Chern classes on differential K-theory

Chern classes on d i fferential K -theory Ulrich Bunk e ∗ Octob er 26, 20 2 1 Abstract In this note w e g iv e a simple, mo del-indep enden t construction of Ch e rn classes as natural transformations from d ifferential complex K -theory to d i fferen tial in tegral cohomology . W e v erify the exp ecte d b eha viour of these Chern classes with resp ect to sums and susp ension. Con te n ts 1 Statemen ts 1 2 Pro ofs 5 1 Statemen t s Complex K -theory and in tegral cohomology H Z are generalised cohomology theories whic h hav e a unique differen tia l 1 extensions ( ˆ K , R, I , a, R ) and ( d H Z , R, I , a, R ) with in te- gration. Moreo v er, these extensions are m ultiplicative in a unique wa y . W e refer to [BS] for a description of the a x ioms for differen tial extensions of cohomology theories and a pro of of these statements . The i ’th Chern class is a natural tra ns f ormation of set-value d functors c i : K 0 → H Z 2 i ∗ NWF I - Mathematik, Univ ersit¨ at Regensburg, 93040 Regensburg, GERMANY, ulrich.bunk e@mathema t ik.uni-regens bur g.de 1 In our previous work instead of ”differential coho m ology” we used the term ”smo oth coho mology”. W e were con vinc e d by D. F r eed that differen tia l cohomology is the b etter name. 1 on the category of top ological spaces. The pro duct H Z ev := Q i ≥ 0 H Z 2 i is a functor with v alues in comm utative g r a de d rings. W e consider subfunctor H Z ev, ∗ 1 := 1 + Q i ≥ 1 H Z 2 i ⊆ Q i ≥ 0 H Z 2 i whic h takes v alues in the subgroup of units. The total Chern class c := 1 + c 1 + c 2 + · · · : K 0 − → H Z ev, ∗ 1 is a natural tra n sformation of gr oup-value d functors. Let Ω ∗ cl ( . . . , K ∗ ) ⊆ Ω ∗ ( . . . , K ∗ ) denote the graded r ing v alued functors on smo oth mani- folds of smo oth differential fo rms with co e fficien ts in K ∗ and its subfunctor of c losed forms. W e use the p o w ers of the Bott elemen t in K 2 in order to identify the functors Ω 0 ( . . . , K ∗ ) ∼ = Ω ev ( . . . ) , Ω − 1 ( . . . , K ∗ ) ∼ = Ω odd ( . . . ) . W e therefore hav e na tural transformations a : Ω odd → ˆ K 0 , R : ˆ K 0 → Ω ev cl , where a only preserv es the additiv e structure, while R is m ultiplicative . W e consider the symmetric formal p o w er series in infinitely man y v ariables ˜ c h := X i ≥ 1 ( e x i − 1) ∈ Q [[ x 1 , x 2 , . . . ]] . W e write c h i for the homog e neous comp onen t of degree i . The n there are p olynomials C i ∈ Q [ s 1 , s 2 , . . . ] of degree i (where s i has degree i ) suc h that C i ( c h 1 , . . . , c h i ) = σ i is the i th elemen tary symmetric function in the x i . The p olynomial C i induces a natural transformation C i : Ω ev → Ω 2 i whic h maps the ev en form ω = ω 0 + ω 2 + ω 4 + . . . , ω 2 k ∈ Ω 2 k ( M ) to C i ( ω ) := C i ( ω 2 , . . . , ω 2 i ) ∈ Ω 2 i ( M ) . The follo wing theorem states that the Chern classes ha ve unique lifts to the differential extensions whic h are, in addition, compatible with the group structures. Theorem 1.1 1. F or every i ≥ 1 ther e exists a unique natur al tr ansforma t ion of set- value d functors on smo oth manifolds ˆ c i : ˆ K 0 → d H Z 2 i 2 such that the fol lowing diag r am c ommutes: Ω ev C i / / Ω 2 i ˆ K 0 R O O I   ˆ c i / / d H Z 2 i R O O I   K 0 c i / / H Z i (1) 2. The total class ˆ c = 1 + ˆ c 1 + · · · : ˆ K 0 → d H Z ev, ∗ 1 pr eserves the gr oup structur e. Lifts o f t he Chern classes ha ve previously b een constructed in [Ber08]. The goal of the presen t pa p er is to give a m uc h simpler, mo del-indep enden t treatmen t. F urther new, but not v ery deep, p oin ts of the presen t t heorem a re the assertions ab out unique ness and the second statemen t. Our metho d of pro of is differen t from [Ber08]. It is in fact a sp eciali- sation of a general principle already used in [BS ] and [Bun09] for the construction of lifts of natural transformat io ns b et we en cohomology functors to their differential refinemen ts. In the next t w o paragraphs we connect the differential Chern classes on differen tial K - theory with pr evious constructions of differen tial Chern classes in sp ecific geometric sit- uations. If V := ( V , h V , ∇ V ) is a hermitian v ector bundle with connection ov er a manifold M , then w e ha v e the Cheeger-Simons classes ˆ c C S i ( V ) ∈ d H Z 2 i ( M ) constructed in [CS85]. In the mo del of differen tial K - the ory [BS07] the g eometric bundle is a cycle for a differen tial K -theory class [ V ] ∈ ˆ K 0 ( M ). W e hav e ˆ c i ([ V ]) = ˆ c C S i ( V ) . An ev en geometric family E ov er M (see [Bun02] for this notion) giv es r is e to a Bism ut sup e rconnection A ( E ) on an infinite-dime nsional Hilb ert space bundle H ( E ) ov er M . This sup e rconnection A ( E ) = D ( E ) + ∇ H ( E ) + h i gher terms extends the family of Dirac op erators D ( E ). If the k ernel of D ( E ) is a v ector bundle, then it has an induced metric h k er( D ( E )) and connection ∇ k er( D ( E )) obtained from ∇ H ( E ) b y pro jection. W e th us get an induced geometric bundle H ( E ) = (ke r ( D ( E )) , h k er( D ( E )) , ∇ k er( D ( E )) ) 3 and can define the class ˆ c C S i ( H ( E )) ∈ d H Z 2 i ( M ). One o f the o r ig inal goals of [Bun02], whic h w as not quite ac hiev ed there, w a s to extend this construction to the general case where w e do not hav e a k ernel bundle. Under the a s sumption that index ( D ( E )) ∈ K 0 ( M ) b elongs to the i ’th-step of the Atiy ah-Hirzebruc h filtration (i.e. v anishes after pull- bac k to any i − 1-dimensional complex) in [Bun02, 4 .1 .19] w e constructed a class ˆ c i ( E ) ∈ d H Z 2 i ( M ) 2 suc h that I ( ˆ c i ( E )) = c i ( index ( D ( E ))) . On the other hand, the geometric family E represen ts a differen tial K -theory class [ E , 0] ∈ ˆ K 0 ( M ) in the mo del [BS07], and w e ha ve I ([ E , 0]) = index ( D ( E )). The class ˆ c i ([ E , 0]) ∈ d H Z 2 i ( M ) also satisfies I ( ˆ c i ( E )) = c i ( index ( D ( E ))) and th us giv es a second differential refinemen t of the i ’th Chern class of the index of D ( E ). But in g eneral the class ˆ c i ( E ) differs from ˆ c i ([ E , 0]). This can already b e seen on the lev el of curv atures. Namely , w e hav e R ( ˆ c i ( E )) = R ([ E , 0]) [2 i ] , R ( ˆ c i ([ E , 0])) = C i ( R ([ E , 0])) , where ω [2 i ] denotes the degree-2 i comp onen t o f the fo r m ω . In a sense, the presen t note giv es the right answ er to the problem considered in [Bun02]. Finally we discuss o dd Chern classes. In top ology , the o dd Chern classes c odd i : K − 1 → H Z i are related with the ev en Chern classes b y susp en sion ˜ K 0 (Σ M + ) c i +1 2 / / ∼ =   g H Z i +1 (Σ M + ) ∼ =   K − 1 ( M ) c odd i / / H Z i ( M ) . In the smo oth con text the susp ension isomorphism is replaced b y the in tegratio n R along S 1 × M → M . W e hav e the follo wing o dd counterpart of Theorem 1.1. Theorem 1.2 F or o dd i ∈ N ther e ar e unique natur al tr ansf o rmations ˆ c odd i : ˆ K − 1 → d H Z i such that ˆ K 0 ( S 1 × M ) ˆ c i +1 2 / / R   d H Z i +1 ( S 1 × M ) R   ˆ K − 1 ( M ) ˆ c odd i / / d H Z i ( M ) c ommutes. The tr ansfo rmation in add i tion satisfies I ◦ ˆ c odd i = c odd i ◦ I . 2 Note that in [Bun02] w e index the Chern class es by their degree, where in the present note we adopt the usual co nven tion. 4 Let π : W → B b e a prop er K -o rie n ted map b et we en manifolds. T hen we ha v e an Umk ehr map π ! : K ∗ ( W ) → K ∗− n ( B ), where n = dim ( W ) − dim( B ). An in tegral index theorem is an assertion ab out the Chern classes c ∗ ( π ! ( x )), o r c odd ∗ ( π ! ( x )) fo r x ∈ K ∗ ( W ), e.g. an expression of these classes in terms of the classes c ∗ ( x ) or c odd ∗ ( x ), respective ly . A protot ypical example is giv en in [Mad09]. The construction of differen tial lifts of Chern classes mak es it p ossible to ask for geometric refinemen ts of these kinds of results. An example of such a theorem related to the P faffian bundle w ill b e discussed in a fo rthc oming pap er. 2 Pro ofs Pr o of. Let K 0 ≃ Z × B U b e a represen tative of the homot op y t yp e of the classifying space o f t h e functor K 0 . W e c ho ose b y [BS, Prop 2.1] a sequence of manifolds ( K k ) k ≥ 0 together with maps x k : K k → K 0 , κ k : K k → K k +1 suc h that 1. K k is ho m otop y equiv alen t to an i -dimensional C W - c omplex, 2. κ k : K k → K k +1 is a n em b edding of a closed submanifold, 3. x k : K k → K 0 is k -connected, 4. x k +1 ◦ κ k = x k . Let u ∈ K 0 ( K 0 ) the univ ersal class represen ted b y the iden tity map K 0 → K 0 . By [BS, Prop. 2.6] w e can further c ho ose a seq uence ˆ u k ∈ ˆ K 0 ( K k ) suc h that I ( ˆ u k ) = x ∗ k u and κ ∗ k ˆ u k +1 = ˆ u k for all k ≥ 0. By [BS, Lem. 3.8] and 2 j − 1 < k w e ha v e t h at H 2 j − 1 ( K k , R ) = 0. W e consider the canonical natura l transformation ι R : H Z ∗ → H R ∗ and the de Rham map Rham : Ω ∗ cl → H R ∗ . Since Rham is m ultiplicative w e hav e ι R ( c i ( I ( ˆ u k ))) = C i ( c h ( I ( ˆ u k ))) = C i ( Rham ( R ( ˆ u k ))) = Rham ( C i ( R ( ˆ u k ))) . If we ch o ose k ≥ 2 i , then t he diagra m d H Z 2 i ( K k ) I / / R   H Z 2 i ( K k ) ι R   Ω 2 i cl ( K k ) Rham / / H R 2 i ( K k ) is car t e sian. Hence for k ≥ 2 i there exists a unique class ˆ z i,k ∈ d H Z 2 i ( K k ) suc h that I ( ˆ z i,k ) = c i ( I ( ˆ u k )) , R ( ˆ z i ) = C i ( R ( ˆ u k )) . 5 F urthermore, we ha v e κ k ˆ z i,k +1 = ˆ z i,k . F or k < 2 i w e define z i,k := ( κ ∗ k ◦ · · · ◦ κ ∗ 2 i − 1 ) z i, 2 i . W e no w define the natural transformation ˆ c i . W e start with the observ ation that if ˆ c i exists, t hen it satisfies ˆ c i ( ˆ u k ) = ˆ z i,k . Let ˆ w ∈ ˆ K 0 ( M ). By [BS , Prop. 2.6] w e ha v e K 0 ( M ) ∼ = colim k [ M , K k ], and the underlying class I ( ˆ w ) ∈ K 0 ( M ) can b e written as I ( ˆ w ) = f ∗ x ∗ k u for some k a nd f : M → K k . W e c ho ose a form ρ ∈ Ω odd ( M ) suc h tha t ˆ w = f ∗ ˆ u k + a ( ρ ) . W e consider a fo r m ˜ ρ ∈ Ω odd ([0 , 1] × M ) whic h restricts to ρ o n { 1 } × M and to 0 on { 0 } × M . W e get a class ˜ ˆ w = pr ∗ M ˆ w + a ( ˜ ρ ) ∈ ˆ K 0 ([0 , 1] × M ). Note that ˜ ˆ w |{ 0 }× M = f ∗ ˆ u k , ˜ ˆ w |{ 1 }× M = ˆ w . If ˆ c i exists, t hen w e m ust hav e b y naturality and the homotopy form ula [BS, (1)] ˆ c i ( ˜ ˆ w |{ 0 }× M ) = f ∗ ˆ z i,k , ˆ c i ( ˜ ˆ w |{ 1 }× M ) − ˆ c i ( ˜ ˆ w |{ 0 }× M ) = a ( Z [0 , 1] × M / M R ( ˆ c i ( ˜ ˆ w ))) . F urthermore, by the comm utativity of the upp er square in ( 1 ) w e mus t r equire R ( ˆ c i ( ˜ ˆ w )) = C i ( R ( ˜ ˆ w )) . Therefore we are forced to define ˆ c i ( ˆ w ) := f ∗ ˆ z i,k + a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ))) (2) W e see that if ˆ c i exists, t hen it is automatically unique. Lemma 2.1 The defin i tion of ˆ c i ( ˆ w ) by (2) is indep endent of the choic es of ˜ ρ , ρ and f : M → K k . Pr o of. Let us start with a second c hoice ˜ ρ ′ and write ˜ ˆ w ′ := pr ∗ M ˆ w + a ( ˜ ρ ′ ). Then we can connect ˜ ρ with ˜ ρ ′ b y a family of suc h fo r m s, e.g. the linear path. This pat h can b e considered as a form ¯ ρ on [0 , 1] × [0 , 1] × M . By construction ¯ ρ | [0 , 1] ×{ j }× M is constan t a nd has no comp onen t in the direction of the first v ariable for j = 0 , 1 . This implies that R ( ˜ ˆ w ′ ) | [0 , 1] ×{ j }× M = 0 . (3) W e set ¯ ˆ w := pr ∗ M ˆ w + a ( ¯ ρ ) ∈ ˆ K 0 ([0 , 1] × [0 , 1] × M ). By Stok es theorem w e hav e d Z [0 , 1] × [0 , 1] × M / M C i ( R (( ¯ ˆ w ))) = Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ′ )) − Z [0 , 1] × M / M C i ( R ( ˜ ˆ w )) 6 (these a re the con tributions of the faces { j } × [0 , 1] × M ) since the in t egr a l ov er the other t wo faces [0 , 1] × { j } × M v anishes b y (3). Since a annihilat es exact forms this implies that a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ))) = a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ′ ))) . Assume no w t ha t we hav e c hosen a different ρ ′ . Then a ( ρ ′ − ρ ) = 0 so that b y the exactness axiom [BS , (2)] there ex ists a class ˆ v ∈ ˆ K 1 ( M ) with R ( ˆ v ) = ρ ′ − ρ . Let ˆ e ∈ ˆ K 1 ( S 1 ) be a lift of the generator of K 1 ( S 1 ) ∼ = Z with R ( ˆ e ) = dt . W e consider the form ˜ σ ∈ Ω odd ([0 , 1] × M ) with no dt -comp onen t given by ˜ σ |{ t }× M := Z [0 ,t ] × M / M R ( ˆ e × ˆ v ) , where w e iden tify S 1 ∼ = R / Z and view the interv al [0 , t ] as a subset of S 1 . Then ˜ σ |{ 0 }× M = 0 , ˜ σ |{ 1 }× M = ρ ′ − ρ , d ˜ σ = dt ∧ pr ∗ M R ( ˆ v ) = R ( ˆ e × ˆ v ) . W e now consider ˜ ˆ v := pr ∗ M ˆ w + pr ∗ M a ( ρ ) + a ( ˜ σ ) ∈ ˆ K 0 ([0 , 1] × M ) and calculate mo dulo the ima g e of d Z [0 , 1] × M / M C i ( R ( ˜ ˆ v )) ≡ Z S 1 × M / M C i ( R ( pr ∗ M ( ˆ w )) + pr ∗ M dρ + R ( ˆ e × ˆ v )) ≡ Z S 1 × M / M C i ( R ( pr ∗ M ( ˆ w )) + R ( ˆ e × ˆ v )) ≡ Z S 1 × M / M C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v )) . It follows tha t Rham ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ v ))) = Rham ( Z S 1 × M / M C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v ))) = Z S 1 × M / M Rham ( C i ( R ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v ))) = Z S 1 × M / M ι R ( c i ( I ( pr ∗ M ( ˆ w ) + ˆ e × ˆ v ))) . In other w ords, Rham ( R [0 , 1] × M / M C i ( R ( ˜ ˆ v ))) is an integral class, and this implies a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ v ))) = 0 7 b y [BS, (2)]. If ˜ ρ was the path connecting ρ with 0, then w e cons truct the path ˜ ρ ′ from ρ ′ to 0 b y concatenating ˜ ρ with ˜ σ (in o rde r to concatenate smo othly w e can change ˜ ρ ). Then get ˜ ˆ w ′ := pr ∗ M ˆ w + a ( ˜ ρ ′ ) ∈ ˆ K 0 ([0 , 1] × M ) and a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ′ ))) = a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ))) + a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ v ))) = a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ))) This finishes the verific ation that our construction of c i is indep enden t of the c hoice of ρ . Finally we v erify that ˆ c i ( ˆ w ) is indep en den t of the c hoice of f : M → K k . If w e replace k by k + 1 and f b y κ k ◦ f , then w e ob viously get the same result. F or tw o c hoices f : M → K k and f ′ : M → K k ′ there exis ts k ′′ ≥ max { k , k ′ } suc h that κ k ′′ k ◦ f a nd κ k ′′ k ′ ◦ f ′ are homotopic. Here κ j i : K i → K j denotes for j > i t he comp osition κ j i := κ j − 1 ◦ · · · ◦ κ i . Therefore it remains to sho w that a c hoice f ′ : M → K k homotopic to f : M → K k giv es the same result for ˆ c i ( ˆ w ). Let H : [0 , 1] × M → K k b e a homotop y f rom f to f ′ . Then w e use H in the construction of ˆ c i ( pr ∗ M ˆ w ) ∈ d H Z 2 i ([0 , 1] × M ). If w e let ˆ c ′ i ( ˆ w ) denote the result o f the construction based on the c ho ice of f ′ w e ha v e by the homotopy formula ˆ c ′ i ( ˆ w ) − ˆ c ′ i ( ˆ w ) = a ( Z R ( ˆ c i ( pr ∗ M ˆ w ))) = a ( Z pr ∗ M C i ( ˆ w )) = 0 . ✷ Lemma 2.2 The c onstruction of ˆ c i defines a natur al tr ansformation ˆ c i : ˆ K → d H Z 2 i of set-value d functors on smo oth manifolds. Pr o of. Let g : N → M b e a smo oth map b et w een manifolds. Let ˆ w ∈ ˆ K 0 ( M ) and assume that w e hav e constructed ˆ c i ( ˆ w ) using the choices of f : M → K k , ρ ∈ Ω odd ( M ) and ˜ ρ ∈ Ω odd ([0 , 1] × M ). Then we construct ˆ c i ( g ∗ ˆ w ) using the choices f ◦ g : N → K k and g ∗ ρ ∈ Ω odd ( N ), ( id × g ) ∗ ˜ ρ ∈ Ω odd ([0 , 1] × N ). With these c hoices we ha v e ( id × g ) ∗ ˜ ˆ w = g g ∗ ˆ w ∈ ˆ K 0 ([0 , 1] × N ) and g ∗ ˆ c i ( ˆ w ) = g ∗ f ∗ ˆ z i,k + g ∗ a ( Z [0 , 1] × M / M C i ( R ( ˜ ˆ w ))) = ( f ◦ g ) ∗ ˆ z i,k + a ( Z [0 , 1] × M / M C i ( R (( id × g ) ∗ ˜ ˆ w ))) = ( f ◦ g ) ∗ ˆ z i,k + a ( Z [0 , 1] × M / M C i ( R ( g g ∗ ˆ w ))) = ˆ c i ( g ∗ ˆ w ) . 8 ✷ This finishes the pro of of Assertion 1 of Theorem 1.1. In order to show the second Assertion 2 w e consider the natural transformation ˆ B : ˆ K 0 × ˆ K 0 → d H Z ev giv en b y ˆ B ( ˆ w , ˆ v ) := ˆ c ( ˆ w ) ∪ ˆ c ( ˆ v ) − ˆ c ( ˆ w + ˆ v ) ∈ d H Z ev ( M ) , ˆ w , ˆ v ∈ ˆ K 0 ( M ) . If we apply I w e get I ( ˆ B ( ˆ w , ˆ v )) = I (ˆ c ( ˆ w ) ∪ ˆ c ( ˆ v )) − I ( ˆ c ( ˆ w + ˆ v )) = I ( ˆ c ( ˆ w )) ∪ I ( ˆ c ( ˆ v )) − I ( ˆ c ( ˆ w + ˆ v )) = c ( I ( ˆ w )) ∪ c ( I ( ˆ v )) − c ( I ( ˆ w ) + I ( ˆ v )) = 0 . Let C = 1 + C 1 + C 2 + · · · ∈ Q [[ s 0 , s 1 , . . . ]]. Then we ha ve the iden tit y C ( s 0 + s ′ 0 , s 1 + s ′ 1 , . . . ) = C ( s 0 , s 1 , . . . ) C ( s ′ 0 , s ′ 1 , . . . ) . Indeed, if ˜ c h = X i ≥ 1 ( e x i − 1) , then C ( c h 1 , . . . ) = Y i ≥ 1 (1 + x i ) . If we introduce another set of v a riables x ′ i and set ˜ c h ′ = P i ≥ 1 ( e x ′ i − 1), then C ( c h 1 + ch ′ 1 , c h 2 + ch ′ 2 , . . . ) = Y i ≥ 1 (1 + x i )(1 + x ′ i ) = C ( ch 1 , c h 2 , . . . ) C ( ch ′ 1 , c h ′ 2 , . . . ) . W e now calculate R ( ˆ B ( ˆ w , ˆ v )) = R (ˆ c ( ˆ w ) ∪ ˆ c ( ˆ v )) − R ( ˆ c ( ˆ w + ˆ v )) = R ( ˆ c ( ˆ w )) ∪ R ( ˆ c ( ˆ v )) − R (ˆ c ( ˆ w + ˆ v )) = C ( R ( ˆ w )) ∧ C ( R ( ˆ v )) − C ( R ( ˆ w ) + R ( ˆ v )) = 0 . 9 It follows tha t ˆ B f a c torises ov er the subfunctor H R odd /H Z odd ⊂ H R / Z odd ⊂ d H Z ev , where the inclusion is induced b y a . Let ρ ∈ Ω odd ( M ) and consider ˜ ρ := t pr ∗ M ρ ∈ Ω odd ([0 , 1] × M ). Then w e hav e ˆ B ( ˆ w + a ( ρ ) , ˆ v ) − ˆ B ( ˆ w , ˆ v ) = ˆ B ( pr ∗ M ˆ w + a ( ˜ ρ ) , ˆ v ) |{ 1 }× M − ˆ B ( pr ∗ M ˆ w + a ( ˜ ρ ) , ˆ v ) |{ 0 }× M . Since ˆ B take s v alues in the homotop y in v aria n t subfunctor H R odd /H Z odd w e conclude that ˆ B ( ˆ w + a ( ρ ) , ˆ v ) = ˆ B ( ˆ w , ˆ v ). In a similar manner w e see that ˆ B ( ˆ w , ˆ v + a ( ρ )) = ˆ B ( ˆ w , ˆ v ). Hence ˆ B has a factorisation o ver a natural transformation K 0 × K 0 → H R odd /H Z odd ⊂ H R / Z odd . Suc h a natural transformation b et ween homot o p y inv arian t functors on manifolds must b e represen ted b y a map o f classifying spaces K 0 × K 0 → K ( R / Z , odd ) , where K ( R / Z , odd ) := W i ≥ 0 K ( R / Z , 2 i + 1) is a w edge of Eilen b erg-MacLane spaces, i.e. b y a class in B ∈ H odd ( K 0 × K 0 ; R / Z ). Since K 0 and therefore K 0 × K 0 are ev en spaces w e kno w that H odd ( K 0 × K 0 ; Z ) = 0. It fo llo ws b y the univ ersal co effi cien t formula that H odd ( K 0 × K 0 ; R / Z ) ∼ = Hom ( H odd ( K 0 × K 0 ; Z ) , R / Z ) = 0. W e see that B = 0 and therefore ˆ B = 0. This finishes the pro of of Assertion 2 of Theorem 1.1. ✷ W e now show Theorem 1.2. W e let ˆ e ∈ K 1 ( S 1 ) b e, as ab o v e, the unique elemen t with R ( ˆ e ) = dt , I ( ˆ e ) = e ∈ K 1 ( S 1 ) the canonical generator, and ˆ e |∗ = 0 for a ba s ep oin t ∗ ∈ S 1 . Then we define for o dd i ∈ N and ˆ x ∈ ˆ K − 1 ( M ) ˆ c odd i ( ˆ x ) := Z ˆ c i +1 2 ( ˆ e × ˆ x ) . Note that I ( Z ˆ c i +1 2 ( ˆ e × ˆ x )) = Z c i +1 2 ( e × I ( ˆ x )) . W e ha v e a natural inclusion g H Z ∗ (Σ M + ) ⊂ H Z ∗ ( S 1 × M ) as the subs pa ce of classes whose restriction to {∗ } × M v anishes. Sinc e e |∗ = 0 we see that e × I ( ˆ x ) b elongs to this subspace. The restriction of R to this subspace coincides with the suspension isomorphism g H Z ∗ +1 (Σ M + ) ∼ → H Z ∗ ( M ), R ( e × x ) = x with in v erse x 7→ e × x . Therefore Z c i +1 2 ( e × I ( ˆ x )) = c odd i ( I ( ˆ x )) . 10 In this wa y w e get a natural transformation whic h has the required prop ert y . Since R : ˆ K 0 ( S 1 × M ) → ˆ K − 1 ( M ) is surjectiv e it is clear that ˆ c odd i is unique. ✷ References [Ber08] Alain Berthomieu. A v ersion of smo oth K-theory adapted to the total Chern class, 20 0 8. [BS] U. Bunke a nd Th. Sc hic k. Uniquene ss of smo oth extensions of generalized coho- mology theories. 2 008. [BS07] Ulric h Bunk e and Thomas Schic k. Smo oth K -theory , 20 07. [Bun02] U. Bunk e. Index theory , eta forms, and Deligne cohomology , Memoirs of the AMS , 198 (5 ), 2009. [Bun09] Ulric h Bunk e. Adams o perations in smo o th K-theory , 20 09. [CS85] Jeff Cheeger and James Simons. Differen tial c har acte rs and geometric in v a r ian ts. In Ge ometry and top olo gy (Col le g e Park, Md., 1983/84) , v olume 11 67 of L e ctur e Notes in Math. , pages 50 –80. Springer, Berlin, 1985. [Mad09] I. Madsen. An in tegral Riemann-Ro c h theorem for surface bundles, 200 9 . arXix.org:0901.42 40 11