arian t ˆ K 1 - class on S U (2) . — 5.6.1 . — Le t G b e a group acting on the manifold M . Deﬁnition 5.10 . — A class x ∈ ˆ K ( M ) is c al le d invaria nt , if g ∗ x = x for al l x ∈ G . 5.6.2 . — F or e x ample, the cla ss x S 1 ∈ ˆ K 1 ( S 1 ) deﬁned in 5.6 is no t inv a riant under the action L t , t ∈ S 1 , of S 1 on itself. Note that R ( x S 1 ) = dt is inv ar ian t. Ther efore L ∗ t x S 1 − x S 1 ∈ R / Z . In fact by (27) we hav e L ∗ t x S 1 − x S 1 = − t . Since dt is the only inv ariant form with integral one we see tha t the o nly way to pro duce an in v ariant smo oth reﬁnement of the genera tor of H 1 ( S 1 , Z ) ∼ = Z would b e to p erturb x S 1 b y a class b ∈ H 0 ( S 1 , R / Z ). But b is of cour se ho mo top y inv a riant, hence L ∗ t b = b . W e conclude that the genera to r of H 1 ( S 1 , Z ) (and also every non- trivial multiple) do es no t admit any inv a riant lift. 58 ULRICH BUNKE & THOMAS SCHICK 5.6.3 . — The situation is diﬀerent for simply-connected groups. Let us consider the following exa mple. The group G := S U (2) × S U (2) acts on S U ( 2) by ( g 1 , g 2 ) h := g 1 hg − 1 2 . Let vol S U (2) ∈ Ω 3 ( S U (2)) denote the normalized volume form. F urthermore we let i : ∗ → S U (2) denote the em b edding o f the identit y . Pr op osition 5.11 . — F or k ∈ Z ther e exists a unique class x S U (2) ( k ) ∈ ˆ K 1 ( S U (2)) such that R ( x S U (2) ) = k vo l S U (2) and i ∗ x = 0 . This element is S U (2) × S U (2) - invariant Pr o of . — Assume, that x, y ∈ ˆ K 1 ( S U (2)) sa tisfy R ( x ) = R ( y ). Then we have x − y ∈ ˆ K 1 f lat ( S U (2)) ∼ = K 1 f lat ( S 3 ) ∼ = R / Z . Since i ∗ x = i ∗ y = 0 we hav e in fac t that x = y . Therefore, if the class x S U (2) ( k ) exists, then it is unique. W e show the existence of an in v ariant cla ss in an abstra ct manner. No te that k vol S U (2) represents a cla ss c h ( Y ) for some Y ∈ K 1 ( S 3 ). In terms o f classifying maps, Y for k = 1 is given by the embedding S U (2) → U (2) → U ( ∞ ) ∼ = K 1 . W e hav e the exa ct sequence 0 → Ω ev ( S U (2)) / im ( c h dR ) a → ˆ K 1 ( S U (2)) I → K 1 ( S U (2)) → 0 . Therefore we ca n cho ose a ny class y ∈ ˆ K 1 ( S U (2)) s uch that I ( y ) = Y . Then the contin uo us group co cycle G ∋ t → c ( t ) = t ∗ y − y ∈ Ω ev ( S U (2)) / im ( c h dR ) represents an element [ c ] ∈ H 1 c ( G, Ω ev ( S U (2)) / im ( c h dR )). W e claim that this cohomolog y group is trivial. Note that Ω ev ( S U (2)) / im ( c h dR ) ∼ = Ω 0 ( S U (2)) / Z ⊕ Ω 2 ( S U (2)) / im ( d ) . Since Ω 2 ( S U (2)) / im ( d ) is a rea l top olog ical vector space with a co n tinuous action of the compact gro up G we immediately conclude that H 1 c ( G, Ω 2 ( S U (2)) / im ( d ) ) = 0 by the usual av era ging argument. W e co nsider the exact sequence of G -s pa ces 0 → Z → Ω 0 ( S U (2)) → Ω 0 ( S U (2)) / Z → 0 . Since G is simply-connected we s e e that taking c o n tinuous functions from G × · · · × G with v alues in these spaces, we obtain again exact se q uences of Z -mo dules. It follows that w e have a long exac t seq uence in co n tinuous co homology . The relev ant part reads H 1 c ( G, Z ) → H 1 c ( G, Ω 0 ( S U (2))) → H 1 c ( G, Ω 0 ( S U (2)) / Z ) → H 2 c ( G, Z ) . Since Z is discrete and G is connected we see that H i c ( G, Z ) = 0 for i ≥ 1. Therefore, H 1 c ( G, Ω 0 ( S U (2))) ∼ = H 1 c ( G, Ω 0 ( S U (2)) / Z ) . But Ω 0 ( S U (2)) is ag ain a co ntin uous representation of G on a real vector space so that H 1 c ( G, Ω 0 ( S U (2))) = 0. The claim follows. W e now can choo se w ∈ Ω ev ( S U (2)) / im ( c h dR ) such that t ∗ w − w = t ∗ y − y for all t ∈ G . W e ca n further assume that i ∗ w = i ∗ y by adding a co nstan t. Then we set x S U (2) ( k ) = y − w ∈ ˆ K 1 ( S U (2)). This elemen t ha s the r equired prop erties. SMOOTH K-THEOR Y 59 It is an interesting pr oblem to write down an in v ariant cycle which represents the class x S U (2) . 5.6.4 . — Note that x S U (2) ( k ) = k x S U (2) (1). Let Σ ⊂ S U (2) b e a n embedded ori- ent ed hyper surface. Then R ( x S U (2) (1)) | Σ = 0 so that ( x S U (2) ) | Σ ∈ ˆ K 1 f lat (Σ). Since x S U (2) (1) ev aluates trivially on points we have in fact ( x S U (2) (1)) | Σ ∈ ker  ˆ K 1 f lat (Σ) → ˆ K 1 f lat ( ∗ )  ∼ = R / Z . This num b er can b e determined by integration ov er Σ. F or mally , let p : Σ → {∗} be the pro jection. If w e cho o se some smo oth K -orientation, then w e can ask for ˆ p ! ( x S U (2) (1)) | Σ ∈ ˆ K 1 f lat ( ∗ ) ∼ = R / Z . The hypersurface Σ decomp oses S U (2) in tw o parts S U (2) ± Σ . Let S U (2) + Σ be the part such that ∂ S U ( 2) + Σ has the orientation given b y Σ. W e choose a K -or ien tation o of the pro jection q : S U (2 ) + Σ → ∗ whic h has a pro duct structure such that σ ( o ) = 0 and ˆ A c ( o ) = 1. In order to get the la tter equality we choo s e a S pin c -structure coming from a spin structure. The smo oth K - orientation of q induces a smo oth K -or ien tation of p . Then q : S U (2) + Σ → ∗ provides a zero -bo rdism o f Σ, a nd of ( x S U (2) (1)) | Σ . Therefore, we hav e by Prop osition 5.18 ˆ p ! ( x S U (2) (1)) | Σ = " ∅ , Z S U (2) + Σ R ( x S U (2) (1)) # = − [ vol ( S U (2) + Σ )] R / Z , where [ λ ] R / Z denotes the cla ss o f λ ∈ R . Note that the iden tiﬁcation ˆ K 1 f lat ( ∗ ) ∼ = R / Z is induced b y a : R ∼ = Ω odd ( ∗ ) / im ( d ) → K 1 f lat ( ∗ ) given by λ 7→ [ ∅ , − λ ]. This explains the minus sign in the seco nd equality a bove. 5.7. In v arian t classes on homogeneous spaces. — 5.7.1 . — Some of the ar g umen ts from the S U (2)- c a se g eneralize. Let G b e a co mpa ct connected and simply-connected Lie group and G/H be a homogenous space. Given Y ∈ K ( G/H ) we can ﬁnd a lift y ∈ ˆ K ( G/H ). W e form the co cycle G ∋ g 7→ c ( g ) := g ∗ y − y ∈ Ω( G/H ) / im ( c h dR ). Since Ω( G/H ) / i m ( c h dR ) is the quotient of a vector space by a lattice and G is co nnected and simply-co nnected we can use the a rgu- men ts as in the S U (2)-case in or der to conclude that H 1 c ( G, Ω( G/H ) / im ( c h dR )) = 0. Therefore we ca n cho ose the lift y such that g ∗ y = y for all g ∈ G . In particular , R ( y ) ∈ Ω( G/H ) is now an inv ariant fo r m represe nting ch ( Y ). Note that an inv ar ian t form is in genera l not determined b y this condition. 5.7.2 . — If we sp ecialize to the case that G/H is symmetric, then inv a riant forms exactly repr e sen t the co homology . In this case we see that t wo choices of inv ar ia n t lifts y 0 , y 1 of Y hav e the same curv ature so that y 1 − y 0 ∈ ˆ K f lat ( G/H ). Since the y i also hav e the sa me index, we indeed hav e y 1 − y 0 ∈ H ( G/H, R ) / im ( c h dR ). W e have th us shown the following lemma. 60 ULRICH BUNKE & THOMAS SCHICK L emma 5.12 . — Assume t hat G/H is a symmetric sp ac e with G c onne ct e d and simply c onne cte d. Then every Y ∈ K ( G/H ) has an invariant lift y ∈ ˆ K ( G/H ) which is uniquely determine d up to H ( G/H , R ) / i m ( c h dR ) . 5.7.3 . — W e ca n a pply this in cer tain cases. First we write S 2 n +1 ∼ = S pin (2 n + 2) /S pin (2 n + 1), n ≥ 1. Note that K 1 ( S 2 n +1 ) ∼ = Z . Since H ev ( S 2 n +1 , R ) / im ( c h dR ) = R / Z is concentrated in degree zer o w e hav e the following result. Cor ol lary 5.13 . — L et n ≥ 1 . F or e ach k ∈ Z ther e is a unique x S 2 n +1 ( k ) ∈ ˆ K 1 ( S 2 n +1 ) which is invariant, has index k ∈ Z ∼ = K 1 ( S 2 n +1 ) , and evaluates trivial ly on p oints. 5.7.4 . — In the even-dimensional case we write S 2 n ∼ = S pin (2 n + 1) /S pin (2 n ), n ≥ 1. Note that K 0 ( S 2 n ) ∼ = Z ⊕ Z and H odd ( S 2 n , R ) / im ( c h dR ) = 0. Cor ol lary 5.14 . — F or e ach k ∈ Z ther e is a unique x S 2 n ( k ) ∈ ˆ K 0 ( S 2 n ) which is invariant and has index k ∈ Z ∼ = ˜ K 0 ( S 2 n ) , and evaluates trivial ly on p oints 5.7.5 . — W e write CP n := S U ( n + 1) /S ( U (1) × U ( n )). Then H odd ( CP n , R ) / im ( c h dR ) = 0. T her efore we conclude: L emma 5.15 . — F or e ach Y ∈ K 0 ( CP n ) ther e is a unique S U ( n + 1) -invariant class y CP n ( Y ) ∈ ˆ K 0 ( CP n ) such that I ( y CP n ( Y )) = Y . 5.7.6 . — Let G b e a connected and simply-connected Lie g roup. Let T ⊂ G be a maximal torus. Then w e have a G -map P : G/T × T → G , P ([ g ] , t ) := g tg − 1 , where G ac ts on the left-hand side b y g ([ h ] , t ) := ([ g h ] , t ), and by co njugation on the right - hand s ide. Le t x ∈ ˆ K ∗ ( G ) b e an inv aria n t element. It is an interesting question how P ∗ x lo oks lik e. Let us consider the s pecial case G = S U (2) and x S U (2) = x S U (2) (1) ∈ ˆ K 1 ( S U (2)). In this ca se we hav e T = S 1 and G/T ∼ = CP 1 . First we compute the curv ature of P ∗ x S U (2) . F or this we must compute P ∗ vol S U (2) which is given by W eyl’s integration formula. W e hav e P ∗ vol S U (2) = v ol CP 1 ∧ 4 sin 2 (2 π t ) dt . There is a unique clas s z ∈ ˆ K 1 ( S 1 ) with cur v ature 4 sin 2 (2 π t ) dt such that 0 ∗ z = 0. F urthermor e , there is a unique class < L > ∈ ˆ K 0 ( CP 1 ) with curv ature vol CP 1 which is in fact the c la ss < L > considered in 5.5.1 asso ciated to the cano nical line bundle L on CP 1 . The pro duct < L > ∪ z has now the same curv ature as P ∗ x S U (2) . W e co nclude that P ∗ x S U (2) − < L > ∪ z ∈ H ev ( CP 1 × S 1 , R ) / im ( c h dR ) . SMOOTH K-THEOR Y 61 Now note that H ev ( CP 1 × S 1 , R ) / im ( c h dR ) ∼ =  H 0 ( CP 1 , R ) ⊗ H 0 ( S 1 , R ) ⊕ H 2 ( CP 1 , R ) ⊗ H 0 ( S 1 , R )  / im ( c h dR ) ∼ = R / Z ⊕ R / Z . The ﬁrst co mponent can b e determined by ev a luating the diﬀerence P ∗ x S U (2) − < L > ∪ z a t a p oint . Since x S U (2) is trivial on p oints, this ﬁr s t comp onent v anishes. The second comp onent ca n be determined by ev alua ting P ∗ x S U (2) − < L > ∪ z at CP 1 × { 0 } . Note that P ∗ CP 1 ×{ 0 } x S U (2) = 0 , since P | CP 1 ×{ 0 } is constant. F urthermore, 0 ∗ z = 0 implies tha t < L > ∪ z | CP 1 ×{ 0 } = 0 . Thus w e hav e shown (using S 2 ∼ = CP 1 ): L emma 5.16 . — P ∗ x S U (2) = x S 2 (1) ∪ z 5.8. Bordism. — 5.8.1 . — A zero b o rdism of a geometric family E over B is a g eometric family W ov er B with b oundary suc h that E = ∂ W . The notion of a geometric family with bo undary is explained in [ Bun ]. It is impo r tan t to note that in our set-up a geometric family with b o undary alwa ys has a pro duct structure. Pr op osition 5.17 . — If E admits a zer o b or dism W , then in ˆ K ∗ ( B ) we have the identity [ E , 0] = [ ∅ , Ω( W )] . Pr o of . — Since E a dmits a zero b ordism we hav e in dex ( E ) = 0 so that E admits a taming E t . This taming induces a b oundar y taming W bt . The obstruction against extending the b oundary taming to a taming of W is index ( W bt ) ∈ K ( B ) [ Bun , Lemma 2.2.6 ]. Let us assume for s implicit y that E is no t zero -dimensional. Otherwis e we may hav e to stabilize in the following a ssertion. Using 2.4.10 w e can a djust the taming E t such that ind ex ( W bt ) = 0 . At this p oin t we employ a version o f the relative index theorem [ Bun95 ] index ( W bt ′ ) = inde x ( W bt ) + i ndex (( E × [0 , 1 ]) bt ) , (28) where E t and E t ′ deﬁne the bo undary taming ( E × [0 , 1]) bt . If i ndex ( W bt ) = 0, then we can extend the b oundary taming W bt to a taming W t . W e now apply the identit y [ Bun , Thm. 2 .2.13]: Ω( W ) = dη ( W t ) − η ( E t ) . Note that this equality is mo re precis e than needed s ince it holds on the level of forms without fac to ring by im ( d ). W e see that ( E , 0) is paired with ( ∅ , Ω( W )). This implies the as sertion. 62 ULRICH BUNKE & THOMAS SCHICK 5.8.2 . — Let p : W → B be a pro per submersion from a ma nifold with b oundar y W which restricts to a submer sion q := p | ∂ W : V := ∂ W → B . W e a ssume that p has a top ologica l K -o rient atio n and a smo oth K -o rient ation represented b y o p which reﬁnes the top ological K - o rient atio n. W e assume that the geometric data of o p has a pro duct s tructure near V (see [ Bun , Section 2.1] for a deta iled discussion of such pro duct structures). Recall o p = ( g T v p , T h p, ˜ ∇ p , σ p ). By the assumption of a pro duct structure we ha ve a quadruple ( g T v q , T h q , ˜ ∇ q , σ q ) and an isomorphism of a neighbourho o d of p | ∂ W : ∂ W → B with the bundle E × [0 , 1) pr E → E p → B such that the geometric data are rela ted as follows. 1. T v p |E × [0 , 1) ∼ = pr ∗ E T v q ⊕ pr ∗ [0 , 1) T [0 , 1 ) and g T v p |E × [0 , 1) = pr ∗ E g T v q + pr ∗ [0 , 1) dr 2 , where r ∈ [0 , 1) is the co or dinate. 2. T h p |E × [0 , 1) = pr ∗ E T h q . 3. ( σ p ) |E × [0 , 1) = p r ∗ E σ q . 4. The S pin c -structure on T v q and the ca nonical S p in c -structure on T [0 , 1 ) induce a S pin c -structure on the vertical bundle T v ∼ = pr E T v E ⊕ pr ∗ [0 , 1) T [0 , 1 ) of E × [0 , 1 ) in a canonical way s o that the asso ciated spinor bundle is S ( T v ) = pr ∗ E S c ( T v q ) or pr ∗ E S c ( T v q ) ⊗ C 2 depending on the dimension of T v q . In particular, the connection ˜ ∇ q gives rise to a connection ˜ ∇ pro d . The pr oduct structure identiﬁes the restricted S pin c -structure of T v p |E × [0 , 1) with this pr oduct S pin c -structure such that ˜ ∇ |E × [0 , 1) beco mes ˜ ∇ pro d . F rom this description w e deduce that ˆ A c ( ˜ ∇ ) |E × [0 , 1) = pr ∗ E ˆ A c ( ˜ ∇ q ) , ˆ A c ( o p ) |E × [0 , 1) = pr ∗ E ˆ A c ( o q ) . It is now easy to s e e that the restriction of representativ es (with pro duct structure) preserves equiv alence a nd gives a well-deﬁned restriction of smo oth K -orientations. W e hav e the following version of b ordism inv ariance of the push-forward in smo oth K -theor y . Pr op osition 5.18 . — F or y ∈ ˆ K ( W ) we set x := y | V ∈ ˆ K ( V ) . Then we have ˆ q ! ( x ) = [ ∅ , p o ! R ( y )] . SMOOTH K-THEOR Y 63 Pr o of . — Let y = [ E , ρ ]. W e compute using (17), Pr opo sition 5.17, Stokes’ theorem, Deﬁnition 3.1 5 , a nd the adiabatic limit λ → 0 at the ma r k ed equalit y ˆ q ! ( x ) = [ q λ ! E | V , Z V /B ˆ A c ( o q ) ∧ ρ + ˜ Ω( λ, E | V ) + Z V /B σ ( o q ) ∧ R ( x )] = [ ∅ , Ω( p λ ! E ) + Z V /B ˆ A c ( o q ) ∧ ρ + ˜ Ω( λ, E | V ) + Z V /B σ ( o q ) ∧ R ( x )] ! = [ ∅ , Z W/B  ˆ A c ( o p ) ∧ Ω( E ) − ˆ A c ( o p ) ∧ dρ − dσ ( o p ) ∧ R ( y )  ] = [ ∅ , Z W/B ( ˆ A c ( o p ) − dσ ( o p )) ∧ R ( y )] = [ ∅ , p o ! R ( y )] 5.9. Z /k Z -in v arian ts. — 5.9.1 . — Here we ass ocia te to a family of Z / k Z -ma nifolds ov er B a class in ˆ K f lat ( B ). Deﬁnition 5.19 . — A ge ometric family of Z /k Z -manifolds is a triple ( W , E , φ ) , wher e W is a ge ometric famil y with b oun dary, E is a ge ometric family without b ound- ary, and φ : ∂ W ∼ → k E is an isomorp hism of the b oundary of W with k c opies of E . W e deﬁne u ( W , E , φ ) := [ E , − 1 k Ω( W )] ∈ ˆ K ( B ). L emma 5.20 . — We have u ( W , E , φ ) ∈ ˆ K f lat ( B ) . This class is a k -torsion class. It only dep ends on the underlying diﬀer ential-top olo gic al data. Pr o of . — W e ﬁrst compute b y 5.17 k u ( W , E , φ ) = k [ E , − 1 k Ω( W )] = [ k E , − Ω( W )] = [ ∅ , 0] = 0 This implies that R ( u ( W , E , φ )) = 0 so that u ( W , E , φ ) ∈ ˆ K f lat ( B ). Indep endence of the geo metric data is now shown by a homotopy arg umen t. 5.9.2 . — W e now explain the rela tion of this construction to the Z /k Z -index of F re e d- Melrose [ FM92 ]. L emma 5.21 . — L et B = ∗ and dim( W ) b e even. Then u ( W , E , φ ) ∈ ˆ K 1 f lat ( ∗ ) ∼ = R / Z . L et i k : Z /k Z → R / Z t he emb e dding which sends 1 + k Z t o 1 k . Then i k ( index a ( ¯ W )) = u ( W , E , φ ) , 64 ULRICH BUNKE & THOMAS SCHICK wher e i k ( index a ( ¯ W )) ∈ Z /k Z is the index of the Z /k Z -manifold ¯ W (the notation of [ FM92 ] ). Pr o of . — W e r ecall the deﬁnition of ind ex a ( ¯ W ). In our languag e is can be stated as follows. Since i ndex ( E ) = 0 we can c ho ose a taming E t . W e let k copies o f E t induce the b oundary taming W bt . W e hav e index a ( ¯ W ) = i ndex ( W bt ) + k Z . In fact it is eas y to see that a c hang e of the taming E t leads to change of the index index ( W bt ) by a multiple of k . W e ca n now prov e the Lemma using [ Bun , Thm. 2.2.18]. u ( W , E , φ ) = [ E , − 1 k Ω( W )] = [ ∅ , − η ( E t ) − 1 k Ω( W )] = [ ∅ , − 1 k index ( W bt )] = a  1 k index ( W bt )  = i k ( index a ( ¯ W )) ∈ R / Z . 5.10. S p in c -b ordism in v arian ts. — 5.10. 1. — Let π b e a ﬁnite group. W e construct a tr a nsformation φ : Ω S pin c ( B U ( n ) × B π ) → ˆ K f lat ( ∗ ) . Let f : M → B U ( n ) × B π repres e nt [ M , f ] ∈ Ω S pin c ( B U ( n ) × B π ). This ma p deter- mines a covering p : ˜ M → M and an n -dimensiona l complex v ector bundle V → M . W e choose a Riemannian metric g T M and a S p in c -extension ˜ ∇ of the Levi-Civ ita connection ∇ T M . These structures determine a smo oth K - orientation of t : M → ∗ . W e further ﬁx a metric h V and a connection ∇ V in or der to deﬁne a g eometric bundle V := ( V , h V , ∇ V ) and the ass o cia ted geometric family V (see 2 .1.4). The pull-back of g T M and ˜ ∇ via ˜ M → M ﬁxes a smoo th K - orientation o f ˜ t : ˜ M → ∗ . W e deﬁne the geometric families M := t ! V and ˜ M := ˜ t ! ( p ∗ V ) o ver ∗ . Then w e set φ ([ M , f ]) := [ ˜ M ⊔ ∗ | π |M op , 0] ∈ ˆ K f lat ( ∗ ) . By a homotopy arg umen t we see that this class is indep enden t of the choice of geo m- etry . W e now a r gue that it only depends on the bo rdism class of [ M , f ]. The co nstruction is additiv e. Let now [ M , f ] b e zero-b ordant by [ W, F ]. Then we hav e a zer o b ordism ˜ W o f ˜ M ov er W . Note that the bundles also ex tend ov er the bo rdism. The lo cal index form o f ˜ W ⊔ B | π |W v anishes. W e conclude by 5.17, that [ ˜ M ⊔ B | π | · M op , 0] = 0 . SMOOTH K-THEOR Y 65 In this construction we can replac e E π → B π by any ﬁnite covering. 5.10. 2. — This co nstruction allows the following mo diﬁcation. Let ρ ∈ Rep ( π ) 0 be a virtual z e r o-dimensional representation of π . It deﬁnes a ﬂat vector bundle F ρ → B π . T o [ M , f ] we asso ciate the geo metr ic family M ρ := t ! ( L ), where L is the geometric family a sso ciated to the ge o metric bundle V ⊗ ( pr 2 ◦ f ) ∗ F ρ . W e deﬁne φ ρ : Ω S pin c ∗ ( B U ( n ) × B π ) → ˆ K f lat ( ∗ ) such that φ ρ [ M , f ] := [ M ρ , 0]. Here we need not to as s ume that π is ﬁnite. This is the cons truction of ρ -inv ariants in the s moo th K -theory picture. The ﬁrst c o nstruction is a s pecial case of the second with the representation ρ = C ( π ) ⊕ ( C | π | ) op . 5.10. 3. — W e now discuss a pa rametrized version. Let B b e some compact manifold and X b e so me to polo g ical spa ce. Then we can deﬁne the parametrize d b ordism g roup Ω S pin c ∗ ( X/B ). Its cycles are pairs ( p : W → B , f : W → X ) of a prop e r top ologica lly K -or ien ted submer s ion p and a co n tinuous map f . The b ordism relation is deﬁned corresp ondingly . There is a natural transformation φ : Ω S pin c ∗ (( B U ( n ) × B π ) /B ) → ˆ K ∗ f lat ( B ) . It asso ciates to x = ( p : W → B , f : W → B U ( n ) × B π ) the c la ss [ ˜ W ⊔ B | π | · W op , 0]. In this formula p : ˜ W → W is again the π -cov ering classiﬁed by pr 2 ◦ f . W e deﬁne the geometric family W using some choice of geometric s tr uctures a nd the twisti ng bundle V , where V is class iﬁed by the ﬁrs t comp onen t o f f . The family ˜ W is obtained fro m ˜ W a nd p ∗ V using the lifted geometric str uctures. Again, the class φ ( x ) is ﬂat and independent of the choices of geometr y . Using 5.17 one chec ks that φ pa sses through the b ordism relation. Again there is the following mo diﬁcation. F or ρ ∈ Rep ( π ) 0 we can deﬁne φ ρ : Ω S pin c ∗ (( B U ( n ) × B π ) /B ) → ˆ K ∗ f lat ( B ) . It a sso ciates to x = ( p : W → B , f : W → B U ( n ) × B π ) the clas s [ W ρ ] of the geomet- ric manifold W with twistin g bundle V ⊗ ( pr 2 ◦ f ) ∗ F ρ . These classes a re K -theoretic higher ρ -inv a riants. It seems promising to use this picture to draw geometr ic co nse- quences using these inv ariants. 5.11. The e -in v ariant. — 5.11. 1. — A framed n -manifold M is a manifold with a tr iv ialization T M ∼ = M × R n . More genera l, a bundle of framed n -manifolds ov er B is a ﬁbre bundle π : E → B with a trivialization T v π ∼ = E × R n . Pr op osition 5.22 . — A bund le of fr ame d n -m anifolds π : E → B has a c anonic al smo oth K -orientation which only dep ends on the homotopy class of t he fr aming. 66 ULRICH BUNKE & THOMAS SCHICK Pr o of . — The framing T v π ∼ = E × R n induces a vertical Riemannian metric g T v π and an isomorphism S O ( T v π ) ∼ = E × S O ( n ). Hence we g et an induced v ertica l orientation and a S pin -str ucture which determines a S pin c -structure, and thus a K -or ien tation of π . W e ch o ose a horizontal distribution T h π which gives r ise to a connection ∇ T v π . Since o ur S pi n c -structure comes from a S p in -structure, this connection extends naturally to a S pin c -connection ˜ ∇ o f trivial c e ntral curv a ture. The trivial connection ∇ tr iv on T v π induced by the framing a lso lifts naturally to the trivia l S pin c -connection ˜ ∇ tr iv . The qua druple o := ( g T v π , T h π , ˜ ∇ , ˜ ˆ A c ( ˜ ∇ , ˜ ∇ tr iv )) deﬁnes a smo oth K -orientation of π which reﬁnes the given underlying top ological K -or ien tation. W e cla im that this orientation is indep endent of the choice o f the v ertical dis - tribution T h π . Indeed, if T h π is a s econd horizontal distr ibution with asso ciated S pin c -connection ˜ ∇ ′ , then we set o ′ := ( g T v π , T h π ′ , ˜ ∇ ′ , ˆ A c ( ˜ ∇ ′ , ˜ ∇ tr iv )) . Since ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ tr iv ) − ˜ ˆ A c ( ˜ ∇ , ˜ ∇ tr iv ) = ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ) we have o ∼ o ′ in view of the Deﬁnition 3.1 .9. Let us now consider a second fra ming o f T v π which is ho motopic to the ﬁrst. In induces a s econd trivial c o nnection ˜ ∇ ′ triv and a metric g ′ T v π . W e therefo r e get a co nnection ˜ ∇ ′ and and a second representativ e of a smo oth K -or ientation o ′ := ( g ′ T v π , T h π , ˜ ∇ ′ , ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv )). In fact, the homotopy b etw ee n the framings provides a connection ˜ ∇ h,tr iv on I × E . Since this connection is ﬂat we see that ˜ ˆ A c ( ˜ ∇ ′ triv , ˜ ∇ tr iv ) = 0. F rom ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv ) = ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ) + ˜ ˆ A c ( ˜ ∇ , ˜ ∇ tr iv ) + ˜ ˆ A c ( ˜ ∇ tr iv , ˜ ∇ ′ triv ) we get ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv ) − ˜ ˆ A c ( ˜ ∇ , ˜ ∇ tr iv ) = ˜ ˆ A c ( ˜ ∇ ′ , ˜ ∇ ) and thus o ∼ o ′ . Since ˜ ∇ tr iv is ﬂat we hav e ˆ A c ( o ) − dσ ( o ) = ˆ A ( ˜ ∇ ) − d ˜ ˆ A ( ˜ ∇ , ˜ ∇ tr iv ) = 1 . Assume that the ﬁbre dimension n satisﬁes n ≥ 1. According to Lemma 3.16 the curv ature of ˆ π ! (1) is given by R ( ˆ π ! (1)) = Z E /B ( ˆ A c ( o ) − dσ ( o )) ∧ 1 = Z E /B 1 ∧ 1 = 0 SMOOTH K-THEOR Y 67 Deﬁnition 5.23 . — If π : E → B is a bund le of fr ame d manifolds of ﬁbr e dimension n ≥ 1 , then we deﬁne a diﬀer ential top olo gic al invariant e ( E → B ) := − ˆ π ! (1) ∈ ˆ K − n f lat ( B ) . In the follo wing we will explain in some detail that this is a higher genera lization of the Adams e -inv ariant. The stable homo top y groups of the spher e π n := π s n ( S 0 ) hav e a decrea sing ﬁltration · · · ⊆ π 2 n ⊆ π 1 n ⊆ π 0 n = π n related to the MSpin-based Adams Noviko v sp ectral sequence. The e -inv a riant is a homomorphism e : π 1 4 n − 1 /π 2 4 n − 1 → R / Z . A closed framed 4 n − 1-dimensiona l manifold M represents a clas s [ M ] ∈ π 4 n − 1 under the Pon trjagin-Thom identiﬁcation of framed b ordism with stable homotopy . In the indicated dimension π 4 n − 1 = π 1 4 n − 1 so that [ M ] is a ctually a b oundary of a compact 4 n -dimensional S pin -ma nifold N . As explained in [ APS75 ] (see also [ Lau99 ]) the e -inv a r iant e [ M ] can b e calculated as follows. One chooses a connection ∇ T N on T N which restr icts to the trivial connection ∇ tr iv on T M given by the framing. T hen e ([ M ]) =  Z N ˆ A ( ∇ )  R / Z . W e no w co ns ider q : M → ∗ as a bundle o f framed manifolds ov er the p oint and iden tify R / Z ∼ → ˆ K − 4 n +1 f lat ( ∗ ) by [ u ] 7→ a ( u ) = [ ∅ , − u ], u ∈ R . L emma 5.24 . — Under these identiﬁc ations we have e ( M → ∗ ) = e ([ M ]) . Pr o of . — W e choose a metric g T M on M which induces the representativ e o := ( g T M , 0 , ˜ ∇ , ˜ ˆ A c ( ˜ ∇ , ∇ tr iv )) of the smo oth K -orientation on q . The S pi n -structure of N induces a S pin c - structure. W e choose a Riemannia n metric g T N on N with a pro duct struc- ture near the boundar y which extends g T M and induces the S pi n - and S pin c - connections ∇ N and ˜ ∇ N . Note that ˜ ˆ A c ( ˜ ∇ N , ˜ ∇ T N ) extends ˜ ˆ A c ( ˜ ∇ , ˜ ∇ tr iv ). Therefore o N := ( g T N , 0 , ˜ ∇ N , ˜ ˆ A c ( ˜ ∇ N , ˜ ∇ T N )) repres en ts a smo oth K -orientation of p : N → ∗ which ex tends the orientation o o f q : M → ∗ . W e ca n now apply the b ordism formula 68 ULRICH BUNKE & THOMAS SCHICK Prop osition 5.18 in the marked step and get e ( M → ∗ ) = − ˆ q ! (1) ! = a ( p ! ( R (1))) = " Z N/ ∗ ( ˆ A c ( o N ) − dσ ( o N )) ∧ 1 # R / Z = " Z N/ ∗ ˆ A c ( ˜ ∇ N ) − d ˜ ˆ A ( ˜ ∇ N , ˜ ∇ T N ) # R / Z = " Z N/ ∗ ˆ A c ( ˜ ∇ T N ) # R / Z = " Z N/ ∗ ˆ A ( ∇ T N ) # R / Z = e ([ M ]) . Using the metho d of Subsection 5.3 or the AP S index theorem it is now easy to repro duce the r esult of [ APS75 ] e ([ M ]) =  η 0 ( M ) − Z M ˆ A ( ˜ ∇ , ˜ ∇ tr iv )  R / Z . 6. The Chern c haracter and a smo oth Grothendie c k-Ri emann-Ro c h theorem 6.1. Smo oth rational cohomology. — 6.1.1 . — Let Z k − 1 ( B ) b e the gro up of s moo th sing ular c ycles o n B . The picture of ˆ H ( B , Q ) as Cheeger- Simons diﬀerential characters ˆ H k ( B , Q ) ⊂ H om ( Z k − 1 ( B ) , R / Q ) is most appropr iate to deﬁne the int egr ation map. By deﬁnition (see [ CS85 ]) a homomorphism φ ∈ Hom ( Z k − 1 ( B ) , R / Q ) is a diﬀerential character if and only if ther e exists a form R ( φ ) ∈ Ω k d =0 ( B ) such that φ ( ∂ c ) =  Z c R ( φ )  R / Q (29) for all s moo th k -chains c ∈ C k ( B ). It is shown in [ CS85 ] that R ( φ ) is uniquely determined by φ . In fact, the map R : ˆ H k ( B , Q ) → Ω k d =0 ( B ) is the cur v ature tra ns - formation in the sense of Deﬁnition 1.1. Assume that T is a closed oriented manifold of dimension n with a triangulation. Then we hav e a ma p τ : Z k − 1 ( B ) → Z k − 1+ n ( T × B ). If σ : ∆ k − 1 → B is a smo oth SMOOTH K-THEOR Y 69 singular simplex, then the triangulatio n o f T × ∆ k − 1 gives rise to a k − 1 + n ch ain τ ( σ ) : = id × σ : T × ∆ → T × B . The integration ( ˆ pr 2 ) ! : ˆ H ( T × B , Q ) → ˆ H ( B , Q ) is now induced by τ ∗ : H om ( Z k − 1+ n ( T × B ) , R / Q ) → Hom ( Z k − 1 ( B ) , R / Q ) . Alternative deﬁnitions of the in tegra tion (for prop er oriented submersions) are given in [ HS05 ], [ GT00 ]. Another constr uctio n of the integration has been given in [ DL05 ], where a lso a pro jection formula (the analog of 4.5 for smo oth cohomo logy) is pr o ved. This picture is used in [ K¨ o7 ] in particular to establish functorialit y . W e will also need the following b ordism form ula which we pr o ve using yet a n- other characterization of the push-forward. W e consider a prop er oriented submersion q : W → B such tha t dim ( T v q ) = n . Let x ∈ ˆ H r ( W , Q ) and f : Σ → B b e a smo oth map fro m a closed oriented manifold of dimension r − n − 1. W e get a pull-back diagram U F − − − − → W   y   y q Σ f − − − − → B . The orientations of Σ a nd T v q induce an or ien tation of U . No te that f ∗ ˆ q ! ( x ) and F ∗ x are ﬂat classes for dimensio n reasons. Ther efore F ∗ x ∈ H r − 1 ( U, R / Q ) and f ∗ ˆ q ! ( x ) ∈ H r − n − 1 (Σ , R / Q ). The compatibility of the push-forward with cartesian diagrams implies the following relation in R / Q : < f ∗ ˆ q ! ( x ) , [Σ] > = < F ∗ x, [ U ] > . If we let f : Σ → B v ar y , then these nu mbers completely characterize the push-for w ard ˆ p ! ( x ) ∈ ˆ H r − n ( B , Q ). W e will use this fac t in the argument b elow. 6.1.2 . — Let now p : V → B b e a prop er o rient ed submersion from a manifold with bo undary such that ∂ V ∼ = W and p | W = q . Ass ume that x ∈ ˆ H ( V , Q ). L emma 6.1 . — In ˆ H ( B , Q ) we have the e quality ˆ q ! ( x | W ) = − a Z V /B R ( x ) ! . Pr o of . — Assume that x ∈ ˆ H r ( V , Q ). Let f : Σ → B b e as ab o ve and form the cartesian diagr am Z z − − − − → V   y   y p Σ f − − − − → B . 70 ULRICH BUNKE & THOMAS SCHICK The oriented manifold Z has the bo unda r y ∂ Z ∼ = U . Using (29 ) at the marked equality we calculate < f ∗ ˆ q ! ( x | W ) , [Σ] > = < F ∗ x | W , [ U ] > = < ( z ∗ x ) | U , [ U ] > ! =  Z Z R ( z ∗ x )  R / Q = " Z Σ Z Z/ Σ R ( z ∗ x ) # R / Q = " Z Σ f ∗ Z V /B R ( x ) # R / Q = − < f ∗ a Z V /B R ( x ) ! , [Σ] > . This implies the asse rtion. 6.2. Construction of the Chern c haracter. — 6.2.1 . — W e start by r e c a lling the c la ssical smo oth characteristic cla sses of Chee g er- Simons. A complex v ector bundle V → B ha s Cher n cla sses c i ∈ H 2 i ( B , Z ) , i ≥ 1. If we add the geo metric data of a hermitean metric and a metric co nnectio n, then we get the geometric bundle V = ( V , h V , ∇ V ). In [ CS85 ] the Chern class es hav e b een reﬁned to smo oth integral c o homology-v alued Chern cla sses ˆ c i ( V ) ∈ ˆ H 2 i ( B , Z ) (see 1 .2.1 for an in tro duction to s moo th ordinar y c o homology). In particular , the class ˆ c 1 ( V ) ∈ ˆ H 2 ( B , Z ) classiﬁes isomorphism cla s ses of hermitean line bundles with connection. The embedding Z ֒ → Q induces a natural ma p ˆ H ( B , Z ) → ˆ H ( B , Q ), and w e let ˆ c Q ( V ) ∈ ˆ H 2 ( B , Q ) denote the image of ˆ c 1 ( V ) ∈ ˆ H 2 ( B , Z ) under this map. 6.2.2 . — The smo oth Chern character ˆ c h which we will construct is a natura l trans- formation ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) of smo oth co ho mology theories. In particular , this means that the following diagra ms commut e (compare Deﬁnition 1.3) Ω( B ) / im ( d ) a / / ˆ K ( B ) I / / ˆ c h   K ( B ) c h   Ω( B ) / im ( d ) a / / ˆ H ( B , Q ) I / / H ( B , Q ) , ˆ K ( B ) R / / ˆ c h   Ω d =0 ( B ) ˆ H ( B , Q ) R / / Ω d =0 ( B ) . (30) SMOOTH K-THEOR Y 71 In a ddition we req uire that the even and odd Chern c har acters are related by susp ension, whic h in the smo oth case amounts to the comm utativity of the following diagram ˆ K 0 ( S 1 × B ) ( ˆ p r 2 ) !   ˆ c h / / ˆ H ev ( S 1 × B , Q ) ( ˆ pr 2 ) !   ˆ K 1 ( B ) ˆ c h / / ˆ H odd ( B , Q ) . (31) The smo oth K -orientation of pr 2 : S 1 × B → B is as in 4.3.2. The or em 6.2 . — Ther e exists a unique natur al tr ansformation ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) such that (30) and (31) c ommut e. Note that na turalit y means that ˆ c h ◦ f ∗ = f ∗ ◦ ˆ c h for every s moo th map f : B ′ → B . The pr o o f of this theorem o ccupies the remainder o f the pres ent subsection. 6.2.3 . — Pr op osition 6.3 . — If t he smo oth Chern char acter ˆ c h exists, t hen it is unique. Pr o of . — Assume that ˆ c h a nd ˆ c h ′ are tw o smo oth Cher n characters . Consider the diﬀerence ∆ := ˆ c h − ˆ c h ′ . It follows from the diagrams ab ov e that ∆ factor s through an o dd natural transfor mation ¯ ∆ : K ( B ) → H ( B , R / Q ) . Indeed, the le ft diagr am of (30) gives a factorization K ( B ) → ( im : Ω( B ) / im ( d ) → ˆ H ( B , Q )) , and the right squa r e in (30) reﬁnes it to ¯ ∆. 6.2.4 . — W e now use the following top olog ical fact. L et P b e a space of the homotopy t yp e of a co un table C W -complex. It represents a contra v ariant set-v alued functor W 7→ P ( W ) := [ W, P ] on the categ ory of compact manifolds. W e further cons ider some ab elian group V . L emma 6.4 . — A natur al tr ansformation of functors N : P ( B ) → H j ( B , V ) on the c ate gory of c omp act manifolds is ne c essarily induc e d by a class N ∈ H j ( P, V ) . Pr o of . — There exists a countable dir ected diagram M of compact manifolds suc h that hocoli m M ∼ = P in the homotopy categor y . Hence we hav e a short exact s e quence 0 → lim 1 H ( M , V ) → H ( P, V ) → lim H ( M , V ) → 0 . If x ∈ P ( P ) is the tautological class, then the pull-ba c k of N ( x ) to the sy s tem M gives an element in l im H ( M , V ). A preimage in H ( P , V ) induces the natural transformation. 72 ULRICH BUNKE & THOMAS SCHICK In our application, P = Z × B U , and the r elev ant cohomo logy H odd ( Z × B U, R / Q ) is trivial. Ther e fo re ¯ ∆ : K 0 ( B ) → H odd ( B , R / Q ) v anishes 6.2.5 . — Next w e observe that ( ˆ pr 2 ) ! : ˆ K ( S 1 × B ) → ˆ K ( B ) is sur jective. In fact, we hav e ( ˆ pr 2 ) ! ( pr ∗ 1 x S 1 ∪ pr ∗ 2 ( x )) = x (32) b y the pro jection formula 4.5 and ˆ p ! ( x S 1 ) = 1 for p : S 1 → ∗ , where x 1 S ∈ ˆ K ( S 1 ) was deﬁned in 5.6. Hence (31 ) implies that ¯ ∆ : K 1 ( B ) → H ev ( B , R / Q ) v anishes, to o. 6.2.6 . — In view o f Pr opo sition 6.3 it remains to show the existence of the smo oth Chern character. W e ﬁr st construct the even par t ˆ c h : ˆ K 0 ( B ) → ˆ H ev ( B , Q ) using the splitting principle. W e will deﬁne ˆ c h a s a natural tra nsformation o f functors such that the following conditions hold. 1. ˆ c h [ L , 0] = e ˆ c Q ( L ) ∈ ˆ H ev ( B , Q ), where L is the geometric family given by a hermitean line bundle with connection L , and ˆ c Q ( L ) ∈ ˆ H 2 ( B , Q ) is derived from the Cheeger-Simons Chern class which class iﬁes the iso morphism class o f L (6.2.1 ). 2. R ◦ ˆ c h = R 3. ˆ c h ◦ a = a Once this is done, the re s ulting ˆ c h automatically satisﬁes (30). F or this it suﬃces to show that ch ◦ I = I ◦ ˆ c h . W e consider the following diagram ˆ K ( B ) R ) ) ˆ c h / / I   ˆ H ( B , Q ) I   R / / Ω d =0 ( B )   K ( B ) c h / / H ( B , Q ) i / / H ( B , R ) The outer square and the right squar e commute. I t follo ws from 2. that the upper triange commutes. Since i is injectiv e w e conclude that the left squar e commutes, to o. 6.2.7 . — In the construction of the Chern c haracter ˆ c h we will use the splitting principle. If x ∈ ˆ K 0 ( B ), then there exists a Z / 2 Z -g raded hermitean vector bundle with connectio n V = ( V , h V , ∇ V ) such that x = [ V , ρ ] for some ρ ∈ Ω odd ( B ) / im ( d ), where V is the zero - dimensional g eometric family with underlying Dirac bundle V . W e will call V the splitting bundle for x . Let F ( V ± ) → B be the bundle of full ﬂags on V ± and p : F ( V ) := F ( V + ) × B F ( V − ) → B . Then we have a deco mposition p ∗ V ± ∼ = ⊕ L ∈ I ± L for some ordered ﬁnite sets I ± of line bundles ov er F ( V ). F or L ∈ I ± let L denote the bundle with the induced metric and c o nnection, and let L be the corresp onding ze r o-dimensional geo metric family . Then we hav e p ∗ x = SMOOTH K-THEOR Y 73 P L ∈ I + [ L , 0] − P L ∈ I − [ L , 0] + a ( σ ) for some σ ∈ Ω odd ( F ( V )) / im ( d ). The prop erties ab ov e thus uniquely determine p ∗ ˆ c h ( x ). L emma 6.5 . — The fol lowing pul l-b ack op er ations ar e inje ctive: 1. p ∗ : H ∗ ( B , Q ) → H ∗ ( F ( V ) , Q ) , 2. p ∗ : H ∗ ( B , R ) → H ∗ ( F ( V ) , R ) 3. p ∗ : H ∗ ( B , R / Q ) → H ∗ ( F ( V ) , R / Q ) 4. p ∗ : ˆ H ∗ ( B , Q ) → ˆ H ∗ ( F ( V ) , Q ) 5. p ∗ : Ω( B ) → Ω( F ( V )) . Pr o of . — The assertion is a classical consequence of the Ler a y- Hir sc h theore m in the cases 1 ., 2., and 3. In c ase 5 ., it follows from the fact that p is surjectiv e and a submersio n. It remains to discuss the case 4 . Let x ∈ ˆ H ∗ ( B , Q ). Assume that p ∗ x = 0. Then in pa rticular p ∗ R ( x ) = R ( p ∗ x ) = 0 so that from 5. also R ( x ) = 0. Thu s x ∈ H ( B , R / Q ). W e now apply 3 . and see that p ∗ x = 0 implies x = 0. In view of Prop osition 6.3 we s ee that a na tural transfor mation ˆ c h : ˆ K 0 ( B ) → ˆ H ev ( B , Q ) is uniquely determined by the conditions 1., 2., and 3 . form ulated in 6 .2.6. 6.2.8 . — Pr op osition 6.6 . — Ther e exists a natur al t r ansformation ˆ c h : ˆ K 0 ( B ) → ˆ H ev ( B , Q ) which satisﬁes the c onditions 1. to 3. formulate d in 6.2.6. W e g ive the pro of of this Prop osition in the next couple o f subsections. Let x := [ E , ρ ] ∈ ˆ K 0 ( B ), a nd V → B b e a s plitting bundle for x with bundle o f ﬂags p : F ( V ) → B . W e c ho ose a g eometry V := ( V , h V , ∇ V ) and let V denote the asso ciated geometric family (5) . In or der to avoid stabilizations we can a nd will always assume that E has a non-zer o dimensional comp onent. Then we have p ∗ I ( x ) = X ǫ ∈{± 1 } ,L ∈ I ǫ ǫI ([ L , 0 ]) . W e deﬁne F := F B ,ǫ ∈{± 1 } ,L ∈ I ǫ L ǫ . Then we ca n ﬁnd a taming ( p ∗ E ⊔ F ( V ) F op ) t , and p ∗ x = X ǫ ∈{± 1 } ,L ∈ I ǫ ǫ ([ L , 0]) − a ( p ∗ ρ − η (( p ∗ E ⊔ F ( V ) F op ) t )) . (5) It was suggested by the referee that one should use the Chern ch aracter ˆ c h ( V ) ∈ ˆ H ev ( B , Q ) constructed in [ CS85 ]. The Ansatz would b e ˆ c h ( x ) : = ˆ c h ( V ) + η (( E ⊔ B V op ) t ) . In order to sho w that this is indep enden t of the choice of V one would need to s how an equation like ˆ c h ( V ) − ˆ c h ( V ′ ) = a ( η (( V op ⊔ V ′ ) t )) . Since after all we know that the Chern c haracter exists this equation i s true, but we do not know a simple dir ect pro of. Therefore we opted for the v ariant to give a complete and independent proof . 74 ULRICH BUNKE & THOMAS SCHICK W e now set p ∗ ˆ c h ( x ) = ˆ c h ( p ∗ x ) := X ǫ ∈{± 1 } ,L ∈ I ǫ ǫ exp( ˆ c Q ( L )) + a ( η (( p ∗ E ⊔ F ( V ) F op ) t )) − a ( p ∗ ρ ) . This construction a priori dep ends on the choices of the representativ e of x , the splitting bundle V → B , and the taming ( E ⊔ F ( V ) F op ) t . 6.2.9 . — In this paragraph we show that this construction is independent o f the choices. Pr op osition 6.7 . — Assu me t hat ther e exists a class z ∈ ˆ H ev ( B , Q ) such that p ∗ z = X ǫ ∈{± 1 } ,L ∈ I ǫ ǫ exp( ˆ c Q ( L )) + a ( η (( p ∗ E ⊔ F ( V ) F op ) t )) − a ( p ∗ ρ ) for one set of choic es. Then z is determine d by x ∈ ˆ K 0 ( B ) . Pr o of . — If ( E ′ , ρ ′ ) is a no ther repre s en tative of x , then we hav e index ( E ) = index ( E ′ ). Therefore w e can take the same splitting bundle for E ′ . The following Lemma (tog ether with Lemma 6.5 ) shows that z doe s not depend on the choice of the repr esen tative o f x . L emma 6.8 . — We have a ( η (( p ∗ E ⊔ F ( V ) F op ) t ) − p ∗ ρ ) = a ( η (( p ∗ E ′ ⊔ F ( V ) F op ) t ) − p ∗ ρ ′ ) Pr o of . — In fact, by Lemma 2 .21 there is a taming ( E ′ ∪ E op ) t such that ρ ′ − ρ = η (( E ′ ∪ E op ) t ). Therefore the assertion is equiv alent to a  η  ( p ∗ E ⊔ F ( V ) F op ) t  − η  ( p ∗ E ′ ⊔ F ( V ) F op ) t  + p ∗ η  ( E ′ ⊔ F ( V ) E op ) t  = 0 . But this is true since this sum of η - forms represents a r ational cohomology class of the form c h dR ( ξ ). This fo llo ws from 2.4.1 0 and the fact p ∗ E ⊔ F ( V ) F op ⊔ F ( V ) p ∗ E ′ op ⊔ F ( V ) F ⊔ F ( V ) p ∗ E ′ ⊔ F ( V ) p ∗ E op admits another taming with v anishing η -form (as in the pro of of Lemma 2.11). 6.2.1 0. — Next we discuss what ha ppens if we v a ry the splitting bundle. Thus let V ′ → B b e another Z / 2 Z -g r aded bundle which r epresents i ndex ( E ). Let p ′ : F ( V ′ ) → B b e the asso ciated splitting bundle. L emma 6.9 . — Assume that we have classes c, c ′ ∈ ˆ H ( B , Q ) such that p ∗ c = X ǫ ∈{± 1 } ,L ∈ I ǫ ǫ exp( ˆ c Q ( L )) + a  η  ( p ∗ E ⊔ F ( V ) F op ) t  − p ∗ ρ  and p ′∗ c ′ = X ǫ ∈{± 1 } ,L ∈ I ′ ǫ ǫ exp( ˆ c Q ( L ′ )) + a  η  ( p ′∗ E ⊔ F ( V ′ ) F ′ op ) t  − p ′∗ ρ  . Then we have c = c ′ . SMOOTH K-THEOR Y 75 Pr o of . — Note that the right-hand s ides depend on the geometr ic bundles V , V ′ since they dep end o n the induced connections on the line bundle summands. W e ﬁrst discuss a sp ecial case, na mely that V ′ is obtained from V by s ta bilization, i.e. V ′ = V ⊕ B × ( C m ⊕ ( C m ) op ). In this ca se there is a natura l embedding i : F ( V ) ֒ → F ( V ′ ) which is induced by e x tension of the ﬂags in V b y the standa rd ﬂag in C m . W e can factor p = p ′ ◦ i . F urthermore, there exists subsets S ǫ ⊂ I ′ ǫ of line bundles (the last m line bundles in the natural o rder) and a natural bijection I ′ ǫ ∼ = I ǫ ⊔ S ǫ . If L ∈ S ǫ , then i ∗ L is trivial with the trivial connection. W e th us hav e p ∗ ( c ′ − c ) = a [ i ∗ η (( p ′∗ E ∪ F ′ op ) t ) − η (( p ∗ E ∪ F op ) t )] It is ag ain easy to s ee that this diﬀerence of η -for ms represents a rational cohomo logy class in the image o f ch dR . Therefore, p ∗ ( c ′ − c ) = 0 and hence c = c ′ b y Lemma 6 .5 . Since the bundle V represents the index of E , tw o choices a re alwa ys stably isomor - phic as hermitean bundles. Using the s pecial case ab o ve we can reduce to the case where V and V ′ only diﬀer by the connection. W e arg ue as follows. W e hav e p ∗ R ( c ′ − c ) = R ( p ∗ ( c ′ − c )) = 0 by an explicit computation. Ther efore c ′ − c ∈ H odd ( B , R / Q ). Since an y tw o connections o n V ca n be connected by a family we conclude that p ∗ ( c ′ − c ) = 0 by a ho mo top y argument. The a ssertion now fo llows. This ﬁnishes the pro of of Prop osition 6.7 . 6.2.1 1. — In order to ﬁnish the cons tr uction of the Chern character in the even case it remains to verify the existence clause in Pro pos ition 6.7. Let x := [ E , ρ ] ∈ ˆ K ( B ) b e such that E has a non-zer o dimensiona l co mponent. Let V → B b e a splitting bundle and p : F ( V ) → B be as above. L emma 6.10 . — We have z := X ǫ ∈{± 1 } ,L ∈ I ǫ ǫ exp( ˆ c Q ( L )) + a [ η (( p ∗ E ∪ F op ) t ) − p ∗ ρ ] ∈ im ( p ∗ ) . Pr o of . — W e use a May er-Vietoris sequence arg ument. Let us ﬁr st recall the May er- Vietoris seq uence for s moo th rational cohomology . Let B = U ∪ V be an open covering of B . Then we hav e the ex a ct sequence · · · → H ( U ∩ V , R / Q ) → ˆ H ( B , Q ) → ˆ H ( U, Q ) ⊕ ˆ H ( V , Q ) → ˆ H ( U ∩ V , Q ) → H ( B , Q ) → . . . which contin ues to the left a nd right by the Mayer-Vietoris sequences of H ( . . . , R / Q ) and H ( . . . , Q ). W e choo se a ﬁnite covering o f B by contractible s ubsets. Let U b e one of these. Note that index ( E ) | U ∈ Z . Thus x | U = [ U × W, θ ] for some form θ and Z / 2 Z -g raded vector s pa ce W . Then we ha ve b y 1. and 3. that c U : = ˆ c h ( x | U ) = dim( W ) − a ( θ ). This ca n b e seen using the splitting bundle F ( B × C n ). Moreov er, p ∗ c U = p ∗ [dim( W ) − a ( θ )] = z | p − 1 U b y Prop osition 6.7. 76 ULRICH BUNKE & THOMAS SCHICK Assume now that we hav e already constructed c V ∈ ˆ H ( V , Q ) such that p ∗ c V = z | p − 1 V , wher e V is a union V of these subsets. Let U be the nex t one in the list. W e show that we can extend c V to c V ∪ U . W e ha ve ( c U ) | U ∩ V = ( c V ) | U ∩ V b y the injectivit y of the pull-ba c k p ∗ : ˆ H ( U ∩ V , Q ) → ˆ H ( p − 1 ( U ∩ V ) , Q ), Lemma 6.5 . The May er-Vietoris sequence implies that w e ca n extend c V b y c U to U ∪ V . 6.2.1 2. — W e now co nstruct the o dd part of the Chern character. In fact, by (31) and (32) we a r e fo r ced to deﬁne ˆ c h : ˆ K 1 ( B ) → ˆ H odd ( B , Q ) b y ˆ c h ( x ) := ( ˆ pr 2 ) ! ( ˆ c h ( x S 1 ∪ x )) . L emma 6.11 . — The diagr ams (30) and (31) c ommute. Pr o of . — The even case of (30) has been chec ked already . The diagram (31) com- m utes b y co nstruction. The o dd case of (30) follows from the Pro jection formula 4.5 and the ev en case. This ﬁnishes the pro of of Theorem 6 .2 6.3. The Chern cha racter is a rational is o morphism and m ultipl i cativ e. — 6.3.1 . — Note that ˆ H ( B , Q ) is a Q -v ector s pace, and that the sequence (1) is an exact sequence o f Q -vector spaces. The Chern character extends to a r ational version ˆ c h Q : ˆ K Q ( B ) → ˆ H ( B , Q ) , where ˆ K Q ( B ) := ˆ K ( B ) ⊗ Z Q . Pr op osition 6.12 . — ˆ c h Q : ˆ K Q ( B ) → ˆ H ( B , Q ) is an isomorphism. Pr o of . — By (30) we hav e the comm utative diag ram K Q ( B ) c h Q   c h dR / / Ω( B ) / im ( d ) a / / ˆ K Q ( B ) ˆ c h Q   I / / K Q ( B ) c h Q   / / 0 H ( B , Q ) / / Ω( B ) / im ( d ) / / ˆ H ( B , Q ) I / / H ( B , Q ) / / 0 , whose horizontal sequence s a re ex a ct. Since ch Q : K Q ( B ) → H ( B , Q ) is an isomor- phism we conclude that ˆ c h Q is a n isomorphism by the Fiv e Lemma. SMOOTH K-THEOR Y 77 6.3.2 . — W e can extend ˆ K Q to a smo oth cohomolog y theory if we deﬁne the structure maps as follows: 1. R : ˆ K Q ( B ) → Ω d =0 ( B ) is the rational extension of R : ˆ K ( B ) → Ω d =0 ( B ). 2. I : ˆ K Q ( B ) I ⊗ id Q → K ( B ) Q c h Q → H ( B , Q ), 3. a : Ω( B ) / im ( d ) a → ˆ K ( B ) ···⊗ 1 → ˆ K Q ( B ). The co mm utative diagr ams (30) now imply: Cor ol lary 6.13 . — The r ational Chern char acter induc es an isomorphism of smo oth c ohomolo gy the ories r eﬁning the isomorphism c h Q : K Q → H Q (in t he sense of Deﬁ- nition 1.3). 6.3.3 . — Pr op osition 6.14 . — The smo oth Chern char acter ˆ c h : ˆ K ( B ) → ˆ H ( B , Q ) is a ring homomorphism. Pr o of . — Since the target of ˆ c h is a Q -vector space it suﬃces to show that ˆ c h Q : ˆ K Q ( B ) → ˆ H ( B , Q ) is a ring homomorphism. Using that ˆ c h Q is an isomor phism of smo oth extensions of rational coho mology we ca n use the rational Chern character in order to transp ort the pro duct o n ˆ K Q ( B ) to a sec o nd pro duct ∪ K on ˆ H ( B , Q ). It remains to show that ∪ a nd ∪ K coincide. Hence the following Lemma ﬁnishes the pro of of P rop osition 6.14. 6.3.4 . — L emma 6.15 . — Ther e is a unique pr o duct on smo oth r ational c ohomolo gy. Pr o of . — Assume that we have tw o pro ducts ∪ k , k = 0 , 1. W e co nsider the bilinear transformation B : ˆ H ( B , Q ) × ˆ H ( B , Q ) → ˆ H ( B , Q ) given by ( x, y ) 7→ B ( x, y ) := x ∪ 1 y − x ∪ 0 y . W e ﬁrst consider the cur v ature. Since a pro duct is co mpatible with the curv ature (1.2, 2.) we get R ( B ( x, y )) = R ( x ∪ 1 y ) − R ( x ∪ 0 y ) = R ( x ) ∧ R ( y ) − R ( x ) ∧ R ( y ) = 0 . Therefore, by (1) the bilinear for m factors over an odd transformation B : ˆ H ( B , Q ) × ˆ H ( B , Q ) → H ( B , R / Q ) . F urthermor e , for ω ∈ Ω( B ) / im ( d ) we have by 1 .2, 2. B ( a ( ω ) , y ) = a ( ω ) ∪ 1 y − a ( ω ) ∪ 0 y = a ( ω ∧ R ( y )) − a ( ω ∧ R ( y )) = 0 . Similarly , B ( x, a ( ω )) = 0. Again by (1) B has a factorization ov e r a natural bilinear transformation ¯ B : H ( B , Q ) × H ( B , Q ) → H ( B , R / Q ) . 78 ULRICH BUNKE & THOMAS SCHICK W e consider the r estriction ¯ B p,q of ¯ B to H p ( B , Q ) × H q ( B , Q ). The functor from ﬁnite C W -complexes to sets W → H p ( W , Q ) × H q ( W , Q ) is repres en ted by a pro duct of Eilenberg MacLane spaces P p,q := H Q p × H Q q . The spaces H Q p , and hence P has the homotopy type of count able C W -complexes . Therefore we can apply Lemma 6 .4 and conclude that ¯ B p,q is induced by a cohomolo g y class b ∈ H ( P p,q , R / Q ). W e ﬁnish the pr oo f of Lemma 6.15 by s howing that b = 0. T o this end we analy se the candida tes for b and show that they v anish either for degree reasons, o r using the fact that ¯ B p,q is bilinea r . Consider a homomor phism of Q -vector spaces w : R / Q → Q . It induces a transfo r - mation w ∗ : H ( B , R / Q ) → H ( B , Q ). In par ticular we can consider w ∗ b ∈ H ( P p,q , Q ). 1. First of all if p, q ar e b oth even, then w ∗ b ∈ H odd ( P p,q, , Q ) v anishes since P p,q do es not have o dd-degre e r ational cohomolog y a t all. 2. Assume no w that p, q ar e b oth o dd. The odd ra tio na l cohomo logy of P p,q is additively gener ated b y the classes 1 × x q and x p × 1 , where x p ∈ H p ( H Q p , Q ) and x q ∈ H q ( H Q q , Q ). It follows tha t w ∗ b = c · x p × 1 + d · 1 × x q for some r ational constant s c, d . Consider o dd classes u p ∈ H p ( B , Q ) and v q ∈ H q ( B , Q ). The form of b implies that w ∗ ◦ ¯ B p,q ( u p , v q ) = c · u p × 1 + d · 1 × v q . This can only b e bilinear if all c a nd d v anish. Hence b = 0. 3. Finally we cons ider the case that p is even a nd q is o dd (o r vice versa, q is even and p is o dd). In this case b is an even class. The even cohomolo gy of P p,q is additively generated by the class e s x n p × 1, n ≥ 0 . Therefore w ∗ b = P n ≥ 0 c n x n p × 1 for some rational consta n ts c n , n ≥ 0. Let u p ∈ H p ( B , Q ) and v q ∈ H q ( B , Q ). Then we hav e w ∗ ◦ ¯ B p,q ( u p , v q ) = X n ≥ 0 c n u n p . This is only bilinear if c n = 0 for all n ≥ 0, hence w ∗ b = 0. Since we can cho ose w ∗ : R / Q → Q arbitrary we conclude that b = 0 . This also ﬁnishes the pro of of the Pro positio n 6 .14. 6.4. Riemann Ro c h theorem . — SMOOTH K-THEOR Y 79 6.4.1 . — Let p : W → B be a pro per submersio n with a smo oth K -orientation o . The Riemann Ro ch theorem as s erts the commutativit y o f a dia gram ˆ K ( W ) ˆ c h − − − − → ˆ H ( W, Q )   y p !   y ˆ p A ! ˆ K ( B ) ˆ c h − − − − → ˆ H ( B , Q ) . Here ˆ p A ! is the comp osition of the cup pr oduct with a smoo th rationa l coho mo logy class ˆ ˆ A c ( o ) and the push-forward in smo oth rational cohomology . The Riemann Ro c h theor em reﬁnes the c har a cteristic class v ers ion of the ordinar y index theor em for families. W e will ﬁrst give the details of the deﬁnition of the push-forward ˆ p A ! . In order to show the Riemann Ro ch theorem we then sho w that the diﬀerence ∆ := ˆ c h ◦ ˆ p ! − ˆ p A ! ◦ ˆ c h v anishes. This is prov ed in several steps. First we use the compatibilites of the push-forward with the transformations a, I , R in o rder to show that ∆ factors over a map ¯ ∆ : K ( W ) → H ( B , R / Q ) . In the next step w e show that ∆ is natura l with resp ect to the pull-back of ﬁbre bundles, and that it do es neither dep end on the smo oth no r on the topo logical K - orientations of p . W e then show that ∆ v anishes in the specia l case that B = ∗ . The ar g umen t is based on the b ordism inv aria nce Pro pos ition 5 .18 a nd some calculation of ra tio nal S pin c -b ordism gro ups. Finally we use the functor ialit y o f the push-forward Pro position 3.23 in order to reduce the case o f a genera l B to the sp ecial case of a po in t. 6.4.2 . — W e consider a proper submersio n p : W → B with closed ﬁbr e s with a smo oth K -orientation r epresented by o = ( g T v p , T h p, ˜ ∇ , σ ). In the following we deﬁne a r eﬁnemen t ˆ ˆ A ( o ) ∈ ˆ H ev ( W , Q ) of the form ˆ A c ( o ) ∈ Ω ev ( W ). The geometric data of o deter mines a connection ∇ T v p (see 2.2.4, 3.1.3) and hence a g eometric bundle T v p := ( T v p, g T v p , ∇ T v p ). According to [ CS85 ] we can deﬁne Pon trjagin clas ses ˆ p i ( T v p ) ∈ ˆ H 4 i ( W , Z ) , i ≥ 1 . The S pin c -structure g iv es ris e to a hermitean line bundle L 2 → W with connection ∇ L 2 (see 3 .1.6). A choice o f a lo cal spin structure amo un ts to a choice of a loca l square ro ot L of L 2 (this bundle was considered alr eady in 3.1.3) such that S c ( T v p ) ∼ = S ( T v p ) ⊗ L as hermitean bundles with connections. W e set L 2 := ( L 2 , h L 2 , ∇ L 2 ). In particular, we hav e 1 2 π i R ˜ ∇ L 2 = 2 c 1 ( ˜ ∇ ) . 80 ULRICH BUNKE & THOMAS SCHICK Again using [ CS85 ] we get a class ˆ c 1 ( L 2 ) ∈ ˆ H 2 ( W , Z ) with curv ature R (ˆ c 1 ( L 2 )) = 2 c 1 ( ˜ ∇ ). 6.4.3 . — Inserting the classes ˆ p i ( T v p ) into that ˆ A -series ˆ A ( p 1 , p 2 , . . . ) ∈ Q [[ p 1 , p 2 . . . ]] we can deﬁne ˆ ˆ A ( T v p ) := ˆ A ( ˆ p 1 ( T v p ) , ˆ p 2 ( T v p ) , . . . ) ∈ ˆ H ev ( W , Q ) . (33) Let ˆ c Q ( L 2 ) ∈ ˆ H 2 ( W , Q ) denote the image of ˆ c 1 ( L 2 ) under the natura l map ˆ H 2 ( W , Z ) → ˆ H 2 ( W , Q ). Deﬁnition 6.16 . — We deﬁne ˆ ˆ A c ( o ) := ˆ ˆ A ( T v p ) ∧ e 1 2 ˆ c Q ( L 2 ) ∈ ˆ H ev ( W , Q ) . Note that R ( ˆ ˆ A c ( o )) = ˆ A c ( o ). L emma 6.17 . — The class (6) ˆ ˆ A c ( o ) − a ( σ ( o )) ∈ ˆ H ev ( W , Q ) only dep ends on the smo oth K -orientation r epr esente d by o . Pr o of . — This is a co nsequence o f the homotopy formula Lemma 2.22. Giv en tw o representativ es o 0 , o 1 of a smo oth K -orientation we can choose a r epresentativ e ˜ o of a smo oth K -orientation on id R × p : R × W → R × B whic h restricts to o k on { k } × B , k = 0 , 1. The co nstruction of the c la ss ˆ ˆ A c ( o ) is compatible with pull-back. Ther efore b y the deﬁnition of the transgres sion form 3.4 we hav e ˆ ˆ A c ( o 1 ) − ˆ ˆ A c ( o 0 ) = i ∗ 1 ˆ ˆ A c ( ˜ o ) − i ∗ 0 ˆ ˆ A c ( ˜ o ) = a " Z [0 , 1] × W /W R ( ˆ ˆ A c ( ˜ o )) # = a h ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) i . By the deﬁnition of equiv alence of representatives of smo oth K -or ien tations we have σ ( o 1 ) − σ ( o 0 ) = ˜ ˆ A c ( ˜ ∇ 1 , ˜ ∇ 0 ) . Therefore ˆ ˆ A c ( o 1 ) − a ( σ ( o 1 )) = ˆ ˆ A c ( o 0 ) − a ( σ ( o 0 )) . (6) This class is denoted by A ( p ) in the abstract and 1.1.9. SMOOTH K-THEOR Y 81 6.4.4 . — W e use the cla ss ˆ ˆ A c ( o ) ∈ ˆ H ev ( W , Q ) in order to deﬁne the push-forward ˆ p A ! := ˆ p ! ([ ˆ ˆ A c ( o ) − a ( σ ( o ))] ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B , Q ) , (34) where ˆ p ! : ˆ H ( W, Q ) → ˆ H ( B , Q ) is the push-forward in smo oth rationa l cohomolog y (see 6.1.1 ) ﬁxed by the underlying ordinary orientation of p . By Lemma 6.17 a lso ˆ p A ! only depe nds to the smo oth K -or ien tation of p and not on the choice of the representativ e. If f : B ′ → B is a smo oth map then w e co nsider the pull-bac k diagram W ′ p ′   F / / W p   B ′ f / / B . The smo oth K -orientation o of p induces (see 3.2 .4) a smo oth K - o rient atio n o ′ of p ′ . W e hav e ˆ ˆ A ( o ′ ) = F ∗ ˆ ˆ A ( o ) a nd ˆ p ′ A ! ◦ F ∗ = f ∗ ◦ ˆ p A ! . 6.4.5 . — As in 3.3 .3 w e consider the compos ition of proper smo othly K -oriented submersions W q 9 9 p / / B r / / A . The co mposition q := r ◦ p has an induced smo oth K -orientation (Deﬁnition 3.21 and Lemma 3.22). In this situation w e have push-fo rw ar ds ˆ p A ! , ˆ r A ! and ˆ q A ! in smo oth rational coho mo logy g iv en by (3 4 ). L emma 6.18 . — We have t he e quality ˆ r A ! ◦ ˆ p A ! = ˆ q A ! of maps ˆ H ( W, Q ) → ˆ H ( B , Q ) . Pr o of . — W e cho ose r epresentativ e s of smo oth K -o r ien tations o p of p and o r of r , and w e let o λ q := o p ◦ λ o r be the comp osition. W e consider the clas s (see Deﬁnition 3.21) ˆ ˆ A c ( o λ q ) − a ( σ ( o λ q )) = ˆ ˆ A c ( o λ q ) − a  σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ λ q ) − dσ ( o p ) ∧ p ∗ σ ( o r )  . By L e mma 6.1 7 and Lemma 3.22 this clas s is indep enden t o f λ . If we let λ → 0, then the co nnection ∇ T v q tends to the direct sum co nnection ∇ T v p ⊕ p ∗ ∇ T v r . F urthermo re, 82 ULRICH BUNKE & THOMAS SCHICK the tra ns gression ˜ ˆ A c ( ˜ ∇ adia , ˜ ∇ λ q ) tends to zero . Therefor e lim λ → 0 [ ˆ ˆ A c ( o λ q ) − a ( σ ( o λ q ))] = ˆ ˆ A c ( o p ) ∪ p ∗ ˆ ˆ A c ( o r ) − a  σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r )  = ( ˆ ˆ A c ( o p ) − a ( σ ( o p ))) ∪ p ∗ ( ˆ ˆ A c ( o r ) − a ( σ ( o r ))) . F or x ∈ ˆ H ( W, Q ) we get using the pro jection formula and the functorialty ˆ q ! = ˆ r ! ◦ ˆ p ! for the push-forward in smo o th ratio na l co homology ˆ r A ! ◦ ˆ p A ! ( x ) = ˆ r ! h ˆ ˆ A c ( o r ) − a ( σ ( o r )) i ∪ ˆ p ! h ˆ ˆ A c ( o p ) − a ( σ ( o p )) i ∪ x  = ˆ q !  p ∗ h ˆ ˆ A c ( o r ) − a ( σ ( o r )) i ∪ h ˆ ˆ A c ( o p ) − a ( σ ( o p )) i ∪ x  = ˆ q !  ( ˆ ˆ A c ( o a q ) − a ( σ ( o a q ))) ∪ x  = ˆ q A ! ( x ) . 6.4.6 . — Recall Deﬁnition 3.18 that the smo oth K -or ien tation deter mines a pus h- down ˆ p ! : ˆ K ( W ) → ˆ K ( B ) . W e can now formulate the index theorem. The or em 6.19 . — The fol lowing squar e c ommutes ˆ K ( W ) ˆ c h − − − − → ˆ H ( W, Q )   y ˆ p !   y ˆ p A ! ˆ K ( B ) ˆ c h − − − − → ˆ H ( B , Q ) . Pr o of . — W e consider the diﬀerence ∆ := ˆ c h ◦ ˆ p ! − ˆ p A ! ◦ ˆ c h . It suﬃces to show that ∆ = 0. 6.4.7 . — Le t x ∈ ˆ K ( W ). L emma 6.20 . — We have R (∆( x )) = 0 . Pr o of . — This Lemma is essentially equiv alent to the lo c al index theorem. W e ha ve b y Deﬁnition 3.15 and Lemma 3.16 R ( ˆ c h ◦ ˆ p ! ( x )) = R ( ˆ p ! ( x )) = p ! ( R ( x )) = Z W/B  ˆ A c ( o ) − dσ ( o )  ∧ R ( x ) . SMOOTH K-THEOR Y 83 On the other hand, since R  ˆ ˆ A c ( o ) − a ( σ ( o ))  = ˆ A c ( o ) − dσ ( o ) w e get R  ˆ p A ! ◦ ˆ c h ( x )  = Z W/B  ˆ A c ( o ) − dσ ( o )  ∧ R ( ˆ c h ( x )) = Z W/B  ˆ A c ( o ) − dσ ( o )  ∧ R ( x ) . Therefore R (∆( x )) = 0. 6.4.8 . — L emma 6.21 . — We have I (∆( x )) = 0 Pr o of . — This is the usua l index theo rem. Indeed, I ( ˆ c h ◦ ˆ p ! ( x )) = c h ◦ I ( ˆ p ! ( x )) = Z W/B ˆ A c ( T v p ) ∪ c h ( I ( x )) and I  ˆ p A ! ◦ ˆ c h ( x )  = Z W/B ˆ A c ( T v p ) ∪ I ( ˆ c h ( x )) = Z W/B ˆ A c ( T v p ) ∪ c h ( I ( x )) . The equality of the right-hand sides prov es the Lemma. Alternatively o ne could observe that the Lemma is a consequence of Lemma 6.2 0. 6.4.9 . — Le t ω ∈ Ω( W ) / im ( d ). L emma 6.22 . — We have ∆( a ( ω )) = 0 . Pr o of . — W e hav e by Prop osition 3.19 ˆ c h ◦ ˆ p ! ( a ( ω )) = ˆ c h ◦ a ( p ! ( ω )) = a Z W/B  ˆ A c ( o ) − dσ ( o )  ∧ ω ! . On the other hand, b y (30) and h ˆ ˆ A c ( o ) − a ( σ ( o )) i ∪ a ( ω ) = a  R  ˆ ˆ A ( o ) − a ( σ ( o ))  ∧ ω  = a  ˆ A c ( o ) − dσ ( o )  ∧ ω  , ˆ p A ! ◦ ˆ c h ( a ( ω )) = ˆ p A ! ( a ( ω )) = a Z W/B  ˆ A c ( o ) − dσ ( o )  ∧ ω ! . 6.4.1 0. — Let o 0 , o 1 represents tw o smo oth reﬁnement s of the s ame top olog ical K - orientation of p . Ass ume that ∆ k is deﬁned with the c hoice o k , k = 0 , 1. L emma 6.23 . — We have ∆ 0 = ∆ 1 . 84 ULRICH BUNKE & THOMAS SCHICK Pr o of . — W e can assume that o k = ( g T v p , T h p, ˜ ∇ , σ k ) for σ k ∈ Ω odd ( W ) / im ( d ). Then we hav e for x ∈ ˆ K ( W ) ∆ 1 ( x ) − ∆ 0 ( x ) = − a Z W/B ( σ 1 − σ 0 ) ∧ R ( x ) ! + Z W/B a ( σ 1 − σ 0 ) ∪ ˆ c h ( x ) = − a Z W/B ( σ 1 − σ 0 ) ∧ R ( x ) ! + Z W/B a h ( σ 1 − σ 0 ) ∧ R ◦ ˆ c h ( x ) i = 0 since R ◦ ˆ c h ( x ) = R ( x ) a nd a ◦ R W/B = R W/B ◦ a . 6.4.1 1. — It follows from Lemma 6.20 and (1) tha t ∆ factorizes throug h a transfo r - mation ∆ : ˆ K ( W ) → H ( B , R / Q ) . By Lemma 6.22 and 2.20 the map ∆ factors ov er a map ¯ ∆ : K ( W ) → H ( B , R / Q ) . This ma p o nly dep ends on the topolo g ical K -or ie ntation of p . It is our goal to show that ¯ ∆ = 0. 6.4.1 2. — Next we wan t to show that the transformation ¯ ∆ is natur a l. F or the moment we write ∆ p := ¯ ∆. Let f : B ′ → B b e a smo oth map a nd form the cartesian diagram W ′ p ′   F / / W p   B ′ f / / B . The ma p p ′ is a prop er submersion with closed ﬁbres whic h has an induced top olog ical K -or ien tation. L emma 6.24 . — We have t he e quality of maps K ( W ) → H ( B ′ , R / Q ) ∆ p ′ ◦ F ∗ = f ∗ ◦ ∆ p . Pr o of . — This follows from the natura lit y of ˆ c h , ˆ p ! , and ˆ p A ! with res pect to the base B . 6.4.1 3. — L emma 6.25 . — If p r 2 : S 1 × B → B is the trivial bund le with the top olo gic al K -orientation given by t he b ounding spin structur e, then ∆ pr 2 : K 0 ( S 1 × B ) → H odd ( B , R / Q ) vanishes. SMOOTH K-THEOR Y 85 Pr o of . — The o dd Chern character is deﬁned suc h that for x ∈ K 0 ( S 1 × B ) we have ˆ c h 1 (( ˆ pr 2 ) ! x ) = ( ˆ pr 2 ) ! ˆ c h 0 ( x ) (see (31)). With the c hoice of the smo oth K -orientation of p r 2 given in 4.3.2 we hav e ˆ ˆ A ( o ) − a ( σ ( o )) = 1 so that ˆ p A ! = ˆ p ! . This implies the Lemma. 6.4.1 4. — T he group H 2 ( W , Z ) acts simply tra nsitiv e on the set of S pin c -structures of T v p . Let Q → W be a unitary line bundle clas s iﬁed b y c 1 ( Q ) ∈ H 2 ( W , Z ) . W e choo se a hermitean connection ∇ Q and for m the geo metr ic line bundle Q := ( Q, h Q , ∇ Q ). Let o := ( T v p, T h p, ˜ ∇ , ρ ) re pr esen t a smo oth K -orientation reﬁning the given topo - logical K -orientation of p . Note that ˜ ∇ is completely determined by the Cliﬀord connection on the Spinor bundle S c ( T v p ). The spinor bundle of the shift of the top o- logical K -orientation by c 1 ( Q ) is given by S c ( T v p ) ′ = S c ( T v p ) ⊗ Q . W e construct a corres ponding smo oth K -o r ien tation o ′ = ( T v p, T h p, ˜ ∇ ⊗ ∇ Q , ρ ). W e let ˆ p ! and ˆ p ′ ! denote the co r resp onding push-forwards in s moo th K -theory . Let Q b e the g eometric family ov er W with zero- dimensional ﬁbr e g iv en b y the bundle Q (see 2.1.4 ). The push-forwards ˆ p ! and ˆ p ′ ! are now related as follows: L emma 6.26 . — ˆ p ′ ! ( x ) = ˆ p ! ([ Q , 0] ∪ x ) , ∀ x ∈ ˆ K ( W ) . Pr o of . — Let x = [ E , ρ ]. By an inspection of the constructions leading to Deﬁnition 3.7 we see that p ′ λ ! E = p λ ! ( Q × W E ) . F urthermor e we have c 1 ( ˜ ∇ ⊗ ∇ Q ) = c 1 ( ˜ ∇ ) + c 1 ( ∇ Q ) so that ˆ A c ( o ′ ) = ˆ A c ( o ) ∧ e c 1 ( ∇ Q ) . On the other hand, since Ω( Q ) = e c 1 ( ∇ Q ) we hav e [ Q , 0] ∪ [ E , ρ ] = [ Q × W E , e c 1 ( ∇ Q ) ∧ ρ ] Using the explicit formula (17) we get ˆ p ′ ! ([ E , ρ ]) − ˆ p ! ([ Q , 0] ∪ [ E , ρ ]) = [ ∅ , ˜ Ω ′ ( λ, E ) − ˜ Ω( λ, E )] for all sma ll λ > 0. Since b oth transgr ession forms v anish in the limit λ = 0 we get the desir e d result. In the notation of 6.4.2 we have L ′ = L ⊗ Q . Therefore ˆ c Q ( L ′ 2 ) = ˆ c Q ( L 2 ) + 2 ˆ c Q ( Q ) and hence we can express ˆ p ′ ,A ! according to (34) a s ˆ p ′ A ! ( x ) = ˆ p ! h ˆ ˆ A c ( o ) ∪ e ˆ c Q ( Q ) − a ( σ ( o ))  ∪ x i . 86 ULRICH BUNKE & THOMAS SCHICK 6.4.1 5. — As b efore, let p : W → B b e a prop er o rien ted submersion which admits topo logical K - orientations. L emma 6.27 . — If ∆ p = 0 for some top olo gic al K - orientation of p , then it vanishes for every top olo gic al K -orientation of p . Pr o of . — W e ﬁx the K -orientation of p such that ∆ p = 0 and let p ′ denote the s a me map with the top ologica l K -or ien tation shifted by c 1 ( Q ) ∈ H 2 ( W , Z ) . W e co ntin ue to use the no tation of 6.4.1 4. W e choose a representativ e o of a smoo th K -o rien tation of p reﬁning the top ological K -orientation. F or simplicity we take σ ( o ) = 0. F urthermore, we take o ′ as above. Using ˆ c h ([ Q , 0]) = e ˆ c Q ( Q ) and the multiplicativit y of the Chern character we get ˆ p ′ A ! ◦ ˆ c h ( x ) − ˆ c h ◦ ˆ p ′ ! ( x ) = ˆ p ! h ˆ ˆ A c ( o ) ∪ e ˆ c Q ( Q ) ∪ ˆ c h ( x ) i − ˆ c h ◦ ˆ p ! ([ Q , 0] ∪ x ) = ˆ p ! h ˆ ˆ A c ( o ) ∪ ˆ c h ([ Q , 0]) ∪ ˆ c h ( x ) i − ˆ p A ! ◦ ˆ c h ([ Q , 0] ∪ x ) = ˆ p A ! ◦ ˆ c h ([ Q , 0] ∪ x ) − ˆ p A ! ◦ ˆ c h ([ Q , 0] ∪ x ) = 0 . 6.4.1 6. — W e now consider the sp ecial ca se that B = ∗ and W is an o dd-dimensiona l S pin c -manifold. Since H ( ∗ , R / Q ) ∼ = R / Q we get a homomor phism ∆ p : K ( W ) → R / Q . Pr op osition 6.28 . — If B ∼ = ∗ , then ∆ p = 0 Pr o of . — First note that ∆ p is triv ia l on K 1 ( W ) for deg ree r easons. It therefore suﬃces to study ∆ p : K 0 ( W ) → R / Q . Let x ∈ K 0 ( W ) b e classiﬁed by ξ : W → Z × B U . It gives r ise to an element [ ξ ] ∈ Ω S pin c dim( W ) ( Z × B U ) of the S pin c -b ordism group of Z × B U . L emma 6.29 . — If [ ξ ] = 0 , then ∆ p = 0 . Pr o of . — Assume that [ ξ ] = 0. In this case there ex ists a compac t S pin c -manifold V with b oundary ∂ V ∼ = W (as S pi n c -manifolds), and a map ν : V → Z × B U such that ν | ∂ V = ξ . W e can choo se a Z / 2 Z -graded v ector bundle E → V whic h r epresent s the class of ν in K 0 ( V ). W e reﬁne E to a geometric bundle E := ( E , h E , ∇ E ) and form the asso ciated geometric family E with zero-dimensional ﬁbre. W e choose a r e pr esen tative ˜ o of a smooth K -orientation of the map q : V → ∗ which reﬁnes the top ologica l K -orientation given by the S pin c -structure and which has a pr o duct structure near the b oundary . F o r simplicity we assume that σ ( ˜ o ) = 0. The r estriction of ˜ o to the b oundary ∂ V deﬁnes a smoo th K -or ie ntation of p . SMOOTH K-THEOR Y 87 W e let ˆ y := [ E , 0] ∈ ˆ K ( V ), and we deﬁne ˆ x := ˆ y | ∂ V such that I ( ˆ x ) = x . By Prop osition 5.18 w e have ˆ c h ◦ ˆ p ! ( ˆ x ) = ˆ c h ◦ ˆ p ! ( ˆ y | W ) = ˆ c h ([ ∅ , q ! ( R ( ˆ y ))]) = − a  Z V ˆ A c ( ˜ o ) ∧ R ( ˆ y )  . On the o ther hand, the b ordism form ula for the push-forward in s mo oth rational cohomolog y , Lemma 6.1, gives ˆ p A ! ◦ ˆ c h ( ˆ x ) = ˆ p !  ˆ ˆ A c ( o ) ∪ ˆ c h ( ˆ x )  = ˆ p !  ˆ ˆ A c ( ˜ o ) | W ∪ ˆ c h ( ˆ y ) | W  = − a  Z V ˆ A c ( ˜ o ) ∧ R ( ˆ y )  . These tw o for m ulas imply that ∆ p = 0. 6.4.1 7. — W e now ﬁnish the pro of of P rop osition 6.28. W e claim that there exists c ∈ N suc h that c [ ξ ] = 0. In view of Lemma 6.29 we then have 0 = ∆ cp = c ∆ p , and this implies the Prop osition since the target R / Q of ∆ p is a Q -vector spac e . Note that the g raded ring Ω S pin c ∗ ⊗ Q is concen tra ted in even degrees. Using that Ω S O ∗ ⊗ Q is concentrated in even degrees, o ne can see this as follows. In [ Sto68 , p. 352] it is shown that the homomor phism S pin c → U ( 1) × S O induces an injection Ω S pin c ∗ → Ω S O ∗ ( B U (1)). Since H ∗ ( B U (1) , Z ) ∼ = Z [ z ] with deg ( z ) = 2 lives in even degrees, we see using the Atiy a h-Hirzebruch sp ectral sequence that Ω S O ( B U (1)) ⊗ Q lives in even degr e e s , to o. This implies that Ω S pin c ∗ ⊗ Q is concent ra ted in even degrees. Since H ∗ ( Z × B U, Z ) is also concentrated in even degr ees it follows again from the A tiyah-Hirzebruch sp ectral sequence that Ω S pin c ∗ ( Z × B U ) ⊗ Q is concentrated in even degrees. Since [ ξ ] is of o dd degree we conclude the claim that c [ ξ ] = 0 for an appropr iate c ∈ N . This ﬁnishes the pro of of Prop osition 6.2 8. 6.4.1 8. — W e now consider the genera l ca se. Let p : W → B b e a prop er submersion with clos e d ﬁbres with a top ological K -orientation. Pr op osition 6.30 . — We have ∆ p = 0 . W e give the pro of in the next couple of subsections. 6.4.1 9. — F or a closed o rien ted manifold Z let PD : H ∗ ( Z, Q ) ∼ → H ∗ ( Z, Q ) denote the Poincar ´ e duality isomo rphism. L emma 6.31 . — The gr oup H ∗ ( B , Q ) is gener ate d by classes of the form f ∗  PD ( ˆ A c ( T Z ))  , wher e Z is a close d S pi n c -manifold and f : Z → B . 88 ULRICH BUNKE & THOMAS SCHICK Pr o of . — W e consider the sequence o f transfor mations of homology theor ies Ω S pin c ∗ ( B ) α → K ∗ ( B ) β → H ∗ ( B , Q ) . The transfor mation α is the K -o rien tation of the S pin c -cob ordism theory , and β is the homologica l Chern ch ar acter. W e consider all groups as Z / 2 Z -gra ded. The homolog- ical Chern character is a ra tional is o morphism. F urther mo re one knows by [ BD82 ], [ BHS ] that Ω S pin c ∗ ( B ) α → K ∗ ( B ) is surjective. It follows that the comp osition β ◦ α : Ω S pin c ( B ) ⊗ Q → H ∗ ( B , Q ) is surjective. An explicit desc r iption of β ◦ α is given as follows. Let x ∈ Ω S pin c ( B ) be r e pr esen ted b y a ma p f : Z → B fro m a closed S pin c -manifold Z to B . Let PD : H ∗ ( Z, Q ) ∼ → H ∗ ( Z, Q ) denote the Poincar´ e duality iso morphism. Then we have β ◦ α ( x ) = f ∗  PD ( ˆ A c ( T Z ))  . 6.4.2 0. — F or the pr o o f of P rop osition 6.30 w e ﬁrst consider the ca se that p has even-dimensional ﬁbres, and that x ∈ K 0 ( W ). By Lemma 6.3 1 , in order to show that ∆ p ( x ) = 0, it suﬃces to show that a ll ev aluations ∆ p ( x )  f ∗ ( PD ( ˆ A c ( T Z )))  v anish. In the following, if x deno tes a K -theor y class, then ˆ x denotes a smo oth K -theory class such that I ( ˆ x ) = x . W e choose a representativ e o q of a s moo th K - orientation which reﬁnes the top o- logical K -orientation of the map q : Z → ∗ induced by the S pin c -structure on T Z . F urthermor e , we co nsider the diagr am with a cartesian squa re V s $ $ r   F / / W p   Z q   f / / B ∗ . SMOOTH K-THEOR Y 89 In the pres en t case ∆ p ( x ) ∈ H odd ( B , R / Q ), and we ca n assume that Z is o dd- dimensional. W e c alculate ∆ p ( x )  f ∗ ( PD ( ˆ A c ( T Z )))  = f ∗ ∆ p ( x )  PD ( ˆ A c ( T Z ))  Lemma 6 . 24 = ∆ r ( F ∗ x )  PD ( ˆ A c ( T Z ))  = ( ˆ A c ( ∇ T Z ) ∪ ∆ r ( F ∗ x ))[ Z ] = Z Z ˆ A c ( o ) ∧ ∆ r ( F ∗ x ) = ˆ q !  ˆ ˆ A c ( o q ) ∪ ∆ r ( F ∗ x )  = ˆ q A ! (∆ r ( F ∗ ˆ x )) = ˆ q A ! h ˆ c h ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ r A ! ◦ ˆ c h ( F ∗ ˆ x ) i = ˆ q A ! ◦ ˆ c h ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ c h ( F ∗ ˆ x ) P r oposition 6 . 28 = ˆ c h ◦ ˆ q ! ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ c h ( F ∗ ˆ x ) = ˆ c h ◦ ˆ s ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ c h ( F ∗ ˆ x ) = ∆ s ( F ∗ x ) P r oposition 6 . 28 = 0 . W e th us hav e shown that 0 = ∆ p : K 0 ( W ) → H odd ( B , R / Q ) if p has even-dimensional ﬁbres. 6.4.2 1. — If p has o dd-dimensional ﬁbres and x ∈ K 1 ( W ), then we can choos e y ∈ K 0 ( S 1 × W ) such that ( ˆ pr 2 ) ! ( y ) = x . Since p ◦ pr 2 has even-dimensional ﬁbres we get using the Lemmas 6 .1 8 and 3.23 ∆ p ( x ) = ˆ c h ◦ ˆ p ! ◦ ( ˆ pr 2 ) ! ( ˆ y ) − ˆ p A ! ◦ ˆ c h ◦ ( ˆ pr 2 ) ! ( ˆ y ) Lemma 6 . 25 = ˆ c h ◦ ( \ p ◦ pr 2 ) ! ( ˆ y ) − ˆ p A ! ◦ ( ˆ pr 2 ) A ! ◦ ˆ c h ( ˆ y ) = ˆ c h ◦ ( \ p ◦ pr 2 ) ! ( ˆ y ) − ( \ p ◦ pr 2 ) A ! ◦ ˆ c h ( ˆ y ) = ∆ p ◦ pr 2 ( y ) = 0 . Therefore 0 = ∆ p : K 1 ( W ) → H odd ( B , R / Q ) if p has o dd-dimensional ﬁbr e s . 90 ULRICH BUNKE & THOMAS SCHICK 6.4.2 2. — Let us now co nsider the case tha t p has ev en-dimensiona l ﬁbre s , and that x ∈ K 1 ( W ). I n this case we consider the diagram S 1 × W Pr 2 − − − − → W   y t := id S 1 × p   y p S 1 × B pr 2 − − − − → B . W e choose a class y ∈ K 0 ( S 1 × W ) suc h that ( Pr 2 ) ! ( y ) = x . W e further cho ose a smo oth reﬁnement ˆ y ∈ ˆ K 0 ( S 1 × W ) of y and set ˆ x := ( ˆ Pr 2 ) ! ( ˆ y ). Then w e calculate using the Lemmas 6.18 and 3.2 3 ∆ p ( x ) = ˆ c h ◦ ˆ p ! ( ˆ x ) − ˆ p A ! ◦ ˆ c h ( ˆ x ) = ˆ c h ◦ ˆ p ! ◦ ( ˆ Pr 2 ) ! ( ˆ y ) − ˆ p A ! ◦ ˆ c h ◦ ( ˆ Pr 2 ) ! ( ˆ y ) Lemma 6 . 25 = ˆ c h ◦ ˆ p ! ◦ ( ˆ Pr 2 ) ! ( ˆ y ) − ˆ p A ! ◦ ( ˆ Pr 2 ) A ! ◦ ˆ c h ◦ ( ˆ y ) = ˆ c h ◦ ( \ p ◦ Pr 2 ) ! ( ˆ y ) − ( \ p ◦ Pr 2 ) A ! ◦ ˆ c h ( ˆ y ) = ˆ c h ◦ ( \ pr 2 ◦ t ) ! ( ˆ y ) − ( \ pr 2 ◦ t ) A ! ◦ ˆ c h ( ˆ y ) = ˆ c h ◦ ˆ pr 2! ◦ ˆ t ! ( ˆ y ) − ˆ pr A 2! ◦ ˆ t A ! ◦ ˆ c h ( ˆ y ) Lemma 6 . 25 = ( ˆ pr 2 ) A ! h ˆ c h ◦ ˆ t ! ( ˆ y ) − ˆ t A ! ◦ ˆ c h ( ˆ y ) i = ( ˆ pr 2 ) A ! ◦ ∆ t ( y ) = 0 . Therefore 0 = ∆ p : K 1 ( W ) → H ev ( B , R / Q ) if p has even-dimensional ﬁbres. 6.4.2 3. — In the ﬁnal c a se p has o dd-dimensiona l ﬁbres and x ∈ K 0 ( W ). In this case we cons ider the se q uence of pro jections S 1 × S 1 × W pr 23 → S 1 × W pr 2 → W . W e choose a class y ∈ K 0 ( S 1 × S 1 × W ) such that ( pr 2 ◦ pr 23 ) ! ( y ) = x . W e further choose a smo oth r eﬁnemen t ˆ y ∈ ˆ K 0 ( S 1 × S 1 × W ) of y and set ˆ x := ( \ pr 2 ◦ pr 23 ) ! ( ˆ y ). Then we calculate using the a lready known cases a nd the Lemmas 6.18 and 3.23, ∆ p ( x ) = ˆ c h ◦ ˆ p ! ( ˆ x ) − ˆ p A ! ◦ ˆ c h ( ˆ x ) = ˆ c h ◦ ˆ p ! ◦ ( ˆ pr 2 ) ! ◦ ( ˆ pr 23 ) ! ( ˆ y ) − ˆ p A ! ◦ ˆ c h ◦ ( ˆ pr 2 ) ! ◦ ( ˆ pr 23 ) ! ( ˆ y ) = ˆ c h ◦ ( \ p ◦ pr 2 ) ! ◦ ( ˆ pr 23 ) ! ( ˆ y ) − ˆ p A ! ◦ ˆ c h ◦ ( \ pr 2 ◦ pr 23 ) ! ( ˆ y ) = ( \ p ◦ pr 2 ) A ! ◦ ˆ c h ◦ ( ˆ pr 23 ) ! ( ˆ y ) − ˆ p A ! ◦ ( \ pr 2 ◦ pr 23 ) A ! ◦ ˆ c h ( ˆ y ) = ( \ p ◦ pr 2 ) A ! ◦ ∆ pr 23 ( ˆ y ) Lemma 6 . 25 = 0 . This ﬁnishes the pro of of Theorem 6 .19. ✷ SMOOTH K-THEOR Y 91 7. Conclusion W e hav e now constructed a geometric mo del for smo oth K- theory , built out of geometric families of Dira c-t yp e op erato rs. W e equipp e d it with a co mpatible multi- plicative structure, and we hav e given a n explicit construction of a push-down map fo r ﬁbre bundles with all the exp ected pro perties. F or the veriﬁcation of these pr o per ties we heavily used lo cal index theor y . W e presented a collection of natural examples of smooth K-theor y classes and show ed in particular that several known sec o ndary analytic-geo metr ic inv ariants ca n be understo o d in this fra mew or k very naturally . This inv olved a lso the considera tion of b ordisms in this fr amew or k. Finally , we constructed a smo oth lift of the Chern c har acter and prov ed a smo oth version of the Gro thendiec k-Riemann-Ro ch theore m. This also inv olved certain con- siderations from homotopy theor y which ar e sp ecial to K-theory . Impo rtan t op en questions concern the construction of equiv aria n t versions of this theory , or even b etter versions whic h work for orbifolds o r similar singular spaces. In a diﬀerent dir ection, we hav e address e d the co nstruction of geometric mo dels of smo oth b ordism theories along s imila r lines in [ BSSW07 ]; using singular b ordism this has also b een a c hieved for smo oth ordinary cohomolo gy in [ BKS09 ]. References [ABS64] M. F. A tiy ah, R. Bott & A. S hapiro – “Cliﬀord mo dules”, T op ol o gy 3 (1964), no. sup pl. 1, p. 3–38. [APS75] M. F. A tiy ah, V . K. P a todi & I. M. Singe r – “Sp ectral asymmetry and Rie- mannian geometry . I”, Math. Pr o c. Cambridge Philos. So c. 77 (1975), p . 43–69. [AS71] M. F. A ti y ah & I. M. Si nger – “The index of elliptic op erators. I V”, Ann. of Math. (2) 93 (1971), p. 119–138. [BC90a ] J.-M. Bismut & J. Cheeger – “F amilies index for manifolds with boundary , sup erconnections, and cones. I. Families of manifolds with b oundary and Dirac op erators” , J. F unct. An al. 89 (1990), n o. 2, p. 313–3 63. [BC90b] , “F amilies ind ex for manifolds with b oundary , sup erconnections and cones. I I. The Chern c haracter”, J. F unct. Ana l. 90 (1990), n o. 2, p. 306–3 54. [BC91] , “Remarks on the index theorem for families of Dirac operators on manifolds with b oundary”, in Diﬀer ential ge ometry , Pitman Monogr. Surve ys Pure Ap pl. Math., vol . 52, Longman Sci. T ech., Harlo w, 1991, p. 59–83. [BD82] P. Ba um & R. G. Douglas – “Index theory , b ordism, and K -homology”, in Op er- ator algebr as and K -the ory (San F r ancisc o, Cali f., 1981) , Con temp. Math., vo l. 10, Amer. Math. Soc., Providence, R.I., 1982, p. 1–31. [Bera] A. B er thomieu – “Direct image for multiplica tive and relativ e K -th eories from transgressio n of the famili es ind ex theorem, part 1”. 92 ULRICH BUNKE & THOMAS SCHICK [Berb] , “Direct image for m ultiplicative and relative K -theories from transgressio n of the families index theorem, p art 2”. [BGV04] N. Berline, E. Getzler & M. Ver gne – He at kernels and Dir ac op er ators , Grundlehren T ext Editions, Springer-V erlag, Berlin, 2004, Corrected reprint of the 1992 original. [BHS] P. Baum, N. Hi gson & T . Schick – “On the Equiva lence of Geometric and Analytic K-Homology”, Pure and Applied Mathematics Q uarterly 3, 1–24, (2007). [Bis85 ] J.-M. B ismut – “The Atiyah-Singer index theorem for families of Dirac operators: tw o heat equation pro ofs” , I nvent. Math. 83 (1985), no. 1, p. 91–151. [Bis05 ] J. M. B ismut – “Eta inv arian ts, diﬀerentia l characters and ﬂat vector bun d les”, Chinese Ann. Math. Ser. B 26 (200 5), no. 1, p. 15–44, With an append ix by K. Corlette and H. Esnault. [BJ83] S. Baaj & P. Julg – “Th´ eori e b iva riante d e K asparo v et op´ erateurs non b orn´ es dans les C ∗ -mod ules hilbertiens”, C. R. A c ad. Sci. Paris S´ er. I Math. 296 (1983), no. 21, p. 875–8 78. [BL95] J.-M. B ismut & J. Lott – “Flat vector bun dles, direct images and higher real analytic torsion”, J. A mer. Math. So c. 8 (1995), no. 2, p. 291–3 63. [Bry93] J.-L. Br ylin ski – L o op sp ac es, char acteristic classes and ge ometric quantiz ation , Progress in Mathematics, vol. 107, Birkh¨ auser Boston Inc., Boston, MA, 1993. [Bun] U. Bunke – “Index theory , eta forms, and Deligne cohomolog y”, to app ear in Memoirs AMS. [Bun95] U. B unke – “A K -theoretic relative index theorem and Callias-t yp e Dirac op era- tors”, M ath. Ann. 303 (1995), n o. 2, p. 241–27 9. [Bun02] , “On the functorialit y of Lott’s secondary analytic ind ex”, K-The ory 25 (2002), no. 1, p. 51–58. [BKS09] U. Bunke, M. Kreck & T. Schick — “A geometric description of smooth coho- mology”, [BM04] U. Bunke & X. Ma – “Index and secondary index theory for ﬂat bundles with dualit y”, in Asp e cts of b oundary pr oblems in analysis and ge ometry , Op er. Theory Adv. Appl., vol . 151, Birkh¨ auser, Basel, 2004, p. 265–341. [BS09] U. B unke & Th. Schick – “Uniqueness of smo oth extensions of generalized coho- mology theories”, a rX iv:0901 .4423 [BSSW07] U. B unke, Th. Schick, I. Schr ¨ oder & M. Wiethaup – “Landweber exact formal group laws and smo oth cohomology theories”, a rXiv:0711.113 4 [CS85] J. Cheeger & J. S imons – “Diﬀerential characters and geometric in va riants”, in Ge ometry and top olo gy (Col l e ge Park, Md., 1983/84) , Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, p. 50–80. [DL05] J. L. Dup ont & R. Ljungmann – “Integratio n of simplicial forms and Deligne cohomolog y”, Math. Sc and. 97 (2005), no. 1, p. 11–39. [FH00] D. S . Freed & M. Hopkins – “On Ramond-R amond ﬁelds and K -theory”, J. High Ener gy Phys. (2000), no. 5, p. P ap er 44, 14. [FM92] D. S. Freed & R. B. Melrose – “A mo d k index theorem”, Invent. Math. 107 (1992), no. 2, p. 283–299. [F re00] D. S. Freed – “Dirac charge quantiza tion and generalized diﬀerentia l cohomology”, in Surveys in diﬀer ential ge ometry , Surv. Diﬀer. Geom., VI I, Int. Press, Somerville, MA, 2000, p. 129–1 94. SMOOTH K-THEOR Y 93 [Gom] K. G omi – “Diﬀerential c haracters and the Steenrod squares” arXiv: math.A T/041 1043 [GT00] K. Gomi & Y. Terashima – “A ﬁb er in tegration form ula for t he smo oth Deligne cohomolog y”, Internat. M ath. R es. Notic es (2000), no. 13, p. 699–708. [HS05] M. J. Hopkins & I. M. Singer – “Quadratic functions in geometry , top ology , and M-theory”, J. Diﬀer ential Ge om. 70 (2005), no. 3, p. 329–452. [K¨ o7] M. K ¨ ohler – Die Inte gr ationsabbildung in Deligne Kohomolo gie und Verkn¨ upfung von Faserb¨ undeln , Master’s Thesis, F ach b ereic h Mathematik, Georg-August- U nive rsit¨ at G¨ ottingen, 2007. [Kar87] M. Ka r oubi – “Homol ogie cyclique et K -th´ eori e”, Ast´ erisque (1987), no. 149, p. 147. [Kar97] , “Sur la K -th´ eorie multipli cative”, in Cyclic c ohomolo gy and nonc ommu- tative ge ometry (Waterlo o, ON, 1995) , Fields Inst. Commun., vo l. 17, Amer. Math. So c., Pro vidence, R I, 1997, p . 59–77. [Kas81] G. Kasp aro v – “The op erator K- functor and extensions of C*-algebras.”, Math. USSR, I zv. 16 (1981), p. 513–572 (English). [Lau99] G. La ures – “The top ologica l q -expansion principle”, T op olo gy 38 (1999), no. 2, p. 387–4 25. [Lot94] J. Lott – “ R / Z index t h eory”, Comm. Anal. Ge om. 2 (1994), no. 2, p. 279–311. [Lot00] , “Secondary analytic indices”, in R e gulators in analysis, ge ometry and num- b er the ory , Progr. Math., vol . 171, Birkh¨ auser Boston, Boston, MA, 2000, p . 231–29 3. [MP97a ] R. B. Melrose & P. Piazza – “F amilies of Dirac op erators, b oundaries and the b -calculus”, J. Diﬀer ential Ge om. 46 (1997), n o. 1, p. 99–18 0. [MP97b] , “An index th eorem for families of Dirac operators on odd -dimensional manifolds with b oundary”, J. Di ﬀer ential Ge om. 46 (1997), n o. 2, p. 287–3 34. [MR07] R. Melr ose & F. Rochon – “P erio dicit y and th e determinant bundle”, Comm. Math. Phys. 274 (2007), no. 1, p. 141–1 86. [MZ04] X. Ma & W. Zhang – “Eta-inv arian ts, torsion forms and ﬂat vector bundles”, arXiv.org:math/0 405599, 2004. [P ek93] O. Pek onen – “A K -theoretic approac h to Chern-Cheeger-Simons inv arian ts”, in Pr o c e e dings of the Winter Scho ol “Ge ometry and Physics” (Srn ´ ı, 1991) , no. 30, 1993, p. 27– 34. [SS] J. Simons & D. Su lliv an – “Axiomatic Characterizatio n of Ordinary Diﬀerential Cohomolog y”. [Sto68] R. E. S tong – Notes on c ob or di sm the ory , Mathematical notes, Princeton Universit y Press, Princeton, N .J., 1968. Ulrich Bunke , NWF I - Mathematik, Johannes-Kepler-Univ ersit¨ at Regensburg, 93040 Regensburg, GERMANY • E-mail : ulric h.bunke@mathe matik.uni-regensburg.de Thomas S chick , M athematische s Institut, Georg-August-Universit¨ at G¨ ottingen, Bunsenstrasse 3, 37073 G¨ ottingen, GERMAN Y • E-mail : schick @uni-math.gwd g.de

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