Cohomotopy invariants and the universal cohomotopy invariant jump formula

COHOMOTOPY INV ARIANTS AND THE UNIVERSAL COHOMOTOPY INV ARIANT JUMP FORMULA CHRISTIAN OKONEK, ANDREI TELEMAN Abstract. Starting from ideas of F uruta, we dev elop a general formalism for the construct ion of cohomotop y inv arian ts asso ciated wi th a certain class of S 1 -equiv ariant non-linear maps b etw een Hilbert bundles. Applied to the Seiberg-Witten map, this formalism yields a new class of coho motop y Seib erg- Witten i n v ariants which hav e clear functorial pr operties with r espect to dif- feomorphisms of 4-manifolds. Our inv arian ts and the Bauer-F u ruta classes are directly compa rable for 4-manifol ds with b 1 = 0; they are equiv alen t when b 1 = 0 and b + > 1, but ar e ﬁner in the case b 1 = 0, b + = 1 (they detect the wa ll-crossing phenomena). W e study f undamen tal prop erties of the new inv arian ts in a ve ry general framework. In particular w e pro v e a univ ersal cohomotop y inv ariant j ump formula and a multiplicativ e prop erty . The f or malism applies to other gauge theoretical problems, e.g. to the theory of gauge theoretical (Hamiltonian) Gromo v-Witten inv ariant s. Contents 1. Int ro duction 2 1.1. Motiv ation 2 1.2. Summary of r esults 4 2. Cohomotopy g roups asso ciated with elements in K ( B ) 6 2.1. Deﬁnition o f S 1 α ∗ B ( X, Y ) 6 2.2. The computation o f S 1 α k ( B + , V + ) 8 2.3. The gro ups α ∗ ( x ) as so ciated with an element x ∈ K ( B ) 10 2.4. The S 1 -equiv aria n t J -map and the descriptio n of α ∗ ( x ) 14 2.5. Stabilization 17 2.6. The cohomo top y Euler class of an element in K ( B ) 1 8 3. Cohomotopy in v ariants asso cia ted with certain non-linear maps betw een Hilber t bundles 19 3.1. The cylinder c onstruction 19 3.2. General prop erties of the inv a riant { µ } 20 3.3. A class o f non-linear ma ps betw een Hilb ert bundles 23 3.4. The Seib erg-Witten map in dimension 4 24 3.5. Finite dimensio na l approximation 27 3.6. Compatibilit y pro per ties 31 4. F undamental prop e rties of the cohomoto py inv ariants 33 4.1. The Hurewicz ima ge of the cohomotopy in v ariant 33 4.2. Cohomotopy inv ariant jump formulae 38 4.3. A pro duct for m ula and a v a nishing theorem 45 5. Appendix 48 1 2 CHRISTIAN OKONEK, ANDREI TELEMAN 5.1. Inductiv e limits of functors 48 5.2. Bundle maps b etw een p ointed sphere bundles 52 References 54 1. Introduction 1.1. Motiv ation. The goa l o f this ar ticle is to develop a ge neral formalism fo r the construction of c o homotopy inv aria n ts ass o cia ted with a certain class of non-linear maps b etw een Hilb ert bundles. The ma in exa mple we have in mind is the Seiber g - Witten map, but the formalism applies to other interesting classes of maps related to gauge theor etical problems a s well. The ﬁrst stable-homotopy Seiber g-Witten inv ar ia n ts hav e b een intro duced in- depe ndently by M. F uruta and S. Bauer. F uruta ﬁr st used “ﬁnite dimensional approximations” of the monop ole map in his work on the 11/8 co njecture [F u1], and then in tro duced a class of reﬁned Seib erg-Witten in v ariants (called “stable ho- motopy version of the Seib erg -Witten inv a riants”) in a g eometric, non-formalized wa y in [F u2]. In this preprint F uruta a c knowledges indep enden t work by B auer [B3]. According to F uruta, the new inv ar iant s b elong to a certain inductive limit of sets of homotopy classes of maps a sso ciated with “ ﬁnite dimensiona l approxi- mations” of the Seiberg -Witten map. The structure and the functorial pr o per ties of this inductiv e limit (with resp ect to diﬀeomorphisms betw een 4-manifolds) hav e not b een w orked out in this a rticle. A precise version of the new inv ariants has bee n int ro duced later by Bauer-F uruta in [BF]: the Bauer-F ur uta classes b elong to ce rtain stable cohomotopy gr oups asso cia ted with a presentation ( E , F ) of the K-theory element ind( 6 D ) deﬁned b y a ﬁxed S pin c -structure. T his ele ment ind( 6 D ) belo ngs to the K-theo ry group K ( B ), where B = H 1 ( X ; R ) /H 1 ( X ; Z ) is the P icard group of the base manifo ld X . In this article w e prop ose a diﬀerent c o nstruction of cohomotopy inv ariants which has the follo wing adv antages: O ur construction yields a larg er class o f inv a riants, which are w ell deﬁned in all c a ses, are alwa ys ﬁner than the clas sical inv ar ia n ts, and have clea r functoria l prop erties. In order to e xplain the adv a n tages of the new formalism in a non-technical w ay , w e consider aga in the Seib erg-Witten case. It is well known that the Seiberg -Witten map µ can b e rega rded as an S 1 - equiv ariant bundle map betw een Hilb ert bundles ov er the torus B (see [BF] and section 3.4 of this article). W e ﬁrst c ho o se the p erturbing form in the second Seibe r g-Witten equation in the “ bad w ay”, i.e. such tha t the equations hav e re- ducible so lutions (so lutions with trivial spinor comp onent); w e make this “bad choice” even in the case b + ( X ) > 1! In “ classical” Seib erg-Witten theory one per turbs the second Seib erg-Witten equa tion using a nontrivial self-dual harmonic form κ ∈ i H + \ { 0 } , and g ets a new map µ κ which deﬁnes a mo duli space whic h do es no t co n tain r eductions. Instead of a constan t p erturbatio n κ , we co nsider a map κ : B → i H + \ { 0 } , and p erturb the Seib erg-Witten map µ (regarded as bundle map ov er B ) using this map. The ass oc iated inv ariant will dep end on the homotopy class [ κ ] ∈ [ B , S ( i H + )]. This leads to the following questions: (1) Does one o btain new inv ariants in this wa y? COHOMOTOPY INV ARIANTS 3 (2) If so, do es one hav e a universal c ohomotopy invariant jump f ormula , i.e . a formula which describ es the jump of the cohomotopy inv ar iant when one passes fro m one homotopy class to another? (3) Use again constant p erturbation forms κ , but let κ v ary in the sphere S ( i H + ) and regard the obtained map ˜ µ a s an S 1 -equiv aria n t bundle map ov er the larg er basis B × S ( i H + ). Do es this universal p ertur bation ˜ µ yield more diﬀeren tial topolo gical informatio n than the individual p erturbations µ κ ? If no t, expr ess the cohomotopy inv ar iant asso ciated with ˜ µ in terms of the inv a riant a s so ciated with µ κ and top ologica l inv ariants of X . These questions are interesting as so on as b 1 ≥ b + − 1 (even for b + > 1!) a nd they are also in teresting for the classic al invariant , b ecause for non-c o nstant pertur bations κ one gets new Seib erg-Witten type mo duli spac e s. The universal wall-cross ing formula [LL], [OO], [OT] for the ful l Seib er g-Witten invariant 1 should be a formal consequence of a universal cohomotopy inv a riant jump formula. These questions will b e completely answered in this a rticle. Another motiv ation for prop osing a new for malism was the need to have wel l deﬁne d inv ariants, with clea r functorial pr op erties . Recall that the classical full Seibe r g-Witten in v a r iant can be regar ded as an element o f [ ∧ ∗ H 1 ( X, Z )] S pin c ( X ) , where S pin c ( X ) denotes the tors or of equiv a lence cla sses o f S pin c -structures. Ther e - fore this inv ariant b elongs to a gr oup which is o b viously functorial with resp ect to pairs ( h, θ ) consisting of a n o rientation preserving homotopy equiv a lence h : X → X ′ , and a bijection θ : S pin c ( X ′ ) → S pin c ( X ) whic h is compatible with the Cher n class maps S pin c ( X ) → H 2 ( X, Z ), S pin c ( X ′ ) → H 2 ( X ′ , Z ) a nd the H 2 ( X, Z ), H 2 ( X ′ , Z )-actions on the tw o sets. Suc h a pair w ill b e called a S pin c -homotopy equiv alence. W e will say that a n assignement X 7→ G ( X ) ∈ A b is top olo gic al ly functorial on the catego ry of s moo th 4- manifolds if it is functor ia l with res pect to S pin c -homotopy equiv alences . It is natural to require that the reﬁned Seib erg- Witten inv aria n t b elongs to a gr o up G ( · ) which is top olog ic ally functoria l, a s it is the case for the class ical inv ariant. In other w ords, we wan t the group to which the inv aria n t b elongs to hav e muc h s tronger functorial prop erties than the inv ar iant it- self. This is imp ortant for pra ctical reasons; for instance, if one wan ts to classify the S pin c -homotopy eq uiv alences X → X which ar e r ealized by diﬀeomorphisms, one will esse ntially need the top olo gic al functoriality of the gr oup to which the inv ariant belo ngs. The deﬁnition of the stable co homotopy gr oup used in [BF] depends on the choice of a pr e s en tation ( E , B × C n ) of ind( 6 D ) ∈ K ( B ) (see [BF] p. 8-9). Since in genera l suc h a pr esentation has homotopically non-trivia l automorphisms, the obtained cohomotopy gr oups cannot b e reg arded as in v ariants of the K-theory el- ement ind( 6 D ) ∈ K ( B ). This makes it diﬃcult to control the functorial pr oper ties of the Bauer-F ur uta s ta ble coho motopy gro ups as deﬁned in [B F] with resp ect to homeomorphisms (or ev en diﬀeomor phisms) of 4-manifolds, and to unders tand in which sense the constructed c lass is well deﬁned. Using Segal co cycles instead of ﬁnite rank presentations ([BF] p. 7 -8) do es not remov e the pro ble m, b ecause of mono dromy phenomena in the spa ce of Seg al 1 The full Seib erg-Witten inv ariant [O T ] is an element in Λ ∗ ( H 1 ( X ; Z )). The numerical Seib erg- Witten inv ar i an t (the original inv ar ian t int ro duced b y Witten) is the degree 0 term of th e full inv ari an t. 4 CHRISTIAN OKONEK, ANDREI TELEMAN co cycles 2 . A similar diﬃcult y concer ns the concept “Thom spectrum of a virtua l bundle”, used by Bauer-F ur uta (see [BF] p. 8) a nd other authors in or der to give a ge ometric interpretation of the Bauer-F ur uta classes. One can indeed as so ciate a Thom sp ectrum to a ﬁxe d pr esen t ation ( E , B × C n ) of a K-theory element x ∈ K ( B ), but unf ortunately not to x itself. F or 4 -manifolds with b 1 = 0, the Bauer-F uruta class giv es a well deﬁned in v ariant, which can easily b e iden tiﬁed with the image of o ur inv ariant under a bounda ry mo rphism o f coho motopy g r oups. The tw o inv aria n ts a re equiv alent when b 1 = 0 , b + > 1 . Note that a n eleg ant constructio n describ ed by F uruta [F u3] leads to a w ell- deﬁned in v ariant belong ing to a group which is C ∞ -functorial for a rbitrary b 1 and b + > 1 . F uruta uses the universal family of Dir ac op er ators asso ciated with a met- ric and a cla ss of S pin c -structures c ∈ S pin c ( X ) in order to remove the a m biguit y in the c hoice of a S pin c -structure τ ∈ c and get a well-deﬁned Segal co cy le. W e will explain this formalis m in section 3.4 . Our new po in t of view has the following adv a ntages: (1) The ne w cohomotopy Seiberg -Witten inv ar iants a re ﬁner than the full clas- sical Seib erg -Witten inv ariants in al l c ases , including the ca se of manifolds with b 1 ≥ b + − 1 and inc luding the inv ariants ass o c iated with non-constant per turbations κ : B → i H 2 + \ { 0 } . In the case b 1 ≥ b + − 1 we prov e a universal cohomotop y inv ariant jump formula; the universal wall-cross ing formula fo r the cla s sical in v ariant is a formal consequence of this result. (2) Our inv a riant b elongs to a cohomo top y gr oup which is intrinsically a s so ci- ated with the bas e 4-manifold, and is to p olo gically functorial in the s ense explained a bove. Remark: An interesting developmen t in cohomo to p y Seib erg-Witten theory con- cerns in v ariants deﬁned fo r families of 4-manifolds parameter ized by a compact space [F u2]. Most parts o f our co nstruction generalize immediately to this situa- tion; note how ev er that in the family ca se the map κ ab ov e has to b e replace d by a section o f a cer ta in sphere bundle. 1.2. Summary o f resul ts. In the ﬁrst s e c tion we constr uct a grade d coho mo top y group asso cia ted with a K-theor y e lemen t x ∈ K ( B ). T o ev ery repres e n tativ e ( E , F ) ∈ x w e asso ciate the g raded gro up S 1 α ∗ B ( S ( E ) + B , F + B ), where S 1 α ∗ B ( · , · ) stands fo r the S 1 -equiv aria n t stable cohomotopy gr o up functor on the catego r y of po in ted S 1 -spaces over B ; it is obta ined by stabilizing with spa c e s of the form ( η ⊕ ξ 0 ) + B , where η is a a co mplex, a nd ξ 0 a r eal bundle. Note that we do not us e all characters of S 1 in the stabilizing proce ss; for this rea s on we do not use the standard notation S 1 ω ∗ B found in the literatur e [CJ]. W e deﬁne α ∗ ( x ) to b e the inductive limit of the graded groups S 1 α ∗ B ( S ( E ) + B , F + B ) with r esp e ct to the category T ( x ) of repr esentativ es ( E , F ) of x . Since T ( x ) is not a small ﬁltering catego ry (see [AM], and section 5 .1 be low) , this limit ca nno t b e o btained using the clas s ical construction. It will b e constructed in tw o steps: First we stabilize the graded group S 1 α ∗ B ( S ( E ) + B , F + B ) with resp ect to standard re pr esentativ e enlargements 2 Con trary to what is often stated in the literature, the space of poss i ble Dirac operators associated with a ﬁxed equiv alence class of S p in c -structures is not contractible (see section 3.4). So even if one considers only S pin c -Dirac op erators, one do es not get a contrac tible space of Segal cocycles. COHOMOTOPY INV ARIANTS 5 ( E , F ) 7→ ( E ⊕ U, F ⊕ U ), and w e obtain a new graded group ˆ α ∗ ( E , F ), which still depe nds on the ﬁxed pair ( E , F ). The groups ˆ α ∗ ( E , F ), ˆ α ∗ ( E ′ , F ′ ) de ﬁned by tw o representatives ( E , F ), ( E ′ , F ′ ) of x a re non-c anonic al ly isomor phic. The group α ∗ ( x ) will b e the q uotien t of ˆ α ∗ ( E , F ) by the equiv alence relation ge ner ated by the inductiv e limit of the auto morphism groups Aut( E ⊕ U ) × Aut( F ⊕ U ). W e give an explicit descr iptio n of α ∗ ( x ) as a q uo tien t of the gr o up ˆ α ∗ ( E , F ) asso cia ted with any r epresentativ e ( E , F ) by the a ction o f the imag e o f the J -homomorphis m S 1 J : K − 1 ( B ) → S 1 α 0 ( B ) × in the gr o up of units S 1 α 0 ( B ) × of the gr ound r ing S 1 α 0 ( B ) := S 1 α 0 B ( B + B , B + B ). In other words, we ar e able to control the eﬀect of bundle automorphisms in our inductive limit and we obtain a gra de d group which is intrinsically ass oc ia ted with the K -theory elemen t x . W e b elieve that this construction is of indep endent int erest from the p oint o f v iew of homotopy theo r y . A wa y to understand the role of the g raded group α ∗ ( x ) is the follo wing: Because of the presence of homotopica lly non-tr iv ial bundle automorphisms, one cannot de- ﬁne the pr o jectivization P ( x ) of a K- theory e le men t x ∈ K ( B ) (neither in the cate- gory of top olog ical spaces nor in the categ ory o f sp ectra). The gr ade d gr oup α ∗ ( x ) plays the r ole of what should b e the c ohomotopy gr oup of a f ormal pr oje ctivizatio n of the K-the ory element x . In the second sec tio n we ﬁr st int ro duce a distinguished clas s o f non- linear maps µ betw een Hilb ert bundles ov er a c ompact base B . The C -linear part of the lin- earization of s uc h a map µ at the zero section is a linear F redho lm o p era tor, so it deﬁnes a K-theory element x ∈ K ( B ). The g o al o f the sec tio n is the construction of an inv a riant { µ } ∈ α ∗ ( x ). This inv ariant is co nstructed using ﬁnite di mensional appr oximations of the map µ . In order to get these approximations we ma k e use of the retra ctions ρ A : A + \ S ( A ⊥ ) → A + asso ciated with ﬁnite dimensional subs pa ces A of a Hilbert space A , as in [BF]. This metho d to construct ﬁnite dimensional approximations a pplies to a v er y large class of non-linear maps, whereas F uruta’s original method base d on L 2 -orthogo nal pro jections on direct sums o f eige nspaces (see [F u2]) is limited to maps whose linearizatio ns are elliptic diﬀerential op erators. The main diﬀerence betw een our deﬁnition and the constructio n of the Ba uer- F uruta cla s ses given in [BF], is tha t (1) our co nstruction uses only space s ﬁb ered over the base B . In par ticular we av oid using Thom space s , (2) w e treat the re al and the complex summands in our ﬁnite dimensional approximations separately . Therefore, fr o m this po in t o f view, our construction is closer to t he original ide as of F ur u ta [F u2]. Having the ﬁnite dimensional approximation, a repr esentativ e of the inv aria n t is an element in a gro up of the for m S 1 α ∗ B ( S ( E ) + B , F + B ) obtained b y a simple ge o metric co ns truction, which we call the cylinder c onst ruction 3 . The class obtained in this way carr ies mo r e infor mation than the o ne deﬁned in [B F]. In 3.4 we show that the Seib erg-Witten map asso ciated with a Spin c -structure τ on a Riemannian 4-manifold M with b + ( M ) > 0 yields a non-linear F redholm map sw κ (depending o n a t wisting map κ : B = H 1 ( X ; R ) /H 1 ( X ; Z ) → i H 2 + \ { 0 } ) which belo ngs to our disting uished clas s of maps. Hence the g eneral theor y applies and 3 After completing the ﬁrst ve rsion of this article we found out ab out the results [C] on Leray- Sc hauder index theory . Under the assumption that the group-action is fr ee on a neighborhoo d of the zero-set of the vect or ﬁeld, one can deﬁne a reﬁnemen t of the usual Poinc ar´ e-Hopf vec tor ﬁeld index, which is probably related to our reﬁnement of the Bauer-F uruta class. 6 CHRISTIAN OKONEK, ANDREI TELEMAN a yields a cohomotopy Seib erg-Witten in v ariant { sw κ } ∈ α b + ( M ) − 1 (ind( 6 D )), which only dep ends of the homoto p y class of κ . Our construction of the bundle map sw κ is diﬀerent from a nd somewhat s impler than the one g iven in [BF]. In this s ection we also explain why the spac e of Dira c oper a tors a s so ciated with a ﬁxed e quiv alence class of S pin c -structures is not contractible, so there is no way to distinguish a co n- tractible class of Segal co cycles deﬁning the K -theory element ind( 6 D ). This makes clear wh y the coho motopy gro ups deﬁned in [BF] (whic h dep end essentially on the choice o f a Sega l co cycle) ca nnot be r egarded as intrinsic (functoria l) inv ar iants of the base manifold. In the third sectio n we prov e several fundamen tal pro p erties of the in v ariant { µ } ∈ α ∗ ( x ) in o ur general, abstr act framework: (1) W e study the image of o ur in v ariant under the Hurewicz morphism, and we pro ve that the Poincar´ e dua l o f this image ca n b e identiﬁed with the virtual fundamen tal class of the v anishing lo cus. In other w ords, the full homology inv ar iant a sso ciated with the virtual fundamental class of the “mo duli spa ce” (i.e. the S 1 -quotient of the v a nishing lo cus of µ ) can be ident iﬁed with the Hurewicz image o f the co homotopy inv ariant. Moreov er, the Hurewicz mor phism is an is omorphism when the “exp ected dimension” v anishes. (2) W e pr ov e a formal universal cohomotopy in v a riant jump formula for our reﬁned co homotopy inv a riants. (3) W e prove a general pro duct formula for the inv ariant { µ 1 × µ 2 } asso cia ted with a pr o duct of maps; in this formu la we allow one o f the factors to have zeros on the S 1 -ﬁxed po in t lo c us. When b oth fa ctors are no where v anishing on their ﬁxed p oint loci, we prov e a v anis hing result for the Hur ewicz image of the in v ariant. Spec ia lized to the Seib erg -Witten map, these prop erties automatically yield im- po rtant res ults for the new cohomotopy Seib erg - Witten inv ar iant s. The ﬁrst r esult shows that the cohomo top y Seib erg- Witten inv a riant is a reﬁne- men t of the clas sical ful l Seiber g-Witten inv a riants in all cas es. Com bined with the s econd prop erty , this als o yields a universal inv a riant jump formula fo r the full classical Seib erg-Witten in v ariant in the case b 1 ( X ) ≥ b + ( X ) − 1. The v a nishing r esult in (2 ) r eprov es the classical v anishing theorem for the Seibe r g-Witten in v ariant of a direct sum in the ca se where b oth summa nds X i hav e b + ( X i ) > 0. The third re s ult provides the top olog ical for ma lism for proving a formula for the cohomotopy inv ariant of a connected sum of tw o 4 -manifolds, even in the ca se when one term of the sum ha s b + = 0 . The analytic techniques r e quired for this gener al gluing formula are not discussed in this article. Ac knowledgmen t: The ﬁrst author w ould like to tha nk T ammo tom Dieck for useful and interesting discussio ns at the b eginning of this pro ject. The s econd author is indebted to Mikio F uruta for useful discuss ions and for expla ining the construction of the “universal Seibe r g-Witten map” descr ibed in sec tio n 3.4. 2. Cohomotopy gr oups associa ted wi th elements in K ( B ) 2.1. Deﬁnition of S 1 α ∗ B ( X, Y ) . Let B b e a compa ct topolo gical space endo wed with the tr ivial S 1 -action. Let C B be the categ ory deﬁned in the following w ay: the COHOMOTOPY INV ARIANTS 7 ob jects of C B are vector bundles over B of the for m ξ = η ⊕ ξ 0 , where η is a complex vector bundle endowed with the sta ndard S 1 -action and ξ 0 is a real vector bundle endow ed with the trivial S 1 -action; for tw o ob jects ξ = η ⊕ ξ 0 , ξ = η ′ ⊕ ξ ′ 0 a morphism ν : ξ → ξ ′ is a pair ( i, ζ ) co nsisting of an S 1 - equiv ariant bundle em b edding i = ι ⊕ i 0 : ξ → ξ ′ and a complement ζ = ν ⊕ ζ 0 of i ( ξ ) in ξ ′ . Composition of morphisms is deﬁned in a natural wa y . A mor phis m u = ( i, ζ ) : ξ → ξ ′ deﬁnes a push-for w ard morphism A ( u ) : A ( ξ ) → A ( ξ ′ ), where A ( ξ ) := A ( η ) × A ( ξ 0 ) is the automo rphism gr oup of ξ . W e obtain in this wa y a functor A : C B → G r . In the terminology of section 5.1, the pair ( C B , A ) is a category with a uto morphism push-forward. Let X → B , Y → B b e tw o p oin ted S 1 -spaces over B . The assig nmen t ξ 7→ S 1 π 0 B ( X ∧ B ξ + B , Y ∧ B ξ + B ) (where S 1 π 0 B ( X, Y ) stands for the set of homo top y classes of S 1 -equiv aria n t base po in t preserving maps over B ) is functorial with resp ect to mor phisms in C B : for a morphism u = ( i, ζ ) : ξ → ξ ′ , the push-fo rward class u ∗ ([ f ]) is deﬁned using i ◦ f ◦ i − 1 on i ( ξ ) and id ζ on its complement ζ . Ther efore this assignment deﬁnes a functor S 1 π 0 B ( X ∧ B · , Y ∧ B · ) : C B → S ets . It is not clea r at all that an inductive limit of this functor exists, b ecause O b ( C B ) is neither a ﬁltering nor a small category (see section 5 .1). Prop osition 2.1. L et ξ = η ⊕ ξ 0 ∈ O b ( C B ) , a = ( α, a 0 ) ∈ A ( ξ ) , and u = ( i, ζ ) the standar d morphism η ⊕ ξ 0 = ξ → ˜ ξ := ( η ⊕ η ) ⊕ ( ξ 0 ⊕ ξ 0 ) deﬁne d by ( y, x ) 7→ (( y , 0) , ( x, 0)) . F or every [ f ] ∈ S 1 π 0 B ( X ∧ B ξ + B , Y ∧ B ξ + B ) one has u ∗ ( a ∗ ([ f ]) = u ∗ ([ f ]) . Pro of: Identifying ˜ ξ with ξ ⊕ ξ o ne can wr ite u ∗ ( a ∗ [ f ]) = [ g ] where g is the comp osition (id X ∧ B [ a ⊕ id ξ ] + B ) ◦ ( f ∧ B id ξ + B ) ◦ (id X ∧ B [ a − 1 ⊕ id ξ ] + B ) : X ∧ B [ ξ ⊕ ξ ] + B → Y ∧ B [ ξ ⊕ ξ ] + B . Let R t be the a utomorphism of ξ ⊕ ξ deﬁned by the matrix r t :=  cos( t π 2 ) − sin( t π 2 ) sin( t π 2 ) cos( t π 2 )  . F or an automorphis m b o f ξ note that r t ◦ ( b ⊕ id ξ ) ◦ r − 1 t deﬁnes a homotopy betw een b ⊕ id ξ and id ξ ⊕ b . This shows that g is homotopic to the map g ′ := (id X ∧ B [id ξ ⊕ a ] + B ) ◦ ( f ∧ B id ξ + B ) ◦ (id X ∧ B [id ξ ⊕ a − 1 ] + B ) = f ∧ B id ξ + B which is a repre s en tative of the cla s s u ∗ ([ f ]). W e deﬁne the sta ble cohomotopy g roup S 1 α 0 B ( X, Y ) by S 1 α 0 B ( X, Y ) := lim − → ( n,m ) ∈ N 2 S 1 π 0 B ( X ∧ B [ C n ⊕ R m ] + B , Y ∧ B [ C n ⊕ R m ] + B ) . In this form ula and in the rest o f the paper w e use the notation V for the trivial bundle B × V o ver the base B . This inductiv e limit has a natural Ab elian g roup structure (see [CJ ] p. 168 for the non-equiv aria n t ca se). 8 CHRISTIAN OKONEK, ANDREI TELEMAN Prop osition 2. 2. The functor S 1 π 0 B ( X ∧ B · , Y ∧ B · ) : C B → S ets admits an inductive limit, which c an b e id entiﬁe d with S 1 α 0 B ( X, Y ) . Pro of: Let N 2 be the small category asso ciated with the o r dered set ( N × N , ≤ ) and consider the functor Θ : N 2 → C B which assig ns to a pair ( n, m ) the trivial bundle C n ⊕ R m ov er B , and to a n inequa lit y ( n, m ) ≤ ( n ′ , m ′ ) the standard morphism b etw e en the corresp onding tr iv ial bundles. Using the terminolo g y of section 5.1, N is a small ﬁltering catego r y , and Θ is a coﬁna l functor from N to the catego ry ( C B , A ), which is a c a tegory with automorphism push-forward. By deﬁnition S 1 α 0 B ( X, Y ) is just the limit o f the co mp osed functor S 1 π 0 B ( X ∧ B · , Y ∧ B · ) ◦ Θ. On the other hand, Prop osition 2 .1 shows that the functor S 1 π 0 B ( X ∧ B · , Y ∧ B · ) satisﬁes the “trivial stable a ctions” axioms TSA, ΘSA. The result follows ther efore from Pro po s ition 5 .11 in section 5.1. Note that Prop osition 2.2 implicitly yields a canonical ma p c ξ : S 1 π 0 B ( X ∧ B ξ + B , Y ∧ B ξ + B ) → S 1 α 0 B ( X, Y ) for every ξ ∈ O ( C B ), such that the s ystem ( c ξ ) ξ ∈ O ( C B ) satisﬁes the universal prop- erty of the inductive limit. As in the non-equiv a riant case we put S 1 α p B ( X, Y ) := S 1 α 0 B ( X ∧ B ( R N ) + B , Y ∧ B ( R N + p ) + B ) ( N , N + p ≥ 0) . Each S 1 α p B ( X, Y ) is a bimo dule ov er the ring S 1 α 0 ( B ) := S 1 α 0 ( B + , S 0 ) = S 1 α 0 B ( B + B , B + B ) , and S 1 α ∗ B ( X, Y ) := ⊕ p ∈ Z S 1 α p B ( X, Y ) is a graded bimodule over the graded ring S 1 α ∗ ( B ) = ⊕ S 1 α p ( B ), where S 1 α p ( B ) := S 1 α p ( B + , S 0 ) = S 1 α 0 ( B + , S p ) . Right and left multiplication with elements in S 1 α 0 ( B ) coincide (see [CJ] p. 172). Remark 2.3. I n t he sp e cial c ase when Y is of t he form Y = ζ + B with ζ ∈ C B , one has a c anonic al identiﬁc ation S 1 α 0 B ( X, ζ + B ) = S 1 α 0  X ∧ B [ ζ ′ ] + B  ∞ , V +  , wher e ζ ⊕ ζ ′ = V , and V has the form C k ⊕ R l . In the terminolo gy of [BF] the latter gr oup is a stable c ohomotopy gr oup forme d with r esp e ct to the universum gener ate d by the S 1 -r epr esentations C and R . 2.2. The computation of S 1 α k ( B + , V + ) . Let S 1 → O ( V ) b e an orthog onal rep- resentation of S 1 . O ur next goa l is the computation of the group S 1 α k ( B + , V + ) for k ≥ 0. In particular, we obtain explicit descr iptions of the po sitive summands S 1 α k ( B ) = S 1 α k ( B + , [ R k ] + ) of the graded r ing S 1 α ∗ ( B ). Replacing V by V ⊕ R k , we can reduce the problem to the case k = 0. One has S 1 α 0 ( B + , V + ) = lim − → ( n,m ) ∈ N 2  B + ∧ [ C n ⊕ R m ] + , V + ∧ [ C n ⊕ R m ] +  S 1 0 , where [ · , · ] S 1 0 stands fo r the set of homotopy classes of S 1 -equiv aria n t maps b etw een t wo p ointed S 1 -spaces. COHOMOTOPY INV ARIANTS 9 According to Hausc hild’s splitting theorem (Satz 3.4 in [H]) there is a natural ident iﬁcation  B + ∧ [ C n ⊕ R m ] + , [ V ⊕ C n ⊕ R m ] +  S 1 0 = (1) h B + ∧ [ R m ] + , [ V S 1 ] + ∧ [ R m ] + i 0 ×  B + ∧  [ C n ⊕ R m ] +  [ R m ] +  , V + ∧ [ C n ⊕ R m ] +  S 1 0 where the pr o jection on the ﬁrst factor is given by res triction to the ﬁxed p oint set. There exists a homeomorphis m of S 1 -spaces [ C n ⊕ R m ] +  [ R m ] + ≈ S ( C n ) + ∧ S m +1 . Indeed, one has [ C n ⊕ R m ] +  [ R m ] + ≈ S ( C n ⊕ R m +1 )  S ( R m +1 ) ≈ ≈ S ( C n ) × D ( R m +1 ) ∪ D ( C n ) × S ( R m +1 )  D ( C n ) × S ( R m +1 ) ≈ S ( C n ) + ∧ S m +1 . Using the natural identiﬁcation B + ∧ [ S ( C n ) + ∧ S m +1 ] ≈ S ( C n ) + ∧ [ B + ∧ S m +1 ] ≈ S ( C n ) × [ B + ∧ S m +1 ]  S ( C n ) × {∗} and denoting by ˜ V n the ass oc ia ted bundle S ( C n ) × S 1 V ov er P ( C n ) we ﬁnd  B + ∧  [ C n ⊕ R m ] +  [ R m ] +  , V + ∧ [ C n ⊕ R m ] +  S 1 0 ∼ = ∼ =  S ( C n ) × [ B + ∧ S m +1 ]  S ( C n ) × {∗} , V + ∧ [ C n ⊕ R m ] +  S 1 0 ∼ = ∼ = S 1 π 0 S ( C n )  S ( C n ) × [ B + ∧ S m +1 ] , S ( C n ) × [ V ⊕ C n ⊕ R m ] +  ∼ = ∼ = π 0 P ( C n )  P ( C n ) × [ B + ∧ S m +1 ] , [ ˜ V n ⊕ O P ( C n ) (1) ⊕ n ⊕ R m ] + P ( C n )  ∼ = ∼ = π 0 P ( C n )   P ( C n ) × [ B + ∧ S 1 ]  ∧ P ( C n ) S m ] , [ ˜ V n ⊕ O P ( C n ) (1) ⊕ n ] + P ( C n ) ∧ P ( C n ) S m  . The limit ov er m of this se t is ω 0 P ( C n )  P ( C n ) × [ B + ∧ S 1 ] , [ ˜ V n ⊕ O P ( C n ) (1) ⊕ n ] + P ( C n )  . Now note tha t ˜ V n ⊕ C ⊕ T P ( C n ) ∼ = ˜ V n ⊕ O P ( C n ) (1) ⊕ n . Therefore, a pplying the duality isomor phis m given in Pr op osition 1 2.41 [CJ] to the map π : P ( C n ) → {∗} , one gets ω 0 P ( C n )  P ( C n ) × [ B + ∧ S 1 ] , [ ˜ V n ⊕ O P ( C n ) (1) ⊕ n ] + P ( C n )  ∼ = ω 0 ( B + ∧ S 1 , π ∗ ([ ˜ V n ⊕ C ] + P ( C n ) )) ∼ = ω 0 ( B + ∧ S 1 , T ( ˜ V n ⊕ C )) ∼ = ω 0 ( B + ∧ S 1 , T ( ˜ V n ) ∧ S 2 ) ∼ = ω 0 ( B + , T ( ˜ V n ) ∧ S 1 ) , where T ( · ) stands for the Thom space functor. This shows that lim − → ( n,m ) ∈ N 2  B + ∧  [ C n ⊕ R m ] +  [ R m ] +  , V + ∧ [ C n ⊕ R m ] +  S 1 0 ∼ = ω 0 ( B + , T ( E S 1 × S 1 V ) ∧ S 1 ) where E S 1 × S 1 V is the v ector bundle asso ciated with the univ ersal S 1 -bundle E S 1 → B S 1 = P ∞ and the ﬁbe r V . Using formula (1) we o btain the following 10 CHRISTIAN OKONEK, ANDREI TELEMAN Prop osition 2.4. O ne has a n atur al gr oup isomorphism S 1 α 0 ( B + , V + ) ∼ = ω 0 ( B + , [ V S 1 ] + ) × ω 0 ( B + , T ( E S 1 × S 1 V ) ∧ S 1 ) . (2) wher e the pr oje ction on the ﬁrst factor is given by r estriction t o the ﬁxe d p oint set. In p articular S 1 α k ( B ) ∼ = ω k ( B ) × ω k ( B + , P ∞ + ∧ S 1 ) . Remark 2. 5. The se c ond sum mand in the de c omp osition S 1 α 0 ( B ) ∼ = ω 0 ( B ) × ω 0 ( B + , P ∞ + ∧ S 1 ) is c al le d “the fr e e su mmand” in [CK] . The pr oje ction S 1 α 0 ( B ) → ω 0 ( B ) is given by r estriction to t he ﬁxe d p oint set, henc e it is a ring homomorph ism. Ther efor e the fr e e sum mand ω 0 ( B + , P ∞ + ∧ S 1 ) is an ide al of S 1 α 0 ( B ) , and one has a n atur al ring isomorphism ω 0 ( B ) ≃ S 1 α 0 ( B )  ω 0 ( B + , P ∞ + ∧ S 1 ) . Corollary 2.6. Supp ose that B is a ﬁn ite CW c omplex. R estriction t o the ﬁx e d p oint set deﬁnes an i somorphism lim − → N ∈ N S 1 α k ( B + , [ C N ] + ) ∼ = − → ω k ( B ) . Pro of: Indeed, taking V = C N ⊕ R k , the seco nd summand in (2) is: ω 0 ( B + , T ( E S 1 × S 1 [ C N ⊕ R k +1 ])) = lim − → l ∈ N π 0 ( B + ∧ [ R l ] + , T ( E S 1 × S 1 [ C N ⊕ R k +1+ l ])) Recall tha t the Tho m space o f a rea l vector bundle of rank r ov er a CW complex X admits a CW deco mpos itio n consisting of a single 0-dimensional cell and cells of dimensio n ≥ r . Ther efore, f or N suﬃcien tly large any ma p B + ∧ [ R l ] + → T ( E S 1 × S 1 [ C N ⊕ R k +1+ l ]) is homo topically trivial. 2.3. The groups α ∗ ( x ) asso ciated with an elem en t x ∈ K ( B ) . Fix an ele ment x ∈ K ( B ). W e de ﬁne a categor y T ( x ) in the following wa y: the o b jects of T ( x ) are the pres en tations o f x . F or tw o such pr e sen tations ( E , F ), ( E ′ , F ′ ), a morphism τ : ( E , F ) → ( E ′ , F ′ ) is a system τ = ( i , j, E 1 , F 1 , k ) co nsisting of bundle mo nomor- phisms j : E ֒ → E ′ , i : F ֒ → F ′ , complemen ts E 1 and F 1 of i ( E ) and j ( F ) in E ′ and F ′ resp ectively , and an isomorphism k : E 1 → F 1 . With every ( E , F ) ∈ x we asso ciate the graded g r oup S 1 α ∗ B ( S ( E ) + B , F + B ). In this formula the sphere bundle S ( E ) is deﬁned b y S ( E ) := ( E \ 0 E ) / R > 0 ; alternatively one can use an arbitrary Hermitian metric on E . W e claim that a morphism τ : ( E , F ) → ( E ′ , F ′ ) induces a morphism τ ∗ : S 1 α ∗ B ( S ( E ) + B , F + B ) − → S 1 α ∗ B ( S ( E ′ ) + B , [ F ′ ] + B ) . Note ﬁrst that, for Euclidean or Hermitian vector s paces V , W , one has a con- traction S ( V ⊕ W ) → S ( V ) + ∧ W + induced by the map c : S ( V ⊕ W ) = [ S ( V ) × D ( W )] ∪ S ( V ) × S ( W ) [ D ( V ) × S ( W )] − → − → S ( V ) × D ( W )  S ( V ) × S ( W ) ≃ S ( V ) × W +  S ( V ) × ∞ W = S ( V ) + ∧ W + . COHOMOTOPY INV ARIANTS 11 It is useful to hav e explicit analytic formulae for the contraction map c . One ca n deﬁne W + in tw o equiv alent ways: as the one-p oint compactiﬁca tion of W , and as the quotient D ( W ) /S ( W ). Acco rdingly , the contraction maps c , c ′ : S ( V ⊕ W ) → S ( V ) + ∧ W + are given b y the formulae: c ( v , w ) =     1 k v k v , 1 √ 1 −k w k 2 w  v 6 = 0 ∗ v = 0 , c ′ ( v , w ) = (  1 k v k v , w  v 6 = 0 ∗ v = 0 . (3) T o save on notations we will still write c instead of c ′ when the second deﬁnition of W + is used. In the presence of a mo r phism τ = ( i, j, E 1 , F 1 , k ) : ( E , F ) → ( E ′ , F ′ ) we choos- ing Hermitian metrics on E ′ and F ′ which make the isomor phisms i , j , k isometr ies and the decomp ositio ns E ′ = i ( E ) ⊕ E 1 , F ′ = j ( F ) ⊕ F 1 orthogo nal. W e get a map S ( E ′ ) + B = S ( i ( E ) ⊕ E 1 ) + B c − → S ( i ( E )) + B ∧ B ( E 1 ) + B , which is w ell deﬁned up to homotopy (the section + B on the left is mapp ed ﬁb erwise to the disting uis hed section o n the right). One obta ins morphisms S 1 α ∗ B ( S ( E ) + B , F + B ) ( i,j ) ≃ − − − − → S 1 α ∗ B ( S ( i ( E )) + B , j ( F ) + B ) = = S 1 α ∗ B ( S ( i ( E )) + B ∧ B ( F 1 ) + B , j ( F ) + B ∧ B ( F 1 ) + B ) = S 1 α ∗ B ( S ( i ( E )) + B ∧ B ( F 1 ) + B , ( F ′ ) + B ) k ≃ S 1 α ∗ B ( S ( i ( E )) + B ∧ B ( E 1 ) + B , ( F ′ ) + B ) c ∗ − − → S 1 α ∗ B ( S ( E ′ ) + B , ( F ′ ) + B ) . The co mpos ition of these maps will b e denoted by τ ∗ . One chec ks that τ ∗ is a morphism of S 1 α ∗ ( B )-mo dules and that, for any tw o comp osable morphisms τ , τ ′ , one has ( τ ′ ◦ τ ) ∗ = τ ′ ∗ ◦ τ ∗ . In o ther words, the ass ignmen t ( E , F ) 7→ S 1 α ∗ B ( S ( E ) + B , F + B ) is functor ial, so it deﬁnes a fun ctor a ∗ x : T ( x ) → A b ∗ , where A b ∗ is the category of graded Ab elian groups. Example: Suppose that the stable class ϕ ∈ S 1 α 0 B ( S ( E ) + B , F + B ) is r epresented by a n S 1 -equiv aria n t map f : S ( E ) → F + B ov er B (or, eq uiv alently , by an S 1 - equiv ariant map S ( E ) + B → F + B of po in ted spaces o ver B ). Let U be a complex vector bundle ov er B a nd let τ b e the obvious mor phism ( E , F ) → ( E ⊕ U, F ⊕ U ). Then f deﬁnes a map [ S ( E ) × B U + B ]  S ( E ) × B ∞ U − → F + B × B U + B  F + B × B ∞ U which, comp osed fr o m the rig ht with the contraction S ( E ⊕ U ) → [ S ( E ) × B U + B ]  S ( E ) × B ∞ U and from on left with the contraction F + B × B U + B  F + B × B ∞ U → F + B × B U + B   F + B × B ∞ U ∪ ∞ F × B U + B  = ( F ⊕ U ) + B gives an S 1 -equiv aria n t map S ( E ⊕ U ) → ( F ⊕ U ) + B ov er B . This ma p r e pr esents τ ∗ ( ϕ ) ∈ S 1 α 0 B ( S ( E ⊕ U ) + B , ( F ⊕ U ) + B ). 12 CHRISTIAN OKONEK, ANDREI TELEMAN Let a ∈ Aut ( E ) b e a unitary gauge tra ns formation of the bundle E . Comp osing with the induced automorphisms S ( a ) o f the sphere bundles S ( E ) + B deﬁnes a morphism S 1 α ∗ B ( S ( E ) + B , F + B ) S ( a ) ∗ − − − − → S 1 α ∗ B ( S ( E ) + B , F + B ) . On the other hand, a deﬁnes an element [ a + B ] ∈ S 1 π 0 B ( E + B , E + B ), w ho se stable class { a + B } is a unit in the ground ring S 1 α 0 ( B ) and deﬁnes m ultiplication a utomorphisms S 1 α ∗ B ( S ( E ) + B , F + B ) m ( a ) − − − − → S 1 α ∗ B ( S ( E ) + B , F + B ) . Clearly these automorphisms dep end only on the homotopy class of a . Prop osition 2.7. L et ϕ ∈ S 1 α ∗ ( S ( E ) + B , F + B ) and a ∈ Aut( E ) . L et τ b e the obvious morphism τ : ( E , F ) → ( E ⊕ E , F ⊕ E ) . In S 1 α ∗ ( S ( E ⊕ E ) + B , [ F ⊕ E ] + B ) is holds τ ∗ ( S ( a ) ∗ ( ϕ )) = τ ∗ ( m ( a )( ϕ )) . Pro of: F or simplicity we prove the statement o nly in degree 0 . W e may ass ume that a is a unitary a utomorphism with respe ct to a Hermitian structure o n E . Suppo se that ϕ is repr esented by [ f ] ∈ S 1 π 0 B ( S ( E ) + B ∧ B ξ + B , F + B ∧ B ξ + B ) . W e will prov e that the natural representativ es p, q ∈ S 1 Map B ( S ( E ⊕ E ) + B ∧ B ξ + B ∧ B E + B , ( F ⊕ E ) + B ∧ B ξ + B ∧ E + B ) of τ ∗ ( S ( a ) ∗ ([ f ])), τ ∗ ( m ( a )([ f ])) ar e homo topic, so they deﬁne the same element in S 1 π 0 B ( S ( E ⊕ E ) + B ∧ B ξ + B ∧ B E + B , ( F ⊕ E ) + B ∧ B ξ + B ∧ B E + B ) . W e supp ose for simplicit y that ξ is triv ial, to sav e on notations. Co nsider the contraction map c : S ( E ⊕ E ) + B → S ( E ) + B ∧ B E + B deﬁned by the ﬁr s t formula in (3), and intro duce the maps Ψ , χ : S ( E ) + B ∧ B E + B ∧ B E + B − → F + B ∧ B E + B ∧ B E + B deﬁned by Ψ := [ f ◦ S ( a )] ∧ B id E + B ∧ B id E + B , χ := f ∧ B id E + B ∧ B a + B . Using our deﬁnitions it is easy to se e that p = Ψ ◦ ( c ∧ B id E + B ), q = χ ◦ ( c ∧ B id E + B ). Use the same metho d as in the pro of of P rop osition 2.1 (co njugation with the rotations of E ⊕ E deﬁned b y the matr ic e s r t ) to construct a ho motopy χ = f ∧ B (id E ⊕ a ) + B ≃ f ∧ B ( a ⊕ id E ) + B = f ∧ B a + B ∧ B id E + B := χ ′ . It suﬃces to construct a ho motopy b et ween Ψ ◦ ( c ∧ B id E + B ), and χ ′ ◦ ( c ∧ B id E + B ), and for this it suﬃces to construct a homotopy b etw een the maps Ψ 0 ◦ c and χ ′ 0 ◦ c , where Ψ 0 := [ f ◦ S ( a )] ∧ B id E + B = ( f ∧ B id E + B ) ◦ ( S ( a ) ∧ B id E + B ) , χ ′ 0 := f ∧ B a + B = ( f ∧ B id E + B ) ◦ (id E + B ∧ B a + B ) . Note that ( S ( a ) ∧ B id E + B ) ◦ c = c ◦ S ( a ⊕ id E ), a nd (id S ( E ) ∧ B a + B ) ◦ c = c ◦ S (id E ⊕ a ). In these form ula e we use the fact that a is a unitary . O n the other hand, using again conjugation with the ro tations deﬁned b e the ma trices r t , we see that S ( a ⊕ id) ≃ S (id E ⊕ a ). There fo re Ψ 0 ◦ c = ( f ∧ B id E + B ) ◦ c ◦ S ( a ⊕ id) ≃ ( f ∧ B id E + B ) ◦ c ◦ S (id E ⊕ a ) = COHOMOTOPY INV ARIANTS 13 = ( f ∧ B id E + B ) ◦ (id S ( E ) ∧ B a + B ) ◦ c = χ ′ 0 , which completes the pro of. A similar statement holds fo r the action of an a utomorphism b ∈ Aut( F ). Denote by [ b + B ] ∗ the automorphism of S 1 α ∗ ( S ( E ) + B , F + B ) deﬁned by comp osition with b + B . Prop osition 2.8 . The automorphisms [ b + B ] ∗ , m ( b ) c oincide on S 1 α ∗ ( S ( E ) + B , F + B ) . The pro of us es similar arguments as the pro of of Prop osition 2 .7 but is substan- tially easier . An a utomorphism c ∈ Aut ( U ) deﬁnes a automorphism σ ( c ) of the gr aded gr oup α ∗ ( S ( E ⊕ U ) + B , [ F ⊕ U ] + B ) deﬁned b y f 7→ [id F ⊕ c ] + B ◦ f ◦ S (id E ⊕ c ) − 1 . Corollary 2 .9. L et τ : ( E ⊕ U, F ⊕ U ) → ( E ⊕ U ⊕ E ⊕ U, F ⊕ U ⊕ E ⊕ U ) b e the natur al morphism. Then for any ϕ ∈ α ∗ ( S ( E ⊕ U ) + B , [ F ⊕ U ] + B ) one has τ ∗ ( σ ( c )( ϕ )) = τ ∗ ( ϕ ) . Pro of: Indeed, one has τ ∗ ◦ { [id F ⊕ c ] + B } ∗ = τ ∗ ◦ m ( c ) , τ ∗ ◦ { S (id E ⊕ c ) − 1 } ∗ = τ ∗ ◦ ( m ( c ) − 1 ) . On the o ther hand the morphism τ ∗ is S 1 α 0 ( B )-linear . Consider now the category U B of all ﬁnite rank complex vector bundles ov er B . A morphism ν : U → U ′ in the catego ry U B is a pair ( i , U 1 ) c onsisting of a bundle embedding i : U → U ′ and a complemen t U 1 of i ( U ) is U ′ . This category can b e regar ded in an ob vious way as a category with automor phis m push-forward (see section 5.1). The assignment U 7→ S 1 α ∗ B ( S ( E ⊕ U ) + B , ( F ⊕ U ) + B ) is functorial with resp ect to mor phisms in U B , so it deﬁnes a functor a ∗ E ,F : U B → A b ∗ . Since U B is not a s ma ll catego ry , it is not c le a r whether this functor ha s an inductiv e limit (see sections 2.1, 5 .1). W e put ˆ α ∗ ( E , F ) := lim − → n ∈ N S 1 α ∗ B ( S ( E ⊕ C n ) + B , ( F ⊕ C n ) + B ) . (4) Prop osition 2.1 0. The fun ctor a ∗ E ,F admits an inductive limit which c an b e iden- tiﬁe d with ˆ α ∗ ( E , F ) . Pro of: Let N b e the category asso ciated with the ordered set ( N , ≤ ) and let Θ : N → U B be the coﬁnal functor n 7→ C n (see sectio n 5 .1). By Cor ollary 2.9, the functor a ∗ E ,F satisﬁes the trivial s ta ble action axiom ΘSA. The r esult follows now from Pro po s ition 5 .11 in section 5.1. In particular one has ca nonical morphisms c U : S 1 α ∗ B ( S ( E ⊕ U ) + B , ( F ⊕ U ) + B ) → ˆ α ∗ ( E , F ) for any co mplex bundle U , and the system ( c U ) U is a ∗ E ,F -compatible and satisﬁes the universal prop erty of the inductiv e limit. Note that ˆ α ∗ ( E , F ) ha s a natural str ucture of a gra ded S 1 α ∗ ( B ) bimodule. By Prop ositions 2.7 and 2.8 w e get: Remark 2.11. The action of t he gauge gr oups Aut( E ⊕ U ) , Aut( F ⊕ U ) on ˆ α ∗ ( E , F ) is induc e d by t he c anonic al S 1 α 0 ( B ) × -action deﬁne d by its mo dule struct u r e via t he morphisms Aut( E ⊕ U ) → S 1 α 0 ( B ) × , Aut( F ⊕ U ) → S 1 α 0 ( B ) × deﬁne d by a 7→ a + B . A morphism τ = ( i , j, E 1 , F 1 , k ) : ( E , F ) → ( E ′ , F ′ ) betw een t wo presentations ( E , F ), ( E ′ , F ′ ) of x induces a sequence a morphisms ( E ⊕ C n , F ⊕ C n ) → ( E ′ ⊕ 14 CHRISTIAN OKONEK, ANDREI TELEMAN C n , F ′ ⊕ C n ), so it induces a morphism ˆ τ ∗ : ˆ α ∗ ( E , F ) ≃ − → ˆ α ∗ ( E ′ , F ′ ). It is eas y to see that ˆ τ ∗ is an isomorphism: it suﬃces to note that the exists an isomorphism θ : ( E ′ , F ′ ) → ( E ⊕ U, F ⊕ U ) (with U := E 1 ) such that θ ◦ τ is the s tandard morphism ( E , F ) → ( E ⊕ U , F ⊕ U ), and to apply Pr op osition 2.10. Ther efore we obtain a functor ˆ a ∗ x : T ( x ) → A b ∗ whose as so ciated morphisms ˆ a ∗ x ( τ ) = ˆ τ ∗ are a ll isomorphisms. Acco rding to Prop osition 5.8 an inductive limit of this functor exis ts and ca n b e iden tiﬁed with a quotient o f ˆ α ∗ ( E , F ), for any ﬁxed presentation ( E , F ) of x . Therefore we ca n make Deﬁnition 2 .12. Deﬁne α ∗ ( x ) := lim − → ( E ,F ) ∈ x ˆ α ∗ ( E , F ) . Remark 2.13. This inductive limit is also an inductive limit of t he functor a ∗ x intr o duc e d at the b e ginning of this se ction. The existenc e of the inductive limit of this functor is a n on-trivial s t atement. W e intro duce now the notations A ( E ) := lim − → N ∈ N Aut( E ⊕ C N ) , A ( F ) := lim − → N ∈ N Aut( F ⊕ C N ) . The tw o gr oups A ( E ), A ( F ) act on the g raded group ˆ α ∗ ( E , F ) via the gro up morphisms l : A ( E ) → S 1 α 0 ( B ) × , r : A ( F ) → S 1 α 0 ( B ) × (see Rema rk 2 .11), so the t w o actions co mm ute. Let Z [ A ( E )], Z [ A ( F )] b e the gro up rings of A ( E ), A ( F ), I [ A ( E )], I [ A ( F )] the a ugmen tation ideals, and λ : Z [ A ( E )] → S 1 α 0 ( B ), ρ : Z [ A ( F )] → S 1 α 0 ( B ) the r ing morphisms as so ciated with the g roup morphisms l , r . Using P rop osition 5.8 we get Remark 2.14. F or every pr esentation ( E , F ) ∈ x ther e is a c anonic al isomorph ism α ∗ ( x ) ≃ − → ˆ α ∗ ( E , F ))  λ ( I [ A ( E )]) ˆ α ∗ ( E , F ) + ρ ( I [ A ( F )]) ˆ α ∗ ( E , F ) . In the next section w e will see that A ( E ), A ( F ) are bo th iso morphic to K − 1 ( B ) and we will identify the images λ ( I [ A ( E )]), ρ ( I [ A ( F )]) o f the tw o ideals in S 1 α 0 ( B ) with the image of the ideal I [ K − 1 ( B )] under the r ing morphism Z [ K − 1 ( B )] → S 1 α 0 ( B ) induced by the J - map K − 1 ( B ) → S 1 α 0 ( B ) × . 2.4. The S 1 -equiv arian t J -map and the description of α ∗ ( x ) . Let π : E → B be a Her mitian vector bundle ov er a compa c t bas is, and let a , b ∈ Aut( E ) b e tw o unitary automo rphisms. W e deﬁne a map ∆ E ( a, b ) : S ( E ) + B ∧ B S 1 − → E + B in the following wa y: W e use the mo dels S ( E ) + B ∧ B S 1 ∼ = S ( E ) × [0 , 1]  S ( E ) × { 0 , 1 } , E + B ∼ = D ( E )  B S ( E ) for the t w o spaces, and deﬁne ∆ E ( a, b )([ e, t ]) :=  [(1 − 2 t ) a ( e )] for 0 ≤ t ≤ 1 2 [(2 t − 1) b ( e )] for 1 2 ≤ t ≤ 1 . Consider the co n traction map c E : E + B − → S ( E ) + B ∧ B S 1 COHOMOTOPY INV ARIANTS 15 induced by e 7→ [( 1 k e k e, k e k )]. One ha s { ∆ E ( a, b ) } = { b + B } − { a + B } . (5) Deﬁnition 2.1 5. The J- homomorp hism asso ciate d with a Hermitian bund le E is the morphism J E : π 0 (Aut( E )) → S 1 α 0 B ( B ) × deﬁne d by J E ([ a ]) := { a + B } . W e intro duce the map Θ E : π 0 (Aut( E )) − → S 1 α − 1 B ( S ( E ) + B , E + B ) , Θ E ([ a ]) := { ∆ E (id E , a ) } . Let ∂ E : S 1 α − 1 B ( S ( E ) + B , E + B ) → S 1 α 0 B ( E + B , E + B ) b e the co nnecting morphism in the long exact cohomotopy sequence: · · · → S 1 α − 1 B ( S ( E ) + B , E + B ) ∂ E → S 1 α 0 B ( E + B , E + B ) → S 1 α 0 B ( B + B , E + B ) → . . . (6) asso ciated with E + B and the co ﬁber sequence S ( E ) + B − → D ( E ) + B − → E + B . Since ∂ E acts by comp osition with the contraction c E , we see that the diagra m π 0 (Aut( E )) ✲ Θ E S 1 α − 1 B ( S ( E ) + B , E + B ) ❄ ❄ ∂ E J E · − 1 S 1 α 0 ( B ) × ✲ S 1 α 0 ( B ) = S 1 α 0 B ( E + B , E + B ) (7) is commutativ e . Remark 2.16 . L et ω 0 ( B + , P ∞ + ∧ S 1 ) ⊂ S 1 α 0 ( B ) b e the fr e e summand of the ring S 1 α 0 ( B ) (se e Pr op osition 2.4 ). F or any [ a ] ∈ π 0 (Aut( E )) it holds J E ([ a ]) − 1 ∈ ω 0 ( B + , P ∞ + ∧ S 1 ) . Indeed, ω 0 ( B + , P ∞ + ∧ S 1 ) is the kernel o f the morphism ρ : S 1 α 0 ( B ) → ω 0 ( B ) given by restriction to the ﬁxed p oint set. Ther efore ρ ( J E ([ a ])) = ρ ( { a + B } ) = { ( a + B ) S 1 } = { id B + B } . Prop osition 2.17. One has (1) lim − → N π 0 (Aut( E ⊕ C N )) = K − 1 ( B ) (2) The system of morphisms ( ∂ E ⊕ C N ) N ∈ N deﬁnes an isomorphisms ∂ : lim − → N S 1 α − 1 B ( S ( E ⊕ C N ) + B , [ E ⊕ C N ] + B ) − → ω 0 ( B + , P ∞ + ∧ S 1 ) . Pro of: Let Φ be a complex bundle on B . F or an y automorphism a ∈ Aut(Φ) we construct a bundle Φ a ov er B × S 1 in the following wa y: we co ns ider the bundle Φ × [0 , 1 ] ov er B × [0 , 1] and w e identify Φ × { 0 } with Φ × { 1 } via a . This bundle comes with an obvious iden tiﬁcation Φ a B ×{ 0 } ≃ p ∗ B (Φ) B ×{ 0 } , so the diﬀerence [Φ a ] − [p ∗ B (Φ)] deﬁnes an element k Φ ( a ) ∈ K ( B × S 1 , B × { 0 } ). It is easy to see that the obtained map k Φ : Aut(Φ) → K ( B × S 1 , B × { 0 } ) = K − 1 ( B ) is a g roup 16 CHRISTIAN OKONEK, ANDREI TELEMAN morphism. T aking the limit ov e r N o f the system o f mor phisms k E ⊕ C N we obtain a morphism κ E : lim − → N π 0 (Aut( E ⊕ C N )) → K − 1 ( B ) . Let E ′ be a complement of E a nd ﬁx an iso mo rphism E ′ ⊕ E ∼ = C n . The assignment a 7→ id E ′ ⊕ a deﬁnes an inje ctive morphism i E ′ : lim − → N π 0 (Aut( E ⊕ C N )) → lim − → N π 0 (Aut( C n + N )) . Similarly , we obtain an obvious inje ctive morphis m j E : lim − → N π 0 (Aut( C N )) → lim − → N π 0 (Aut( E ⊕ C N )) . Hence we hav e morphisms lim − → N π 0 (Aut( C N )) j E → lim − → N π 0 (Aut( E ⊕ C N )) i E ′ → lim − → N π 0 (Aut( C n + N )) κ C n − − − → K − 1 ( B ) . The co mpos ition i E ′ ◦ j E is clearly an isomor phism. Moreov er, it is well-kno wn that κ C n is an isomorphism, fo r every n ∈ N . Since i E ′ is injective, we see that κ E = κ C n ◦ i E ′ is injective. On the other hand, κ C n ◦ i E ′ ◦ j E = κ E ◦ j E is a n isomorphism, so κ E is also surjective. F or the second isomorphism, we take the direct limit ov er N in the cohomotopy exact sequence (6) asso cia ted with E ⊕ C N . W e hav e lim − → N S 1 α k B ([ E ⊕ C N ] + B , [ E ⊕ C N ] + B ) = S 1 α k ( B ) . On the other hand, the system of mor phisms deﬁned by restriction to the ﬁxed po in t set (see s ection 2.2) deﬁnes a morphism r k E : lim − → N S 1 α k B ( B + B , [ E ⊕ C N ] + B ) → ω k ( B + , S 0 ) = ω k ( B ) . Using aga in a complement E ′ of E as ab ov e, we obtain morphisms lim − → N ∈ N S 1 α k ( B + , [ C N ] + ) = lim − → N ∈ N S 1 α k B ( B + B , B × [ C N ] + ) → → lim − → N S 1 α k B ( B + B , [ E ⊕ C N ] + B ) → lim − → N S 1 α k B ( B + B , [ C n + N ] + B ) r k C n − − − → ω k ( B ) . The morphism lim − → N ∈ N S 1 α k B ( B + B , B × [ C N ] + ) → lim − → N S 1 α k B ( B + B , [ C n + N ] + B ) is a n iso- morphism, and lim − → N S 1 α k B ( B + B , [ E ⊕ C N ] + B ) → lim − → N S 1 α k B ( B + B , [ C n + N ] + B ) is injectiv e. Moreov er, by Corollar y 2.6, the map r k C n is an isomor phism. Now the same argu- men ts as ab ove show tha t r k E is an isomorphism. The limit of (6) b e comes S 1 α − 1 ( B ) ρ − 1 → ω − 1 ( B ) → lim − → N S 1 α − 1 B ( S ( E ⊕ C N ) + B , [ E ⊕ C N ] + B ) ∂ → S 1 α 0 ( B ) ρ → ω 0 ( B ) But the map ρ − 1 : S 1 α − 1 ( B ) = S 1 α 0 ( B + ∧ S 1 ) → ω 0 ( B + ∧ S 1 ) = ω − 1 ( B ) is a lso induced by restrictio n to the ﬁxed p oint set, so it is surjective by Remark 2.5 applied to the basis B + ∧ S 1 . Therefore ∂ induce s an iso morphism lim − → N S 1 α − 1 B ( S ( E ⊕ C N ) + B , [ E ⊕ C N ] + B ) ∼ = − → k er( ρ ) = ω 0 ( B + , P ∞ + ∧ S 1 ) . COHOMOTOPY INV ARIANTS 17 T aking the inductiv e limit w ith resp ect to N o f the diagra m (7) written for E ⊕ C N , we obtain the co mm utativ e diagra m K − 1 ( B ) ✲ Θ lim − → N S 1 α − 1 B ( S ( E ⊕ C N ) + B , [ E ⊕ C N ] + B ) = ˆ α − 1 ( E , E ) ❄ ❄ ∂ ≃ J · − 1 S 1 α 0 ( B ) × S 1 α 0 ( B ) . ✲ ι ֒ → ω 0 ( B + , P ∞ + ∧ S 1 ) (8) Remark 2. 18. The map ι ◦ ∂ ◦ Θ : K − 1 ( B ) → S 1 α 0 ( B ) satisﬁes the identity [ ι ◦ ∂ ◦ Θ ]( a + b ) = [ ι ◦ ∂ ◦ Θ]( a )[ ι ◦ ∂ ◦ Θ]( b ) + [ ι ◦ ∂ ◦ Θ]( a ) + [ ι ◦ ∂ ◦ Θ]( b ) . It is the “fr e e J-map” in the terminolo gy of Cr abb-Knapp ( [CK] , p. 88, p.93). Corollary 2.19. The map J : K − 1 ( B ) → S 1 α 0 ( B ) × is inje ctive. Pro of: It suﬃces to note that ∂ ◦ Θ is injective b y Corolla ry 2 .5 in [CK]. The group mo rphism J extends to a ring morphism ˜ J : Z [ K − 1 ( B )] → S 1 α 0 ( B ). Question: Do es t he sub gr oup ˜ J ( I [ K − 1 ( B )]) = h{ J ( u ) − 1 | u ∈ K − 1 ( B ) }i = h im ( ∂ ◦ Θ) i ⊂ ω 0 ( B + , P ∞ + ∧ S 1 ) c oincide with the fr e e summand ω 0 ( B + , P ∞ + ∧ S 1 ) ? W e come back to the description of α ∗ ( x ): Using Remar ks 2.11 and 2.14 o ne gets the following descriptions of α ∗ ( x ). Prop osition 2. 20. F or every pr esentation ( E , F ) ∈ x t her e exist c anonic al iso- morphisms α ∗ ( x ) ∼ = ˆ α ∗ ( E , F )  ˜ J ( I [ K − 1 ( B )]) ˆ α ∗ ( E , F ) . Sinc e ˜ J ( I [ K − 1 ( B )]) is c ontaine d in ω 0 ( B + , P ∞ + ∧ S 1 ) , which is an ide al of S 1 α 0 ( B ) , we get epimorphi sms α ∗ ( x ) − → ˆ α ∗ ( E , F )  ω 0 ( B + , P ∞ + ∧ S 1 ) · ˆ α ∗ ( E , F ) . 2.5. Stabilization. In this section we will show that the mo rphism τ ∗ : S 1 α k B ( S ( E ) + B , F + B ) → S 1 α k B ( S ( E ′ ) + B , [ F ′ ] + B ) (9) asso ciated with a morphism τ : ( E , F ) → ( E ′ , F ′ ) in the categor y T ( x ) is an isomorphism as so on as the rank f of F is suﬃcien tly lar ge. In other w ords, for ﬁxed k , the groups α k ( x ) can be computed using only pr esentations ( E , F ) with a priori b ounded r a nks. Prop osition 2. 21. Supp ose that B is a ﬁnite CW c omplex. The stabilization morphism (9) is an isomorphism fo r 2 f ≥ dim( B ) − k . 18 CHRISTIAN OKONEK, ANDREI TELEMAN Pro of: A mo rphism τ deﬁnes a bundle U a nd isomorphisms E ′ ∼ = E ⊕ U , F ′ ∼ = F ⊕ U . The long exa ct sequence ass o c iated with the co ﬁb er sequence ov er B S ( U ) + B − → S E ′ + B c − → S ( E ) + B ∧ B U + B , and the tar get space [ F ′ ] + B contains the segment → S 1 α k − 1 B ( S ( U ) + B , [ F ′ ] + B ) ∂ − → S 1 α k B ( S ( E ) + B ∧ B U + B , [ F ′ ] + B ) c ∗ − − → S 1 α k B ( S E ′ + B , [ F ′ ] + B ) . The morphism τ ∗ is deﬁned by c ∗ via the identiﬁcation S 1 α k B ( S ( E ) + B , F + B ) = S 1 α k B ( S ( E ) + B ∧ B U + B , [ F ′ ] + B ), so it is an isomorphism as so on as S 1 α k − 1 B ( S ( U ) + B , [ F ′ ] + B ) = S 1 α k B ( S ( U ) + B , [ F ′ ] + B ) = 0 . Suppo se for s implicity k ≥ 0. A class u ∈ S 1 α k B ( S ( U ) + B , [ F ′ ] + B ) is represent ed b y an S 1 -equiv aria n t p ointed map ov er B ϕ : S ( U ) + B ∧ B ξ + B = S ( U ) × B ξ +  B S ( U ) × B ∞ ξ − → [ F ′ ⊕ R k ⊕ ξ ] + B , where ξ = η ⊕ ξ 0 is the sum of a co mplex and a real vector bundle. W e may suppo s e that ξ 0 is an oriented bundle, s o that all o ur bundles b ecome o riented bundles. W e will pr ov e that any such map is homotopic to the map ϕ ∞ which maps the left hand s pace ﬁb erwise onto the section ∞ F ′ ⊕ R k ⊕ ξ . Denote by q : P ( U ) → B the bundle pro jection and put ˜ F ′ := q ∗ ( F ′ )(1) , ˜ ξ := q ∗ ( η )(1) ⊕ q ∗ ( ξ 0 ) . A map ϕ as ab ov e induces a p ointed bundle map ˜ ϕ : ˜ ξ + P ( U ) − → [ ˜ F ′ ⊕ R k ⊕ ˜ ξ ] + P ( U ) ov er P ( U ), and the a ssignment ϕ 7→ ˜ ϕ is a bijection. But by Corollar y 5.15 in s ection 5 .2, any suc h p ointed bundle map is homo topic to the ﬁb erwise consta nt bundle map as so on as dim R ( P ( U )) + rk( ˜ ξ ) < r k R ( ˜ F ′ ) + k + rk( ˜ ξ ). This condition is equiv alent to 2 f > dim( B ) − k − 2. Similarly , we will have S 1 α k − 1 B ( S ( U ) + B , [ F ′ ] + B ) = 0 a s so on as 2 f > dim( B ) − k − 1. 2.6. The cohomotopy Eul er class of an ele men t in K ( B ) . Let x ∈ K ( B ) and consider a pres en tation ( E , F ) ∈ x . The map o ( E ,F ) : S ( E ) + B → F + B which sends the section + B of S ( E ) + B to the inﬁnity section of F + B and maps any p oint e b ∈ S ( E b ) to 0 b is an S 1 -equiv aria n t map of p ointed spaces ov er B , hence it deﬁnes an element { o ( E ,F ) } ∈ S 1 α 0 B ( S ( E ) + B , F + B ). One has a canonica l is omorphism (see [CJ ] Prop osition 12.4 0) S 1 α 0 B ( S ( E ) + B , F + B ) ∼ = S 1 α 0 S ( E ) ( S ( E ) + S ( E ) , π ∗ ( F ) + S ( E ) ) , where π : S ( E ) → B is the ob vious pro jection. Under this isomorphism the class { o ( E ,F ) ] } maps to the equiv ariant Euler class of the bundle π ∗ ( F ) over S ( E ). This class is the pull- back o f the equiv ariant Euler cla ss γ ( F ) ∈ S 1 α 0 ( B + B , F + B ) of the bundle F under the pro jection S ( E ) + S ( E ) → B + B . F or any morphism τ = ( i, j, E 1 , F 1 , k ) : ( E , F ) → ( E ′ , F ′ ) in the ca teg ory T ( x ) one has τ ∗ ( { o ( E ,F ) } ) = { o ( E ′ ,F ′ ) } . Therefor e the a ssignment ( E , F ) 7→ −{ o ( E ,F ) } deﬁnes a tautolo gic al element γ ( x ) ∈ α ∗ ( x ). This elemen t will be called the equi- v ariant coho motopy Euler c lass of x . COHOMOTOPY INV ARIANTS 19 3. Cohomotopy inv ariants associa ted with cer t ain non-linear maps between Hilber t bundles 3.1. The cylinder c onstruction. Let ( E , F ) be a pair o f Hermitian v ector bun- dles ov er a compact basis B . Let V , W b e Euclidean vector spaces, a nd let µ : E × V → [ F × W ] + B be an S 1 -equiv aria n t map ov er B . W e supp ose that µ is ﬁb erwise diﬀerentiable and its ﬁber wise diﬀerential is contin uous on E × V . The equiv ariance pro per t y implies that µ (0 E × V ) ⊂  0 F × W  + B . (10) W e a ssume that µ has the following prop erties : P1: (prop erness) There exist p ositive constants c , C such that k µ ( e, v ) k > c for all pairs ( e, v ) ∈ E × V with k ( e , v ) k ≥ C . P2: (restriction to the S 1 -ﬁxed p oint set) (1) There ex is ts a direct sum decomp osition W = H ⊕ W 0 such that µ (0 E y , v ) = h ( y ) + l ( v ) , ∀ y ∈ B , ∀ v ∈ V , where l : V ≃ − → W 0 ⊂ W is a linear isomor phism, which do es not depe nd on y , and h : B → H is a co ntin uous map. (2) There ex is ts ε 0 > 0 such that k h ( y ) k = k p H ( µ (0 E y , v )) k ≥ ε 0 ∀ ( y , v ) ∈ B × V . (11) W e ﬁx an orientation O of H , and s e t b := dim( H ). Cho ose num ber s R ≥ C and ε ≤ min( c, ε 0 ). The restriction µ R of µ to D R ( E ) × D R ( V ) satisﬁes k µ ( e, v ) k ≥ ε ∀ ( e, v ) ∈ ∂ [ D R ( E ) × D R ( V ))] ∪  0 E × D R ( V )  . Therefore, µ R deﬁnes a n S 1 -equiv aria n t mor phism of pair s ov er B µ R,ε :  D R ( E ) × D R ( V ) , ∂ [ D R ( E ) × D R ( V )] ∪ [0 E × D R ( V )]  − → − →  [ F × W ] + B , [ F × W ] + B \ ˚ D ε ( F × W )  . The ﬁrst space D R ( E ) × D R ( V ) of the pair on which µ R,ε is deﬁned can b e regar ded as a “cylinder bundle” ov er B , whos e base is the complex disk bundle D ( E ); the second space of this pair is the union of the bounda ry o f this cylinder bun dle with the cor e 0 E × D R ( V ). Using p ola r co ordinates in D R ( E ) we obtain a map S ( E ) × [0 , R ] → D R ( E ), hence a map ρ : S ( E ) × [0 , R ] × D R ( V ) = S ( E ) × D R ( R ⊕ V ) → D R ( E ) × D R ( V ) , which maps [ S ( E ) × { 0 , R } × D R ( V )] ∪ [ S ( E ) × [0 , R ] × S R ( V )] = S ( E ) × S R ( R ⊕ V ) onto the the second co mponent of the pair o n which µ R,ε is deﬁned. Her e w e used suitable mo de ls D ( R ⊕ V ), S ( R ⊕ V ) for the disc and the spher e in R ⊕ V . Therefore, comp osing µ R,ε with ρ we get a n S 1 -equiv aria n t map of pairs ov er B  S ( E ) × [0 , R ] × D R ( V ) , S ( E ) × ( { 0 , R } × D R ( V ) ∪ [0 , R ] × S R ( V ))  = ( S ( E ) × D R ( R ⊕ V ) , S ( E ) × S R ( R ⊕ V )) →  [ F × W ] + B , [ F × W ] + B \ ˚ D ε ( F × W )  20 CHRISTIAN OKONEK, ANDREI TELEMAN which we denote by the same symbol µ R,ε . Collapsing ﬁb erwise ov er B the second terms of the t w o pairs , and comp osing with the natura l isomorphism [ F × W ] + B  B [ F × W ] + B \ ˚ D ε ( F × W ) ≃ [ F × W ] + B , one gets a n S 1 -equiv aria n t map of p ointed spaces over B µ R,ε : S ( E ) × [ R ⊕ V ] +  B S ( E ) × {∞} = S ( E ) + B ∧ B [ B × ( R ⊕ V )] + B − → [ F × W ] + B . Using the iso morphism l : V ≃ → W 0 and an o rientation pr e serving isomorphism R b ≃ H , w e obtain a n element { µ } ∈ S 1 α b − 1 B ( S ( E ) + B , F + B ) , which is ob viously indep endent of the choice o f the pair ( R, ε ). This e le ment will be called the c ohomotopy invariant o f µ . 3.2. General properti e s of the in v arian t { µ } . 3.2.1. A vanishing pr op ert y. Let µ : E × V → [ F × W ] + B be a map satisfying P1 , P2 . Prop osition 3.1. If µ D C ( E ) × D C ( V ) is nowher e vanishing, t hen { µ } = 0 . Pro of: W e take ε ≤ inf {k µ ( e, v ) k | k e k ≤ C, k v k ≤ C } , and w e note that the  [ F × W ] + B  / B  [ F × W ] + B \ ˚ D ε ( F × W )  -v alued p ointed map induced b y µ R,ε is ﬁber wise constant. 3.2.2. Homotopy invarianc e. Let µ ′ , µ ′′ : E × V → [ F × W ] + B t wo maps sa tisfying prop erties P1 , P2 with consta nts C ′ , c ′ , ε ′ 0 , and C ′′ , c ′′ , ε ′′ 0 . W e suppo se that the prop erty P2 of the t wo maps holds for the same decompositio n W = H ⊕ W 0 of W and for the sa me isomorphism l : V → W 0 . W e introduce the notations ˜ B := B × [0 , 1] , ˜ E := E × [0 , 1] = p ∗ B ( E ) , ˜ F := F × [0 , 1] = p ∗ B ( E ) . Prop osition 3.2. Su pp ose ther e exists C ≥ max( C ′ , C ′′ ) and a c ont inuous S 1 - e quivariant map ˜ µ : D C ( ˜ E ) × D C ( V ) → [ ˜ F × W ] + ˜ B over ˜ B whose r estriction t o ∂ h D C ( ˜ E ) × D C ( V ) i ∪ h 0 ˜ E × D C ( V ) i is nowher e vanishi ng, and such that ˜ µ t =0 = µ ′ , ˜ µ t =1 = µ ′′ . Then { µ ′ } = { µ ′′ } in S 1 α b − 1 B ( S ( E ) + B , F + B ) . Pro of: The stable classes { µ ′ } , { µ ′′ } can be computed using the the cylinder D C ( ˜ E ) × D C ( V ) and tak ing ε ≤ min  ε ′ 0 , ε ′′ 0 , c ′ , c ′′ , inf n k ˜ µ ( y ) k y ∈ ∂ [ D C ( ˜ E ) × D C ( V )] ∪ [0 ˜ E × D C ( V )] o Applying the cylinder co ns truction with para meters C , ε to the map ˜ µ we obtain a homotopy b etw een the cor resp onding r epresentativ es of the classes { µ ′ } , { µ ′′ } . COHOMOTOPY INV ARIANTS 21 3.2.3. A pr o duct formula. Let V i , W i be Euclidean s paces , E i , F i Hermitian bun- dles ov e r a compa c t base B ( i = 1 , 2) and µ i : E i × V i → [ F i × W i ] + B S 1 -equiv aria n t maps ov er B sa tisfying the prop erties P1 , P2 (1) of section 3.1 with constants C , c . Let W i = H i ⊕ W 0 ,i be the corresp onding direct sum decompositio ns, and l i : V i ≃ → W 0 ,i , h i : B → H i the maps given b y P2 (1). Fix orientations o n the H i , and put V := V 1 ⊕ V 2 , W := W 1 ⊕ W 2 , H := H 1 ⊕ H 2 , W 0 := W 0 , 1 ⊕ W 0 , 2 , l := l 1 ⊕ l 2 , and consider the bundles E := E 1 ⊕ E 2 , F := F 1 ⊕ F 2 . W e hav e a pr o duct map µ : E × V = [ E 1 × V 1 ] ⊕ [ E 2 × V 2 ] − → [ F × W ] + B = [ F 1 × W 1 ] + B ∧ B [ F 2 × W 2 ] + B ov er B . This ma p satisﬁes pro per ties P1 , P2 (1) with the map h = ( h 1 , h 2 ) : B → H . Note tha t µ will also satisfy P2 (2) a s so o n as one of the tw o ma ps µ 1 , µ 2 has this prop erty . Suppo se that µ 1 also satisﬁes prop erty P2 (2) with consta n t ε 0 and denote by { µ 1 } ∈ S 1 α b 1 − 1 B ( S ( E 1 ) + B , [ F 1 ] + B ) the co rresp onding stable class. The map µ 2 deﬁnes a map [ E 2 ⊕ V 2 ] + B − → [ F 2 ⊕ W 2 ] + B hence a clas s { µ + 2 } ∈ S 1 α b 2 B ([ E 2 ] + B , [ F 2 ] + B ). One can then fo rm the pro duct { µ 1 } ∧ B { µ + 2 } ∈ S 1 α b − 1 B  S ( E 1 ) + B ∧ B [ E 2 ] + B , F + B  . Consider now the contraction map 1 c : S ( E 1 ⊕ E 2 ) + B → S ( E 1 ) + B ∧ B [ E 2 ] + B int ro- duced in sec tio n 2.3 (see formula (3)). Using the iden tiﬁcations [ E 2 ] + B = D R ( E 2 )  B S R ( E 2 ) = E 2  B E 2 \ ˚ D R ( E 2 ) , we can use as mo del for the co n traction 1 c any map of the form 1 c R R given by 1 c R R ( e 1 , e 2 ) :=  1 k e 1 k e 1 , R e 2  , ( R ≥ R ) . Prop osition 3.3. Under the ab ove assumptions it holds { µ } = 1 c ∗  { µ 1 } ∧ B { µ + 2 }  . Pro of: The class { µ } is repres en ted by the map of pairs µ R : ( S ( E ) × [0 , R ] × D R ( V ) , S ( E ) × ([0 , R ] × S R ( V ) ∪ { 0 , R } × D R ( V ))) − → − → ([ F × W ] + B , [ F × W ] + B \ D ε ( F × W )) which is deﬁned by µ R ( e 1 , e 2 , ρ, v 1 , v 2 ) = [ µ 1 ( ρe 1 , v 1 ) , µ 2 ( ρe 2 , v 2 )] . The class 1 c ∗  { µ 1 } ∧ B { µ + 2 }  is re presented by the map ν R R betw een the same pairs deﬁned by ν R R ( e 1 , e 2 , ρ, v 1 , v 2 ) =  µ 1 ( ρ 1 k e 1 k e 1 , v 1 ) , µ 2 ( R e 2 , v 2 )  . Comp osing µ R , ν R R with the pro jection p : [ F × W ] + B − → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) 22 CHRISTIAN OKONEK, ANDREI TELEMAN we obtain t w o maps m 0 , m 1 : S ( E ) × [0 , R ] × D R ( V ) − → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) ≃ [ F × W ] + B which map S ( E ) × ([0 , R ] × S R ( V ) ∪ { 0 , R } × D R ( V )) onto the inﬁnit y section in the rig ht ha nd bundle. The natural ho motopy b et w een these maps is the map m : [0 , 1] × S ( E ) × [0 , R ] × D R ( V ) − → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) given by m ( t, e 1 , e 2 , ρ, v 1 , v 2 ) =  µ 1  ρ  1 − t + t 1 k e 1 k  e 1 , v 1  , µ 2 ([(1 − t ) ρ + t R ] e 2 , v 2 )  Claim: F or any R ≥ √ 2 C and suﬃciently lar ge R ≥ R it holds (1) the map m is well deﬁned and contin uous at the points ( t, e 1 , e 2 , ρ, v 1 , v 2 ) with e 1 = 0 . (2) the map m maps [0 , 1] × S ( E ) × ([0 , R ] × S R ( V ) ∪ { 0 , R } × D R ( V )) to the inﬁnit y sectio n in the r ight hand bundle. In fact we show that for e 2 ∈ [ E 2 ] y , one has lim u → ( t, 0 E 1 b 1 ,e 2 ,ρ,v 1 ,v 2 ) m ( u ) = ∞ y , so m maps the lo cus e 1 = 0 to the inﬁnit y s ection. Let η R > 0 b e suﬃcie ntly small, such that k µ 1 ( e 1 , v 1 ) k > ε 0 for every ( e 1 , v 1 ) ∈ D η R ( E 1 ) × D R ( V 1 ). One ha s lim e 1 → 0     ρ  1 − t + t 1 k e 1 k  e 1     = ρt . When ρt < η R , the ﬁrst co mponent of m ( t, e 1 , e 2 , ρ, v 1 , v 2 ) will alr eady hav e a no rm larger that ε 0 . When ρt ≥ η R , we obta in (using k e 1 k 2 + k e 2 k 2 = 1 ): lim e 1 → 0 k [(1 − t ) ρ + t R ] e 2 k = (1 − t ) ρ + t R ≥ η R ( 1 t − 1) + t R ≥ 2 p η R R − η R , which will b e larg er than R when R is s uﬃcien tly large . The seco nd par t of the cla im is obvious for the s pa ces [0 , 1] × S ( E ) × [0 , R ] × S R ( V ), [0 , 1 ] × S ( E ) × { 0 } × D R ( V ). F or ρ = R we obtain     ρ  1 − t + t 1 k e 1 k  e 1     2 + k [(1 − t ) ρ + t R ] e 2 k 2 ≥ R 2 ( k e 1 k 2 + k e 2 k 2 ) = R 2 ≥ 2 C 2 , so at lea st one of the tw o norms is ≥ C . Using the cla im, it follows that m des cend to an homo top y betw een tw o r epre- sentativ es of the class es { µ } and 1 c ∗  { µ 1 } ∧ B { µ + 2 }  . An interesting case is the one when also µ 2 satisﬁes prop erty P2 (2). In this case the cylinder construction applies to µ 2 and o ne can write { µ + 2 } = ∂ 2 ( { µ 2 } ) , where { µ 2 } ∈ S 1 α b 2 − 1 B ( S ( E 2 ) + B , [ F 2 ] + B ) is the inv a riant asso cia ted with µ 2 , and ∂ 2 is the connecting morphism in the long exact s equence a s so ciated with the coﬁb er sequence S ( E 2 ) + B − → D ( E 2 ) + B − → [ E 2 ] + B . COHOMOTOPY INV ARIANTS 23 Let 2 c : S ( E 1 ⊕ E 2 ) + B → [ E 1 ] + B ∧ B S ( E 2 ) + B be the standar d contraction. In this case, our multiplication formula b ecomes Corollary 3.4. Supp ose that b oth maps µ 1 , µ 2 satisfy pr op erties P1 , P2 . Then { µ } = 1 c ∗ ( { µ 1 } ∧ B ∂ 2 ( { µ 2 } )) = 2 c ∗ ( ∂ 1 ( { µ 1 } ) ∧ B { µ 2 } ) . Another coro llary is obtained when µ 2 is deﬁned by a pair of linea r isomorphisms E 2 → F 2 , V 2 → W 2 . The corresp onding formula will play an imp ortant role in the pro of of the coherence Lemma 3.1 3 compar ing the in v ariants asso ciated to tw o ﬁnite dimensional approximations of a n admissible bundle map b et w een Hilbe r t bundles. Prop osition 3. 5. L et µ : E × V → F × W b e a map satisfying the pr op erties P1 , P2 with c onstants C , c , ε 0 and maps l : V → W 0 , h : B → H . L et a : E ′ → F ′ b e an isomorphism of c omplex ve ctor bund les over B , and let b : V ′ → W ′ b e an isomorphi sm of r e al ve ct or sp ac es. Put ˜ E := E ⊕ E ′ , ˜ F := F ⊕ F ′ , ˜ V := V ⊕ V ′ , ˜ W := W ⊕ W ′ , and deﬁne ˜ µ ( e, e ′ , v , v ′ ) = ι [ µ ( e, v ) ∧ B ( a ( e ′ ) , b ( v ′ ))] , wher e ι is the obvious id entiﬁc ation ι : [ F × W ] + B ∧ B ( F ′ × W ′ ) + B → [( F ⊕ F ′ ) × ( W ⊕ W ′ )] + B . Then (1) ˜ µ satisﬁes P1 with c onst ants C , γ (for suﬃciently smal l 0 < γ < c ), and P2 with c onstant ε 0 and maps ˜ l := l ⊕ b , ˜ h := h . (2) { ˜ µ } = τ ∗ ( { µ } ) , wher e τ denotes t he obvious morphism ( E , F ) → ( ˜ E , ˜ F ) . The second statement follows directly from Pr op osition 3.3. The ﬁrst s ta temen t (whic h is sp eciﬁc to the cas e when the second factor is a linear iso morphism) is prov e d a s follows: Since the closed s e t µ − 1 ( D c ( F × W )) is contained in the op en disk ˚ D C ( E × V ), there exists r > 0 such that k µ ( e, v ) k > c as so on as k ( e, v ) k ≥ C − r . F or a p oint ( e, e ′ , v , v ′ ) with k ( e, e ′ , v , v ′ ) k ≥ C o ne ha s either k ( e, v ) k ≥ C − r , or k ( e ′ , v ′ ) k ≥ r . In the ﬁrst case one obtains k µ ( e , v ) k > c , whereas in the seco nd w e get k ( a ( e ′ ) , b ( v ′ ) k ≥ c ′ r for a constant c ′ . 3.3. A class of non-l inear maps b e tw een Hilb ert bundles. Supp ose now that V , W ar e rea l Hilber t spa c es, and that E , F are complex Hilber t bundles ov e r the compact basis B , and let µ : E × V → F × W b e a co n tin uous S 1 -equiv aria n t map ov er B which is ﬁberw is e C ∞ , and whose ﬁb erwise der iv atives ar e contin uous on E × V . W e assume that the ﬁb erwise diﬀerentials d y := d 0 y µ y = E y × V − → F y × W , y ∈ B at the origins of the ﬁbers E y × V ar e F redholm. The linear op erator d y has the form d y = ( δ y , l y ), where δ y : E y → F y and l y : V → W are deﬁned by the deriv atives of the restrictions µ E y ×{ 0 V } , µ { 0 E y }×V . Note that the co n tin uous family δ := ( δ y ) y ∈ B of complex F redholm op erator s deﬁnes an element ind( δ ) ∈ K ( B ). Let d : E × V → F × W the ﬁb erwise linear map deﬁned by the family o f F r edholm op erators ( d y ) y ∈ B . W e s uppos e that µ also has the prop erties P 1 : (prop erness) There exis t p ositive constants c , C such that k µ ( e, v ) k > c for all pairs ( e, v ) ∈ E × V with k ( e, v ) k ≥ C . P 2 : (b ehavior near the S 1 -ﬁxed p oint set) 24 CHRISTIAN OKONEK, ANDREI TELEMAN (1) W s plits orthog onally a s W = H ⊕ W 0 , where H is a ﬁnite dimensional subspace, and fo r every y ∈ B one has µ (0 E y , v ) = h ( y ) + l ( v ) ∀ y ∈ B , ∀ v ∈ V , where l : V ≃ − → W 0 ⊂ W is a linea r isometry , and h is a map from B to H . In particular the op erator l y coincides with l , so is independent of y . (2) There ex is ts ε 0 > 0 such that for every y ∈ B one has k h ( y ) k = k p H ( µ (0 E y , v )) k ≥ ε 0 . P 3 : (linear+co mpactness) The diﬀerence k := µ − d is g lobally compact, in the sense that for every R > 0 the ima ge k ( D R ( E × V )) of the disk bundle D R ( E × V ) is rela tiv ely compact in the total space F × W . Note that o ne has the identit y k (0 E y , v ) = h ( y ) ∈ H , ∀ y ∈ B . (12) In the next section we will see that the left hand o f the Seib erg-Witten equations on a 4-manifold M deﬁnes a ma p satisfying prop erties P 1 – P 3 . A diﬀeren t cons tr uction of such a map ca n be found in [BF]. 3.4. The Sei b erg-Witten map i n dimensio n 4. Let M b e closed orien ted 4- manifold, and let L b e a Hermitian line bundle o n M . W e ﬁx the following data: (1) A c losed complement S of the closed subspace iB 1 DR ( M ) = d ( iA 0 ( M )) o f iA 1 ( M ). (2) A closed c o mplemen t V o f the ﬁnite dimensional spa c e i H 1 := S ∩ ker( d : iA 1 ( M ) → i A 2 ( M )) ≃ iH 1 ( M , R ) in S (3) A complement i H 2 of d ( iA 1 ( M )) in ker( d : iA 2 ( M ) → iA 3 ( M )). This complement will come with an isomor phism i H 2 ≃ i H 2 ( M , R ). (4) An aﬃne subspace A o f the space of connections A ( L ) mo deled after S . Therefore, A is a slice to the or bits of the right action of the gaug e group G on the space of connections: a · g := a + 2 g − 1 dg The quotien t ¯ A := A / V is an aﬃne space modeled after iH 1 ( M , R ). Consider the ﬁnite dimensional Lie gr oup G := { u ∈ C ∞ ( M , S 1 ) | u − 1 du ∈ S } . One has an obvious short exact s equence { 1 } − → S 1 − → G λ − → 2 π iH 1 ( M ; Z ) − → { 1 } , where λ is deﬁned by u 7→ [ u − 1 du ] DR . The choice of a po in t x 0 ∈ M deﬁnes a left splitting ev x 0 : G → S 1 whose kernel is isomor phic to 2 π i H 1 ( M ; Z ) and which will be denoted by G x 0 . In the aﬃne space A we have a natural i H 1 -inv ar iant (hence G x 0 -inv ar iant ) subset A 0 deﬁned by A 0 := { a ∈ A| F a ∈ i H 2 } . The cur v ature F a 0 of a connection a 0 ∈ A 0 is indep endent o f a 0 , b e cause it coincides with the representativ e in i H 2 of the de Rham class − 2 π ic DR 1 ( L ); this 2-form will be denoted by F 0 . Note that A 0 is a G x 0 -inv ar iant complete system of repres en tatives COHOMOTOPY INV ARIANTS 25 for the quotien t ¯ A = A / V . The space A /G x 0 can b e regarded as a n a ﬃne bundle ov er the tor us Pic( L ) := ¯ A  G x 0 , which is natura lly a i H 1 ( X ; R ) / 4 π i H 1 ( X ; Z )-tors or. The ﬁb ers of the aﬃne bundle π : A /G x 0 − → P ic( L ) are aﬃne V -spac e s . Since the quotien t A 0 /G x 0 is a section of this aﬃne bundle, we can regar d it as a V -vector bundle over Pic( L ) with A 0 /G x 0 as zer o section. This vector bundle is actually trivial: indeed, the map ( a 0 , v ) 7→ a 0 + v ∈ A is G x 0 -equiv aria n t, and it desc e nds to a trivialization Pic( L ) × V → A /G x 0 . Remark 3.6. Cho osing a R iemannian metric g on M gives c anonic al choic es for the thr e e obje cts S , T , i H 2 ab ove, namely S = ker( d ∗ : iA 1 ( M ) − → iA 0 ( M )) , V := d ∗ ( iA 2 ( M )) , i H 2 = i H 2 g , wher e t he subscript g on t he right denotes the r esp e ct ive g -harmonic sp ac e. With these choic es, A 0 is just the the set of g - Y ang-Mil ls c onne ctions in the slic e A . Let g b e a Riemannian metr ic o n M , c ∈ S pin c ( M ) a n equiv alence class of S pin c -structures, and let τ : Q → P g be a S pin c -structure on M representing the class c . Denote by Σ ± , Σ := Σ + ⊕ Σ − the spinor bundles o f τ , L = det(Σ ± ) the determinant line bundle, and γ : Λ 1 → E nd 0 (Σ) the Cliﬀord map [OT]. Note that the gauge gro up Aut( Q ) of Q a c ts o n the space o f S pin c -structures τ : Q → P g representing c (or, equiv alently , on the space of Cliﬀord maps γ : Λ 1 → E nd 0 (Σ) which ar e c o mpatible with c ). Therefore, the spa ce of S pin c -Dirac op era to rs which are compatible with the pair ( g , c ) ha s a very complicated top ology . Note that, for the constructio n o f a Dirac op erator one needs a concr ete S pin c -structure τ (or, equiv alently , a concre te Cliﬀord map γ ), not only an equiv a le nce class c . The ga uge group G a nd its subgro up G x 0 act fro m the le ft on the vector spa ces of sections A 0 (Σ ± ) by the formula ( g , Ψ) 7→ g − 1 Ψ . Since G x 0 acts freely on the aﬃne quotient space ¯ A we get tw o ﬂat v ec to r bundles ¯ A× G x 0 A 0 (Σ ± ) over Pic( L ) with standard ﬁb ers A 0 (Σ ± ). In order to use o ur gener al formalism we make the following deﬁnitions: B := P ic( L ) , E := ¯ A × G x 0 A 0 (Σ + ) , F := ¯ A × G x 0 A 0 (Σ − ) , W := iA 2 + ( M ) . Let κ : B → i H + g be a s moo th map. The κ -twisted Seiber g-Witten map is the map from A 0 (Σ + ) × A to A 0 (Σ − ) × iA 2 + given by (Ψ , a ) 7→ ( 6 D a Ψ , ( F a − F 0 + κ ( π ( a ))) + − γ − 1 ((Ψ ¯ Ψ) 0 ) . Via the iden tiﬁcation B × V = A /G x 0 this map descends to an S 1 -equiv aria n t map sw κ : E × V − → F × W . The restrictio n of sw κ to the ﬁber over y = [ a 0 ] ∈ B is given b y the formula sw κ (Ψ , v ) =  6 D a 0 Ψ + 1 2 γ ( v )Ψ , d + v + κ ( y ) − γ − 1 ((Ψ ¯ Ψ) 0  . 26 CHRISTIAN OKONEK, ANDREI TELEMAN The lineariza tion of this map at the zero section in the bundle E × V over B is a ﬁb erwise linear bundle map given by d (Ψ , v ) = ( 6 D a 0 Ψ , d + v ) . ov er the ﬁb er [ a 0 ] ∈ B . Hence sw κ decomp oses as sw κ = d + c κ , where c κ is the sum of a quadra tic ma p c and the ﬁb erwise constan t map deﬁned by κ . Denote b y w τ the expec ted dimension of the Seib erg -Witten mo duli space corres p onding to τ : w τ := 1 4 ( c 1 ( L ) 2 − 3 σ ( M ) − 2 e ( M )) W e deﬁne Sob olev L 2 k -completions of the spaces V , W in the usual way . The construction of Sob olev norms on the bundles E , F is mo r e delicate, b ecaus e these bundles are quo tien ts with res p ect to the group G x 0 , which do es not op erate by L 2 k -isometries 4 . F o r a po in t y = [ a 0 ] ∈ B (with a 0 ∈ A 0 ) one identiﬁes the ﬁber s E y , F y with { a 0 } × A 0 (Σ ± ) and uses the cov ariant deriv atives asso ciated with ∇ a 0 to deﬁne the L 2 k -norm on E y . A gauge tra nsformation g ∈ G x 0 deﬁnes a n isometry { a 0 } × A 0 (Σ + ) → { a 0 · g } × A 0 (Σ ± ), so in this wa y o ne obtains a well deﬁned Sob olev norm o n the ﬁb er E y . Lemma 3 .7. With r esp e ct to s uitable Sob olev c ompletions, t he fol lowing holds: (1) sw κ is smo oth. (2) The ﬁb erwise line ar map d is ﬁb erwise F r e dholm of index w τ − b 1 + 1 , a nd c β is a c omp act map. (3) Ther e exists p ositive c onstants c , C such that k (Ψ , v ) k ≥ C ⇒ k sw κ (Ψ , v ) k > c . (4) The map c κ = sw κ − d is c omp act. Therefore the Seib erg- Witten map sw κ satisﬁes a lw ays the prop erties P 1, P 2 (1) and P 3 in section 3.3. It also satisﬁes P 2 (2) for all maps κ : B → i H + g \ { 0 } . The ﬁr st and the third statements in the lemma are ea sy to see. The crucial prop erness asser tion (2) is s tated in [F u1], [F u2]. A pro of of the ana logue statement for another version of the Seiber g -Witten map can be found in [BF]. A detailed pro of for our version, a nd an analogue prop erness pro per ty in a diﬀerent gauge theoretic con tex t can b e found in [B]. S imilar methods can be also used to treat the 3-dimens io nal Casson-Seib erg- Witten theory . The univ ers al Seib e rg-Witten map: With the notations in tro duced ab ov e we ﬁx the par ameters ( g , κ, c , Q ) on M . As we ex plained ab ov e, the family of Dir ac op- erators δ := ( 6 D a 0 ) [ a 0 ] ∈ B (and implicitly the Seib erg- Witten ma p sw κ ) still de p ends on the choice of a S pin c -structure τ : Q → P g in the class c . This parameter v aries in the s pace Γ := Hom M ( Q, P g ) of equiv ariant bundle morphisms Q → P g cov er ing id M . Since this space ha s a c o mplicated top olog y , and our purp ose is to construct an inv a riant which is in trinsically and canonica lly asso ciated with the bas e mani- fold, it is important to understand how the ob jects ( E , F , δ, sw κ ) asso c ia ted with diﬀerent bundle morphisms τ should b e identiﬁed. A construction which has been 4 W e ar e grateful to M arkus Bader f or pointing out this subtili t y to us. COHOMOTOPY INV ARIANTS 27 presented by F ur uta in his talk at the Postnik ov Memorial Confere nc e [F u3 ] solves this problem in an eleg ant wa y 5 : One has a universal family ( 6 D τ a ) ( τ ,a ) ∈ Γ ×A ( L ) of Dirac op era tors, which is intrinsi- cally a sso ciated with the system ( g , κ, c , Q ). An automorphism f ∈ Aut( Q ) deﬁnes automorphisms f ± of Σ ± and an a utomorphism det( f ) ∈ Aut( L ); the relation betw een the Dira c op erators as so ciated with τ and τ ′ := τ ◦ f is 6 D τ ′ det( f ) ∗ ( a ) = f − 1 − ◦ 6 D τ a ◦ f + . The g roup Aut( Q ) acts transitively with c onstant stabiliz e r G ⊂ Aut( Q ) on Γ , and acts with constant sta bilize r S 1 on the pro duct Γ × A ( L ). Now ﬁx a S pi n c (4)- equiv ariant map θ : Q x 0 → P g,x 0 , and put Γ 0 := { τ ∈ Γ , τ x 0 = θ } , Aut ( Q ) θ := { f ∈ Aut( Q ) | θ ◦ f x 0 = θ } , Aut( Q ) 0 := { f ∈ Aut( Q ) | f x 0 = id Q x 0 } . The quotient Aut( Q ) θ / Aut( Q ) 0 can b e iden tiﬁed with S 1 . The universal family ( 6 D τ a ) ( τ ,a ) ∈ Γ ×A ( L ) of Dirac ope r ators descends to a a family 6 D : E → F on the free quotient B := Γ 0 × A ( L )  Aut( Q ) 0 . By c ho osing an element τ ∈ Γ 0 one obtains a n identiﬁcation B ≃ A ( L ) / G x 0 , where G x 0 acts by the form ula ( a · g ) = a + 2 g − 1 dg , but the identiﬁcation is no t canonical. The free actio n o f V := d ∗ ( iA 2 ( M )) on the s e cond factor A ( L ) by transla tio ns induces a free action on B , and the q uo tien t B with resp ect to this action is a b 1 -dimensional torus. The space of pair s ( τ , a ) with a Y ang- Mills deﬁnes a se c tio n B 0 of the V -bundle B → B , whic h therefore b ecomes a trivia l v ec to r bundle with ﬁber V . One can construct a “universal” Seiber g-Witten map sw κ ov er B using the space Γ 0 × A 0 (Σ + ) × A ( L ) as space of conﬁg urations, and Aut( Q ) 0 as gaug e gro up; this map is in trinsically asso ciated with the system ( g , κ, c , Q, θ ). The impo rtant po in t in this construction is that res tricting the universal family 6 D to the torus B 0 ≃ B one obtains a “universal Segal co cycle” 6 D : E → F r epresenting the K- theory element x = ind( 6 D ). F uruta s how ed that the corresp onding sp e ctrum is independent of the choice of θ , up to homo top y . On can apply the construction in [BF] and get – for manifolds with b + ≥ 2 and arbitrary b 1 – a well deﬁned Bauer - F uruta in v ariant b elonging to a homotopy gro up which is functoria l with r espe c t to diﬀeomorphisms. As we explained in the intro duction (see section 1.1), w e believe that for some applications it is useful to hav e in v ariants b elonging to gro ups which are top olo gic al ly functorial , as it is the case in cla ssical Seib erg-Witten and Donaldson theo ries. 3.5. Finite dime ns ional appro ximation. W e will need the fo llowing simple geo- metric constructio n. Let A b e a (real o r complex) Hilber t space, and A ⊂ A a ﬁnite dimensional subspa c e. F ollowing [B F] we intro duce, for every ε > 0 the re tr action ρ ε,A : A + \ S ε ( A ⊥ ) → A + 5 F uruta explained the details of this construction in an e-mail to the second author, and informed us that simil ar ideas hav e b een used b efore. 28 CHRISTIAN OKONEK, ANDREI TELEMAN in the following wa y . F or every a ∈ A \ { 0 } put s ε,a := k a k 2 − ε 2 2 k a k 2 , c ε,a = s ε,a a , r ε,a := k a k 2 + ε 2 2 k a k . Let S ε,a ⊂ R a + A ⊥ be the hyper sphere of R a + A ⊥ deﬁned by the equation k b − c ε,a k 2 + k a ′ k 2 = r 2 ε,a . The hypersphere S ε,a has the prop erties a ∈ S ε,a , S ε ( A ⊥ ) ⊂ S ε,a . Consider also the spherica l calo tte: C ε,a := { ta + a ′ ∈ S ε,a | t > 0 } ⊂ S ε,a . Denote by C ε, ∞ ⊂ [ A ⊥ ] + the exter ior of the sphere S ε ( A ⊥ ) ⊂ A ⊥ (including ∞ ), and by C ε, 0 its interior. Now note that F ε,A := { C ε,a | a ∈ A + } is a foliatio n of A + \ S ε ( A ⊥ ) with closed leav es; the leav es ar e all diﬀeomorphic to the s tandard disk of A ⊥ . The r etraction ρ ε,A assigns the po in t a ∈ A + to any p oint of the lea f C ε,a ⊂ A + . Note that for any z ∈ A one has the implicatio n  z ∈ A + \ S ε ( A ⊥ ) , k z k ≥ ε  ⇒ k ρ ε,A ( z ) k ≥ k z k (13) (equality is obtained when k z k = ε or z ∈ A ). A second imp or tan t prop erty of the retraction ρ ε,A is z ∈ A \ A ⊥ ⇒  ρ ε,A ( z ) = λ ε,z p A ( z ) with λ ε,z ≥ 1  . (14) An y R - linear isometry u of A which leav es the subs pa ce A inv ariant will also leav e inv a riant the folia tion F ε,A . Therefo r e Remark 3. 8. ρ ε,A is e quivariant with r esp e ct to any R -line ar isometry of A which le aves the subsp ac e A invariant. These retra ctions play a fundamental role in the following co nstruction of ﬁnite dimensional approximations. This construction is a reﬁnement of the one developed in [BF]. The main diﬀerence is that w e hav e to work over a bas e B , and that we treat the real and co mplex summands separ a tely . Consider again an S 1 -equiv aria n t map µ : E × V → F × W ov er B satis fying the prop erties P 1 , P 2, P 3 of section 3.3. Recall from section 3.3 that w e denoted by d the linear iz ation of µ at the 0-section and by δ and l the complex and the rea l comp onents of d . W e hav e assumed tha t the R -linear op erator l induces an isometry V → W 0 . A ﬁnite rank subbundle F ⊂ F will be ca lled admissible if it is mapp ed surjectively onto the linear space deﬁned by the family o f cokernels (coker( δ y )) y ∈ B . A ﬁnite dimensional subspace W ⊂ W will b e called admiss ible if it contains H . A pair ( F, W ) will be called admiss ible if F and W are b oth admissible; in this case, for every y ∈ B the pro duct F y × W is ma pped surjectively ont o coker( d y ). F or every admiss ible pair π = ( F , W ) the preimage d − 1 ( F × W ) is a ﬁnite ra nk subbundle of E × V whic h splits as d − 1 ( F × W ) = δ − 1 ( F ) × l − 1 ( W ) . COHOMOTOPY INV ARIANTS 29 W e denote b y W 0 the or thogonal complement of H in W , and put V := l − 1 ( W ) = l − 1 ( W 0 ), E := δ − 1 ( F ) ⊂ E . The pair ( E , F ) represe n ts ind( δ ) ∈ K ( B ). W e get top olo gic al orthogo nal direct sum decomp ositions F = F ⊕ F ⊥ , E = E ⊕ E ⊥ , W = W ⊕ W ⊥ = H ⊕ W 0 ⊕ W ⊥ , V = V ⊕ V ⊥ . The pro duct F × W is a ﬁnite dimensiona l Hilbert subbundle of F × W whose orthogo nal complement is F ⊥ × W ⊥ . The retraction ρ ε,F × W : [ F × W ] + B \ S ε ( F ⊥ × W ⊥ ) − → [ F × W ] + B is deﬁned ﬁb erwise. W e will s ee that, for suﬃcie ntly small ε > 0 and suﬃciently large admissible pa irs π = ( F, W ), the image of the restr ic tion µ E × V do es not int ersect S ε ( F ⊥ × W ⊥ ). Therefor e we ca n deﬁne a map µ ε,π := { ρ ε,F × W ◦ µ } E × V : E × V − → [ F × W ] + B , which b elongs to the cla ss studied in sectio n 3 .1. Such a map will b e called a ﬁnit e dimensional appr oximation of µ . The res ult w e nee d is very m uc h similar to the ﬁrst par t of L e mma 2.3 in [BF]. W e know that the preimage µ − 1 ( D c ( F × W )) is contained in the disk bundle D C ( E × V ) ⊂ D C ( E ) × D C ( V ). The image k ( D C ( E ) × D C ( V )) is relatively compac t in the total space F × W , beca us e k is compact b y prop erty P 3 . Now ﬁx η > 0. Deﬁnition 3.9 . A p air π := ( F , W ) is c al le d η - admissible if it is admissible, and any element of the c omp act set k ( D C ( E ) × D C ( V )) is η -close to an element in F × W b elonging to t he sa me ﬁb er. Lemma 3.10. L et K ⊂ F × W a c omp act set, F a ﬁnite r ank subbund le of F , and W a ﬁnite dimensional subsp ac e of W . The fol low ing c onditions ar e e quivalent: (1) Any p oint k ∈ K is η -close to a p oint of F × W b elonging to the same ﬁb er. (2) Ther e exists a ﬁ nite syst em ( φ 1 , . . . , φ k ) of se ctions of F and a ﬁnite system ( w 1 , . . . , w k ) of ve ctors of W such t hat K ⊂ [ y ∈ B , 1 ≤ i ≤ k B (( φ i ( y ) , w i ) , η ) . Pro of: The implication (2) ⇒ (1 ) is o b vious. F or the second it is conv enient to int ro duce the no tation B (( φ, w ) , η ) := [ y ∈ B B (( φ ( y ) , w ) , η ) , for a section φ ∈ Γ( F ) and a vector w ∈ W . If K s atisﬁes (1) then it is con- tained in the union of op en sets ∪ φ ∈ Γ( F ) ,w ∈ W B (( φ, w ) , η ). It s uﬃces no w to use the compac tnes s o f K . Corollary 3.11. The set of p airs ( F, W ) satisfying the η -admissibility c ondition is non-empty, op en and c oﬁnal. Pro of: Since K := k ( D C ( E ) × D C ( V )) is compa ct in F × W there exists ﬁnite systems φ = ( φ 1 , . . . , φ k ) ∈ Γ( F ) k , w = ( w 1 , . . . w k ) ∈ W k such that K ⊂ [ i B (( φ i , w i ) , η ) . (15) Now ﬁx an admissible pair ( F 0 , W 0 ). Since F has inﬁnite ra nk , it is easy to s ee that any neig h bor ho od of φ contains a system φ ′ which is in g eneral p osition with resp ect 30 CHRISTIAN OKONEK, ANDREI TELEMAN to F 0 in the following sense: for every p o in t y ∈ B the system φ ′ ( y ) is linearly independent in F y , and h φ ′ ( y ) i ∩ F 0 ,y = { 0 y } . Since the condition (15) is obviously op en with resp ect to the pair ( φ , w ), we can choo se such a s ystem φ ′ which still satisﬁes (1 5) and is in gener al p osition with r espec t to F 0 . W e denote by F ′ the ra nk k -subbundle g e nerated by φ ′ , and we put F := F 0 ⊕ F ′ and W := W 0 + h w 1 , . . . w k i . T o prove that η -admissibility is o p en, note that admissibilit y is open, and use Lemma 3.10 to prove that the second condition in the deﬁnition of η - admissibility is also o pen. Finally , to s ee that the set of η -admissible pairs is coﬁnal, we ﬁx an η -admissible pair ( F 0 , W 0 ). F or an ar bitary pair ( F, W ) consider a small deforma- tion F ′ 0 of F 0 for which ( F ′ 0 , W ) is still η -admissible a nd such that F ′ 0 is ﬁb erwis e transversal to F . Then ( F ⊕ F ′ 0 , W + W ) will b e an η -admissible pair which co n tains ( F, W ). Lemma 3 .12. (Finite dimensional appr oximations) L et 0 < η < c 4 . Then (1) F or any η - admissible p air π = ( F, W ) one has im  µ E × V  ∩ S c ( F ⊥ × W ⊥ ) = ∅ , so the ﬁnite dimensional a ppr oximation µ c,π := { ( ρ c,F × W ) ◦ µ } E × V : E × V − → ( F × W ) + B is deﬁne d. (2) The r estriction µ c,π D C ( E ) × D C ( V ) takes values in F × W . (3) F or any η -admissible p air π = ( F , W ) the ﬁnite dimensional appr oximation µ c,π satisﬁes t he c onditions P1 , P2 (se e se ction 3.1 ) with the s ame c on- stants C , c , ε 0 , isometry l : V → W 0 ⊂ W and the same map h : B → H as µ . Pro of: 1. If the intersection im  µ E × V  ∩ S c ( F ⊥ ⊕ W ⊥ ) was not empty , there would exist a p oint ( e, v ) ∈ E × V such that µ ( e, v ) ∈ S c ( F ⊥ × W ⊥ ). Since S c ( F ⊥ × W ⊥ ) ⊂ D c ( F × W ), it follows ( e, v ) ∈ D C ( E ) × D C ( V ). Ther efore µ ( e, v ) = d ( e, v ) + k ( e, v ) ∈ F × W 0 + k ( D C ( E ) × D C ( V )) . But any element in the second set k ( D C ( E ) × D C ( V )) is η -close to an elemen t in F × W by as sumption, so µ ( e, v ) is η -close to F × W . Since η < c 4 , this contradicts µ ( e, v ) ∈ S c ( F ⊥ ⊕ W ⊥ ). 2. The same ar gument s hows that µ ( D C ( E ) × D C ( V )) do es not in tersect the complement of D c ( F ⊥ ⊕ W ⊥ ) in F ⊥ ⊕ W ⊥ . 3. W e ha ve to c hec k that, for an η -admissible pair π = ( F , W ), the ﬁnite di- mensional appr oximation µ c,π has the t w o prop erties P1 , P2 in s ection 3.1. F or a po in t ( e, v ) ∈ E × V with k ( e, v ) k ≥ C it holds k µ ( e, v ) k > c so , by (13), we have k ρ c,F × W ( µ ( e, v )) k ≥ k µ ( e , v ) k > c . (16) On the other hand, for a n y y ∈ B , v ∈ V o ne has µ (0 E y , v ) = h ( y ) + l ( y ) ∈ { 0 F y } × W , hence µ c,π (0 E y , v ) = ρ c,F × W ( µ (0 E y , v )) = µ (0 E y , v ) = h ( y ) + l ( v ) . COHOMOTOPY INV ARIANTS 31 3.6. Compatibility prop erties. Lemma 3.13 . (Coher enc e L emma) L et 0 < η < c 4 , let π = ( F, W ) , ˜ π = ( ˜ F , ˜ W ) b e two η -admissible p airs with π ⊂ ˜ π , and let F ′ , W ′ b e the ortho gonal c omplements of F , W i n ˜ F , ˜ W r esp e ctively. The m ap µ c,π , ˜ π := ι ◦ n [ µ c,π ◦ (p E , p V )] ∧ B [(p F ′ , p W ′ ) ◦ ( δ, l )] + B o : ˜ E × ˜ V → [ ˜ F × ˜ W ] + B satisﬁes pr op erties P1 , P2 with c onstants C , γ (for a suﬃciently smal l γ with 0 < γ < c ), ε 0 , and one has { µ c, ˜ π } = { µ c,π , ˜ π } . Pro of: The ﬁrst statement follows fr om Prop ositio n 3 .5. W e use the same metho d as in the pr oo f of Lemma 2 .3 in [BF] to construct a homotopy b etw een the res tr iction o f the tw o maps to the pro duct D C ( ˜ E ) × D C ( V ) and we will apply the homotopy inv ar ia nce prop erty of our inv a r iant (see P rop osition 3 .2). The main diﬀerence compared to [BF] is that w e hav e to cont rol the restriction to the S 1 -ﬁxed po in t set, but w e do not need a n extension o f the homo top y to the whole ˜ E × ˜ V . F or co mpleteness we include detailed ar gumen ts adapted to o ur situation. Pro of: Denote by E ′ , V ′ the o rthogonal complements of E , V in ˜ E , ˜ V . W e deﬁne the map H : [0 , 4] × [ D C ( ˜ E ) × D C ( ˜ V )] − → [ F × W ] \ h ˜ F ⊥ × ˜ W ⊥ \ ˚ D c ( ˜ F ⊥ × ˜ W ⊥ ) i (17) by the formula 6 H t =        d + [(1 − t ) id F ×W + t p F × W ] ◦ k for 0 ≤ t ≤ 1 , d + p F × W ◦ k ◦  (2 − t ) id ˜ E × ˜ V + ( t − 1) p E × V  for 1 ≤ t ≤ 2 , p F × W ◦ k ◦ p E × V + [ d − ( t − 2 ) p F × W ◦ d ◦ p E ′ × V ′ ] for 2 ≤ t ≤ 3 , p F ′ × W ′ ◦ d + [(4 − t ) p F × W + ( t − 3) ρ c,F × W ] ◦ µ ◦ p E × V for 3 ≤ t ≤ 4 . Claim: H is a wel l deﬁne d, c ontinuous , S 1 -e quivariant m ap over B . This follows from: a) F or a p oint ( t, ˜ e, ˜ v ) ∈ [0 , 4 ] × D C ( ˜ E ) × D C ( ˜ V ), the term ρ c,F × W ( µ (p E × V ( ˜ e, ˜ v ))) is ﬁnite, so the convex combination in the fo urth branch is deﬁned and ﬁnite. Indeed, recall that the retr action ρ c,F × W is ﬁnite on the complement o f the leaf  F ⊥ × W ⊥  \ D c ( F ⊥ × W ⊥ ). Therefore it suﬃces to no te that k ( D C ( E ) × D C ( V )) is η -close to F × W and d ( E × V ) ⊂ F × W , so the p oint µ (p E × V ( ˜ e, ˜ v )) is η -clo se to F × W for ( ˜ e, ˜ v ) ∈ D C ( ˜ E ) × D C ( ˜ V ). Therefore µ (p E × V ( ˜ e, ˜ v )) 6∈  F ⊥ × W ⊥  \ ˚ D c ( F ⊥ × W ⊥ ) . b) The formulae given for the four comp onents o f H agree on the in ters ections of their domains. c) H ta k es v alues in [ F × W ] \ h ˜ F ⊥ × ˜ W ⊥ \ ˚ D c ( ˜ F ⊥ × ˜ W ⊥ ) i . Indeed, for ( t, ˜ e, ˜ v ) ∈ [0 , 4 ] × D C ( ˜ E ) × D C ( ˜ V ) we see as in the proo f of a) that the rig ht hand term o f H t m ust be η -clo se to ˜ F × ˜ W , so H ([0 , 4] × D C ( ˜ E ) × D C ( ˜ V )) av oids  F ⊥ × W ⊥  \ ˚ D c ( F ⊥ × W ⊥ ). 6 The third br anc h of the homotop y was omitted in [BF]. 32 CHRISTIAN OKONEK, ANDREI TELEMAN The map H has the following pr ope r ties: (1) H 0 coincides with the restr ic tion µ D C ( ˜ E ) × D C ( ˜ V ) . (2) H 4 coincides with the map µ c,π , ˜ π comp osed with the inclusion ˜ F × ˜ W ֒ → [ F × W ] + B \ S c ( ˜ F ⊥ × ˜ W ⊥ ). (3) One has H t (0 ˜ E y , ˜ v ) = h ( y ) + l ( ˜ v ) , ∀ t ∈ [0 , 4] ∀ y ∈ B ∀ ˜ v ∈ D C ( ˜ V ) . (18) F ormula (18) follows fr o m (12) and the fa c t that l is an iso metr y , so it commutes with orthog onal pr o jections. (4) H ([0 , 4] × ∂ ( D C ( ˜ E ) × D C ( ˜ V )) ∩ [ ˜ F ⊥ × ˜ W ⊥ ] = ∅ . Indeed, for ( ˜ e, ˜ v ) ∈ ∂ ( D C ( ˜ E ) × D C ( ˜ V )) we get k H 0 ( ˜ e, ˜ v ) k = k µ ( ˜ e, ˜ v ) k ≥ c , where as k µ ( ˜ e, ˜ v ) k is η -clo se to F × W ⊂ ˜ F × ˜ W . Moreover, for t ∈ [0 , 1] it holds k H t ( ˜ e, ˜ v ) − H 0 ( ˜ e, ˜ v ) k = t k (p F ⊥ × W ⊥ ◦ k )( ˜ e, ˜ v ) k ≤ η . F or t ≥ 2 we hav e p F ′ × W ′ ◦ h t = p F ′ × W ′ ◦ d , so H t ( ˜ e, ˜ v ) ca n belong to ˜ F ⊥ × ˜ W ⊥ only when p F ′ × W ′ ◦ d ( ˜ e, ˜ v ) = 0, i.e. when ( ˜ e, ˜ v ) ∈ E × V . F or such a pair we ﬁnd H t ( ˜ e, ˜ v ) = ( d + p F × W ◦ k )( ˜ e, ˜ v ) = µ ( ˜ e, ˜ v ) − (p F ⊥ × W ⊥ ◦ k )( ˜ e, ˜ v ) ∀ t ∈ [1 , 3] , H t ( ˜ e, ˜ v ) ∈ [p F × W ( µ ( ˜ e, ˜ v )) , ρ c,F ( µ ( ˜ e, ˜ v ))] ∀ t ∈ [3 , 4] , so H t ( ˜ e, ˜ v ) is a non-v a nishing vector o f F × W (m ore precisely a p ositive m ultiple of p F × W ( µ ( ˜ e , ˜ v )) = µ ( ˜ e, ˜ v ) − (p F ⊥ × W ⊥ ◦ k )( ˜ e, ˜ v )) for any t ∈ [1 , 4]. These pr ope r ties ha ve the following imp ortant consequence: Remark: The c omp osition ρ c, ˜ π ◦ H is nowher e va nishing on the sp ac e [0 , 4] × n ∂ h D C ( ˜ E ) × D C ( ˜ V ) i ∪ h 0 ˜ E × ˜ V io . This follows from the fact that the v anishing lo cus of the retra ction ρ c, ˜ π is the leaf ˚ D c ( ˜ F ⊥ × ˜ W ⊥ ) ⊂ ˜ F ⊥ × ˜ W ⊥ . On the other hand we have ρ c, ˜ π ◦ H 0 = µ c, ˜ π D C ( ˜ E ) × D C ( ˜ V ) , ρ c, ˜ π ◦ H 4 = µ c,π , ˜ π D C ( ˜ E ) × D C ( ˜ V ) It suﬃces now to apply Pro p ositio n 3 .2. Using P rop osition 3.5 and Le mma 3.13 we obtain Corollary 3. 14. L et µ : E × V → F × W b e an S 1 -e quivariant map over a c omp act CW c omplex B s at isfying P 1 , P 2 , P 3 , and let 0 < η < c 4 . Fix an orientation O of the ﬁnite dimensional summand H of W . The el ements { µ c,π } ∈ S 1 α b − 1 B ( S ( E ) + B , F + B ) asso ciate d with η - admissible p airs π = ( F , W ) deﬁne a unique class { µ } ∈ α b − 1 (ind( δ )) which dep ends only on t he map µ and the orientation O . COHOMOTOPY INV ARIANTS 33 In pa rticular, using ﬁnite dimens io nal appr oximations asso ciated with constants C ′ ≥ C and 0 < c ′ ≤ c (and para meter 0 < η < c ′ 4 ), one o btains the sa me class. Pro of: Let π = ( F, W ), π 1 = ( F 1 , W 1 ) tw o η -admissible pair s. By Lemma 3 .1 3 we can identify the images of the classes µ c,π , µ c,π 1 in α b − 1 (ind( δ )) under the assumption π ⊂ π 1 . The problem is to reduce the general case to this situa tio n. By Corollar y 3.11 w e know that η - a dmissibilit y of π is an op en condition, i.e. it is stable under small deformations. On the other hand, by the ho motopy pro per t y Prop osition 3.2, the imag e of the clas s µ c,π in the g roup α b − 1 (ind( δ )) is stable under small deormations of π . Hence it suﬃces to consider a ge ner ic small deformation F ′ of the subbundle F ⊂ F which is ﬁb erwise transversal to F 1 , such that ( F ′ , W ) is still η - admissible. Then we can put ˜ F := F ′ ⊕ F 1 , ˜ W := W + W 1 and apply t w ic e the compatibility Lemma 3.13. Prop osition 3.1 5. Su pp ose that the r estr iction µ D C ( E ) × D C ( V ) is nowher e vanish- ing. Then { µ } = 0 . Pro of: Since µ D C ( E ) × D C ( V ) is no where v anishing , it is easy to see tha t there exists γ > 0 such that k µ ( e, v ) k > γ for every ( e, v ) ∈ D C ( E ) × D C ( V ). Indeed, if not there would exist a sequence ( e n , v n ) ∈ D C ( E ) × D C ( V ) such that k µ ( e n , v n ) k → 0. Let K ⊂ F × W b e a compact subspace which contains k ( D C ( E ) × D C ( V )). Since d = ( δ, l ) is a contin uous family of F redholm op era to rs, it follows that d − 1 ( K ) ∩ [ D C ( E ) × D C ( V )] is compact. Ther efore ( e n , v n ) n admits a subsequence whic h conv er ges in this intersection. The limit will b e a v anishing p oint of µ , which contradicts the assumption. Use now the constant c ′ := min( γ , c ) (instead o f c ) in the co ns truction of the ﬁnite dimensio nal approximations of µ . The obtained maps µ c ′ ,π are no where v a n- ishing on D C ( E ) × D C ( V ), and our a ssertion follows from the v a nishing prop erty Prop osition 3.1 proved in the ﬁnite dimensiona l case. 4. Fundament al pr o per ties of the cohomotopy inv ariants 4.1. The Hurewicz image of the cohomo top y in v arian t. 4.1.1. The r elative Hur ewicz morphism. Let B be a co mpact spa c e, and let E , F b e Hermitian bundles of ranks e , f o v er B . Let k be an integer and u ∈ S 1 α k B ( S ( E ) + B , F + B ) a s ta ble class . Suppos e for simplicity k ≥ 0. Co nsider a re pre- sentativ e ϕ : S ( E ) + B ∧ B ξ + B → F + B ∧ B [ R k ] + B ∧ ξ + B of this stable c lass, wher e ξ = η ⊕ ξ 0 is the direct sum of a complex vector bundle η and a rea l vector bundle ξ 0 . W e may supp ose tha t the r e a l summand ξ 0 of ξ is ori- ent able. W e choose an orientation of ξ 0 ; in this w ay all our bundles b ecome oriented bundles. The space S ( E ) + B ∧ B ξ + B can b e identiﬁed with the ﬁb erwise quotient { S ( E ) × B ξ + B } / B { S ( E ) × B ∞ ξ } . C o mpo s ing ϕ with the c a nonical pro jection o ne obtains a ma p of pair s ov er B ˜ ϕ : ( S ( E ) × B ξ + B , S ( E ) × B ∞ ξ ) → ([ F ⊕ R k ⊕ ξ ] + B , ∞ F ⊕ R k ⊕ ξ ) . Consider now the pro jection π : P ( E ) → B and the following bundles over P ( E ): ˜ F := π ∗ ( F )(1) , ˜ ξ := π ∗ ( η )(1) ⊕ π ∗ ( ξ 0 ) . 34 CHRISTIAN OKONEK, ANDREI TELEMAN The map ˜ ϕ des c ends to a morphism of p ointed sphere bundles ov er P ( E ) ¯ ϕ : ˜ ξ + P ( E ) − → [ ˜ F ⊕ R k ⊕ ˜ ξ ] + P ( E ) . Denote by s the re a l rank of ξ . Let t ˜ ξ ∈ H s ( ˜ ξ + P ( E ) , ∞ ˜ ξ ; Z ) , t ˜ F ⊕ R k ⊕ ˜ ξ ∈ H 2 f + k + s ([ ˜ F ⊕ R k ⊕ ˜ ξ ] + P ( E ) , ∞ ˜ F ⊕ R k ⊕ ˜ ξ ; Z ) be the Tho m classes of the oriented bundles ˜ ξ , ˜ F ⊕ R k ⊕ ˜ ξ . The formula ¯ ϕ ∗ (t ˜ F ⊕ R k ⊕ ˜ ξ ) = p ∗ P ( E ) ( h ¯ ϕ ) ∪ t ˜ ξ deﬁnes a cohomo lo gy class h ¯ ϕ ∈ H 2 f + k ( P ( E ); Z ) whic h is indep endent of the chosen orientation of ξ 0 and o f the representativ e ϕ of the stable cla ss u . F or k ≤ 0 one has a simila r construction, but uses a [ R − k ] + B factor on the left side. The as signment u = [ ϕ ] 7→ h ¯ ϕ deﬁnes a mor phism h : S 1 α k B ( S ( E ) + B , F + B ) → H 2 f + k ( P ( E ); Z ) , which we call the r elative Hur ewicz m orphism ov er B . Denote by q : ˜ ξ → P ( E ) the bundle pr o jection, and b y ˚ ϕ the section in the pull-back [ q ∗ ( ˜ F ⊕ R k ⊕ ˜ ξ )] + ˜ ξ ov er ˜ ξ deﬁned b y ¯ ϕ . Since the v a nishing lo cus Z ( ˚ ϕ ) of this section is co mpact, one can deﬁne its lo c alize d Euler class class [ ˚ ϕ ] ∈ H d +2 e − 2 − 2 f − k ( ˜ ξ ; Z ), which coincides with the fundamental class [ Z ( ˚ ϕ )] of the co m- pact oriented submanifold [ Z ( ˚ ϕ )] when ˚ ϕ is smo oth a nd transversal to the ze r o section [Br]. Remark 4.1. (Th e ge ometric interpr etation of the Hur ewicz morphi sm) Supp ose that B is an oriente d n -dimensional c omp act manifold. Then P D P ( E ) ( h ( u )) = [ ι ∗ ] − 1 ([ ˚ ϕ ]) , wher e ι ∗ : H n +2 e − 2 − 2 f − k ( P ( E ); Z ) → H n +2 e − 2 − 2 f − k ( ˜ ξ ; Z ) . is the isomorphism induc e d by the zer o se ction of ˜ ξ . If ˚ ϕ is smo oth and t r ansversal to the zer o se ct ion, then P D P ( E ) ( h ( u )) = [ ι ∗ ] − 1 ([ Z ( ˚ ϕ )]) . Pro of: The lo calized Euler clas s [ ˚ ϕ ] ∈ H n +2 e − 2 − 2 f − k ( ˜ ξ ; Z ) is deﬁned a s the cap pro duct ˚ ϕ ∗ (t q ∗ ( ˜ F ⊕ R k ⊕ ˜ ξ ) ) ∩ [ ˜ ξ ], where [ ˜ ξ ] sta nds for the fundamen tal class o f ˜ ξ in cohomolog y with c ompact supp orts [Br]. W e get [ ˚ ϕ ] := ˚ ϕ ∗ (t q ∗ ( ˜ F ⊕ R k ⊕ ˜ ξ ) ) ∩ [ ˜ ξ ] = ¯ ϕ ∗ (t ˜ F ⊕ R k ⊕ ˜ ξ ) ∩ [ ˜ ξ ] = [p ∗ P ( E ) ( h ( u )) ∪ t ˜ ξ ] ∩ [ ˜ ξ ] = = p ∗ P ( E ) ( h ( u )) ∩ ι ∗ ([ P ( E )]) = ι ∗ ( h ( u ) ∩ [ P ( E )]) = ι ∗ ( P D P ( E ) ( h ( u )) . Let ν = ( i, E 1 ) : E → E ′ be a morphism in the category U B of complex v ector bundles ov er B (see section 2.3). Such a morphism induces an iso morphism E ′ ∼ = E ⊕ E 1 . The c omplemen t P ( E ′ ) \ P ( E 1 ) can be identiﬁed with the total space o f the co mplex vector bundle π ∗ ( E 1 )(1) → P ( E ). Multiplication w ith the Thom class t π ∗ ( E 1 )(1) deﬁnes a morphism H ∗ ( P ( E ); Z ) − → H ∗ +2 e 1 ( π ∗ ( E 1 )(1) + P ( E ) , ∞ π ∗ ( E 1 )(1) ; Z ) ∼ = ∼ = H ∗ +2 e 1 ( P ( E ′ ) , P ( E 1 ); Z ) − → H ∗ +2 e 1 ( P ( E ′ ); Z ) , COHOMOTOPY INV ARIANTS 35 which will be denoted b y a ν . Now ﬁx an element x ∈ K ( B ). A morphism τ = ( i, j ; E 1 , F 1 , l ) : ( E , F ) → ( E ′ , F ′ ) in the categ ory T ( x ) deﬁnes morphisms a ( i,E 1 ) : H 2 f + k ( P ( E ); Z ) → H 2 f ′ + k ( P ( E ′ ); Z ) , P ( i ) ∗ : H k ( P ( E ); Z ) → H k ( P ( E ′ ); Z ) . F or a n in teger k ∈ Z w e deﬁne H k ( x ; Z ) := lim − → ( E ,F ) ∈ x H 2 f + k ( P ( E ); Z ) , H k ( x ; Z ) := lim − → ( E ,F ) ∈ x H k ( P ( E ); Z ) . Using the sa me metho ds a s in sections 2.1, 2.3 (stabilizing ﬁrst with resp ect to trivial bundle enlargements) we see that thes e inductive limits exist in A b . Remark 4. 2. (1) One has H ∗ ( x ; Z ) = H ∗ ( B ; Z ) ⊗ Z [ t ] . (2) F or a c omp act n -dimensional CW c omplex B ther e exist isomorphisms H k ( x ; Z ) ≃ M s − k ∈ 2 Z max(0 ,k − 2 ι ( x )+2) ≤ s ≤ n H s ( B ; Z ) , wher e ι ( x ) ∈ Z is the index of x . In p articular, putting n ( x ) := 2 ι ( x ) − 2 + n , one has H n ( x ) ( x ; Z ) = H n ( B ; Z ) . The integer n ( x ) := 2 ι ( x ) − 2 + n will b e called the dimension of the formal pr oje ctivization of x . Remark 4.3. S upp ose t hat B is a c omp act c onne cte d oriente d m anifold of dimen- sion n . The s ystem of Poinc ar´ e duality isomorphisms P D P ( E ) deﬁnes isomorphisms P D x : H k ( x ; Z ) ≃ − → H n ( x ) − k ( x ; Z ) . Remark 4. 4. The system of Hur ewicz morphisms h : S 1 α k B ( S ( E ) + B , F + B ) → H 2 f + k ( P ( E ); Z ) deﬁnes a morphi sms of gr ade d gr oups h x : α ∗ ( x ) → H ∗ ( x ; Z ) . If B is a c omp act c on- ne cte d oriente d manifold, one also gets a morphism P D x ◦ h x : α ∗ ( x ) → H ∗ ( x ; Z ) , which we c al l the homolo gic al Hur ewicz morphism. The res ult below has the following imp or tan t conseq ue nce : for a mo duli pr o blem with v anishing “exp ected dimension”, the co homotopy in v ariant yields the same information as the class ical (co)homolog ical inv ar iant . Recall that our cohomo- topy inv ariant { µ } asso ciated with a map s atisfying prop erties P 1 – P 3 b elongs to α b − 1 ( x ), where x := ind( δ ), b := dim ( H ) (see section 3.3). The ex p e cte d dimension w ( µ ) := 2 ι ( x ) + dim( B ) − b − 1 of the moduli problem asso cia ted with µ v anis he s if and only if b − 1 = n ( x ). Prop osition 4. 5. Supp ose that B is a ﬁnite CW c omplex of dimension n . Then the Hur ewicz morphism h n ( x ) x : α n ( x ) ( x ) − → H n ( x ) ( x ; Z ) = H n ( B ; Z ) . is an isomorphism. 36 CHRISTIAN OKONEK, ANDREI TELEMAN Pro of: Supp ose n ( x ) ≥ 0 for simplicit y . Fix a stabilizing bundle ξ . Using the same metho d a nd the same no tations as in section 4.1 .1 w e see that the set S 1 π 0 ( S ( E ) + B ∧ B ξ + B , F + B ∧ B [ R n ( x ) ] + B ∧ ξ + B ) can b e identiﬁed with the set of p ointed bundle ma ps ¯ ϕ : ˜ ξ + P ( E ) − → [ ˜ F ⊕ R n ( x ) ⊕ ˜ ξ ] + P ( E ) ov er P ( E ). The latter set ca n b e identiﬁed with H dim R ( P ( E ) ( P ( E ); Z ) = H n ( B ; Z ) by Pr opo sition 5.15 via the map ¯ ϕ 7→ h ¯ ϕ . The obta ined bijections S 1 π 0 ( S ( E ) + B ∧ B ξ + B , F + B ∧ B [ R n ( x ) ] + B ∧ ξ + B ) ≃ H n ( B ; Z ) are co mpatible with morphisms ξ → ξ ′ in the ca tegory C B and with mo rphisms ( E , F ) → ( E ′ , F ′ ) in the ca tegory T ( x ). Ther efore we get a bijectio n α n ( x ) ( x ) → H n ( B ; Z ), which c o incides with the Hurewicz map b y the deﬁnition. 4.1.2. A c omp arison t he or em. The main r esult of this sectio n states: the vir tual fundamen tal class of the mo duli space of solutions as so ciated with a map µ satis- fying prope r ties P 1, P 2, P 3 can b e identiﬁed with the image of the co homotopy inv aria n t under the homological Hur ewicz ma p. Applied to Se ib erg -Witten theor y , this implies that the ful l Seib erg -Witten type inv ariant coincides with the Hurewicz image of the cohomotopy Seiberg -Witten inv a r iant. W e begin with the ﬁnite dimens io nal case. Let B b e a co mpact oriented manifold, p : E → B , q : F → B Hermitian bundles ov er B , let V , W b e Euclide a n s paces, and le t µ : E × V → [ F × W ] + B be an S 1 -equiv aria n t map o ver B sa tisfying prop erties P 1, P2 of sec tio n 3.1. The inv ariant { µ } ∈ S 1 α b − 1 B ( S ( E ) + B , F + B ) is deﬁned by a map of pa irs ( S ( E ) × D R ( R ⊕ V ) , S ( E ) × S R ( R ⊕ V )) → ([ F × W ] + B , [ F × W ] + B \ ˚ D ε ( F × W )) induced by the r estriction µ R,ε : D R ( E ) × D R ( V ) → ( F × W ) + B of µ to a suﬃciently large cylinder D R ( E ) × D R ( V ). The v a nishing lo cus of µ (re g arded as section in the bundle ( p ∗ ( F ) × V ) × W → E × V ) is an S 1 -inv ar iant compact s pace contained in the op en subspace ˚ D R ( E ) × ˚ D R ( V ) \ [0 E × B D R ( V )] of the cylinder. Its S 1 -quotient can be iden tiﬁed with the v anis hing lo cus of the s e ction ˚ µ R,ε induced b y µ R,ε on the S 1 -quotient P ( E ) × ˚ D R ( R ⊕ V ) of S ( E ) × ˚ D R ( R ⊕ V ). Using Remark 4.1 one obtains Corollary 4.6. Supp ose t hat B is a c omp act oriente d manifold . Via the isomo r- phism H ∗ ( P ( E ) × ˚ D R ( R ⊕ V ); Z ) ≃ H ∗ ( P ( E ); Z ) the Poinc ar ´ e dual P D P ( E ) ( h ( { µ } )) c oincides with the virtual fundamental class asso ciate d with the se ct ion ˚ µ R,ε . If this se ction is smo oth and tr ansversal to t he zer o se ction, then P D P ( E ) ( h ( { µ } )) c an b e identiﬁe d with t he fundamental class of the vanishing lo cus Z ( ˚ µ R,ε ) ⊂ P ( E ) × ˚ D R ( R ⊕ V ) . Note that µ is no where v anishing outside the cylinder D R ( E ) × D R ( V ), so the v anishing lo ci of µ and µ R,ε can b e identiﬁed. The v anishing lo cus M := Z ( ˚ µ R,ε ) ∼ = Z ( µ ) /S 1 will b e c a lled the “mo duli sp ac e” a sso ciated with the map µ . Let p : E → B , q : F → B be complex Hilb ert bundles ov er B , let V , W b e real Hilb ert spaces , and let µ : E × V → F × W b e an S 1 -equiv aria n t map o ver COHOMOTOPY INV ARIANTS 37 B satis fying prop erties P 1, P 2, P 3 in sectio n 3.3. Denote by π : P ( E ) → B the natural pro jection. The map µ R,ε descends to a smoo th sec tio n ˚ µ R,ε in the bundle π ∗ ( F )(1) × ˚ D R ( R ⊕ V ) × W → P ( E ) × ˚ D R ( R ⊕ V ) , and again o ne can ident ify the mo duli spac e M := Z ( µ ) /S 1 of µ with the v an- ishing locus Z ( ˚ µ R,ε ) of this section. Using the sa me argumen t as in the pro of of Prop osition 3.15, we see that the mo duli spa ce M is compact. Supp ose now that P 4 : B is a c ompact, smo oth, connected, or ien ted manifold, µ is s moo th and the ﬁber wise diﬀerential of k := µ − d at any point is a compact op erator . This c o ndition is always satisﬁed in practica l ga uge theoretica l situa tions; indeed, the map k is usually given by the comp osition of a smo oth map E × V → F 1 × W 1 with a map F 1 × W 1 → F × W ov er B deﬁned by a smo oth family of compact op erators. The condition P 4 implies that ˚ µ R,ε is a smo oth F redholm section on the Banach ma nifo ld P ( E ) × ˚ D R ( R ⊕ V ). In order to give sense to the virtua l fundamen tal cla ss of the mo duli space M we hav e to trivialize the determinant line bundle det(index( D ˚ µ R,ε )) ov er M . Equiv alently , it suﬃces to trivia lize the line bundle det(index( D µ )) ov er Z ( µ ). In these form ula e the symbo l D stands for the family of intrinsic der iv atives of a section at its zer o lo cus, and µ is r egarded as a section of the bundle [ p ∗ ( F ) × V ] × W → E × V . F or a point ( e, v ) ∈ Z ( µ ) with p ( e ) = y o ne has a na tural identiﬁcation det(index( D ( e,v ) µ )) = Λ n ( T y ( B )) ⊗ det(index( d ( e,v ) µ E y ×V )) , where n := dim( B ) and µ E y ×V : E y × V → F y × W is the re s triction of µ to the ﬁb er ov er y . B y the condition P 4, the diﬀerential of this restriction is c ongruent with the op erator d y = ( δ y , l ) modulo a compact op erator . Therefo r e (since the family δ = ( δ y ) y ∈ B has a canonical co mplex orien tation, and B is oriented) one obtains a trivia lization of det(index( D µ )) for every or ie n tation O of coker ( l ) = H . This is precisely the or ien tation parameter in volv ed in the deﬁnition of the cohomo top y inv aria n t { µ } . Fix such an orientation O . Using the results in [Br ], we obtain a virtual fundamental class in C ˇ ech homolog y [ M ] vir ∈ ˇ H w ( M ; Z ), wher e w = w ( µ ) := n + 2 ι (ind ( δ )) − b − 1 = n (ind ( δ )) − ( b − 1 ) is the ex p ected dimension o f our mo duli pro blem (the index of the section ˚ µ R,ε ). Put x := ind( δ ), and no te that the group H w ( x ; Z ) = M 0 ≤ 2 i ≤ w H w − 2 i ( B ; Z ) ⊗ t i . can b e identiﬁed with H w ( P ( E ) × ˚ D R ( R ⊕ V ); Z ) = H w ( P ( E ); Z ). Deﬁnition 4. 7. The ful l homolo gic al invariant of µ is t he image { µ } H of the class [ M ] vir in the gr oup H w (ind( δ ); Z ) . Theorem 4.8. Su pp ose that c onditions P 1 – P 4 hold. Then { µ } H = P D x ◦ h x ( { µ } ) . Pro of: As in section 3.5 c ho ose a ﬁnite dimensional appro ximation µ c,π of π , asso ciated with an η -admissible pair ( F, W ). Deﬁne µ c,π , ∞ : D C ( E ) × D C ( V ) → F × W by µ c,π , ∞ ( e, v ) = µ c,π (p E ( e ) , p V ( v )) + p F ⊥ × W ⊥ ◦ d ◦ p E ⊥ × V ⊥ . 38 CHRISTIAN OKONEK, ANDREI TELEMAN This map takes ﬁnite v alues b y Lemma 3 .12. W e claim that ther e exists a smo oth homotopy H : [0 , 4] × D C ( E ) × D C ( V ) → F × W betw een µ D C ( E ) × D C ( V ) and µ c,π , ∞ in the s pace of S 1 -equiv aria n t F redholm maps ov er B , such that for 0 ≤ t ≤ 4 the map H t has no zero es in ∂ [ D C ( E ) × D C ( V )] ∪ 0 E × D C ( V ). T o obtain such a homotopy it suﬃces to re pla ce ˜ E , ˜ V , ˜ F , ˜ W in the deﬁnition o f the homotopy H used in the pro of o f Lemma 3.13 by E , V , F , W , and to compo se the r e s ulting map fro m the right with a smo oth homeomor phisms θ : [0 , 4] → [0 , 4] having the prop erties θ ( i ) = i , θ ( k ) ( i ) = 0 for i ∈ { 0 , 1 , 2 , 3 , 4 } , k ≥ 1 (to assure diﬀerentiabilit y). Using the homoto p y inv ariance of the virtual clas s [Br], we can ide ntify { µ } H with the image o f the virtual cla ss [ µ c,π , ∞ ] vir in H w ( P ( E ); Z ). On the other ha nd, by the “asso ciativity prop erty” of the virtual class (see Prop osi- tion 14 (4) in [Br]) and Cor ollary 4.6, the latter is just the imag e o f P D P ( E ) ( h ( { µ c,π } ) via the embedding P ( E ) → P ( E ). But P D P ( E ) ( h ( { µ c,π } ) is a repre s en tative of P D x ◦ h x ( { µ } ). 4.2. Cohomotopy inv arian t jump form ul ae. 4.2.1. Gener al r esu lt s. Le t M − → N − → P be a coﬁb er sequence of po in ted S 1 -spaces over a compact basis B . F or every po in ted S 1 -space Y ov er B there is an asso ciated long exa ct sequence of cohomotopy groups · · · → S 1 α k B ( P, Y ) → S 1 α k B ( N , Y ) → S 1 α k B ( M , Y ) ∂ − → S 1 α k +1 B ( P, Y ) → . . . . (19) The connecting mor phism ∂ : S 1 α k B ( M , Y ) = S 1 α k +1 ( M ∧ B S 1 , Y ) − → S 1 α k +1 ( P, Y ) is given by co mp ositio n with the con tr action map c : P → M ∧ B S 1 induced by a ﬁxed homoto p y equiv ale nc e b et w een P and the mapping cone o f the map M → N . F or the coﬁb er sequence S ( ξ ) + B − → D ( ξ ) + B − → ξ + B asso ciated with a vector bundle ξ over a compact bas is B , the morphism ∂ can b e describ ed in the following way . The obvious is omorphisms S ( ξ ) + B ∧ B S 1 ∼ = S ( ξ ) × [0 , 1]  S ( ξ ) × { 0 , 1 } , ξ + B ∼ = S ( ξ ) × [0 , 1]  ∼ (where ∼ is the e q uiv alence r elation generated by ( v , 0 ) ∼ ( v ′ , 0), ( v , 1) ∼ ( v ′ , 1)) allow us to us e S ( ξ ) × [0 , 1 ] /S ( ξ ) × { 0 , 1 } , S ( ξ ) × [0 , 1] / ∼ as mo dels for S ( ξ ) + B ∧ B S 1 and ξ + B . Using these models , the morphism ∂ is given by comp osition with the contraction ma p c ξ : S ( ξ ) × [0 , 1]  ∼ − → S ( ξ ) × [0 , 1]  S ( ξ ) × { 0 , 1 } (20) induced by the identit y of S ( ξ ) × [0 , 1 ]. COHOMOTOPY INV ARIANTS 39 Consider now an o riented b -dimensiona l r eal vector space H a nd the co ﬁb er sequence over B asso cia ted with the tr iv ial bundle H = B × H ov er B : S ( H ) + B ) − → D ( H ) + B − → H + B . Let E b e a Her mitian vector bundle over B . T aking s mash pro duct with S ( E ) + B ov er B yields the following coﬁb er se q uence ov er B S ( E ) + B ∧ B S ( H ) + B → S ( E ) + B → S ( E ) + B ∧ B H + B Since S ( E ) + B ∧ B S ( H ) + B = [ S ( E ) × S ( H )] + B , the asso ciated lo ng exact cohomo- topy seq uence is · · · → S 1 α − 1 B ( S ( E ) + B ∧ B H + B , [ F ⊕ H ] + B ) → S 1 α − 1 B ( S ( E ) + B , [ F ⊕ H ] + B ) → → S 1 α − 1 B ([ S ( E ) × S ( H )] + B , [ F ⊕ H ] + B ) ∂ → → S 1 α 0 B ( S ( E ) + B , [ F ] + B ) → S 1 α 0 B ( S ( E ) + B , [ F ⊕ H ] + B ) → . . . . (21) Note that one has ca nonical base change isomor phisms S 1 α k B ([ S ( E ) × S ( H )] + B , [ F ⊕ H ] + B ) ≃ S 1 α k ˜ B ( S ( ˜ E ) + ˜ B , [ ˜ F ⊕ H ] + ˜ B ) . (22) asso ciated with the pro jection p : ˜ B = B × S ( H ) → B (see [CJ] Prop osition 5.37, P r op osition 12.40 for the non-equiv a r iant cas e). A map κ : B → S ( H ) deﬁnes a section j E κ : S ( E ) + B → [ S ( E ) × S ( H )] + B ov er B of the pro jection [ S ( E ) × S ( H )] + B → S ( E ) + B , so it deﬁnes a splitting of the ex act sequence (4.2.1). Lemma 4.9. L et m ∈ S 1 α − 1 B ([ S ( E ) × S ( H )] + B , [ F ⊕ H ] + B ) , and let κ 0 , κ 1 : B → S ( H ) b e two maps. One ha s the identity ( j E κ 1 ) ∗ ( m ) − ( j E κ 0 ) ∗ ( m ) = d ( κ 0 , κ 1 ) · ∂ ( m ) , wher e wher e d ( κ 0 , κ 1 ) ∈ S 1 α − 1 B ( B + B , H + B ) = S 1 α b − 1 B ( B + B , B + B ) is the diﬀer enc e class of the maps κ 0 , κ 1 r e gar de d as se ctions in the spher e bund le S ( H ) . Pro of: The diﬀerence class d ( κ 0 , κ 1 ) is deﬁned by the map ∆ : B + B ∧ B S 1 = B × [0 , 1]  B B × { 0 , 1 } − → D ǫ ( H )  S ǫ ( H ) = H + B induced by ( b, t ) 7→  [(1 − 2 t ) κ 0 ( b )] for 0 ≤ t ≤ 1 2 [(2 t − 1) κ 1 ( b )] for 1 2 ≤ t ≤ 1 . The connecting mor phism ∂ H in the long exact s equence S 1 α − 1 B ( B + B , H + B ) ∂ H → S 1 α 0 B ( B + B , S ( H ) + B ) → S 1 α 0 B ( B + B , B + B ) → S 1 α 0 B ( B + B , H + B ) is deﬁned via the identiﬁcations S 1 α − 1 B ( B + B , H + B ) = S 1 α 0 B ( B + B ∧ B S 1 , H + B ) S 1 α 0 B ( B + B , S ( H ) + B ) = S 1 α 0 B ( B + B ∧ B S 1 , S ( H ) + B ∧ B S 1 ) , by left c o mpos ition with the contraction c H : H + B → S ( H ) + B ∧ B S 1 . The image of d ( κ 0 , κ 1 ) under ∂ H is just the diﬀerence { κ 1 } − { κ 0 } ∈ S 1 α 0 B ( B + B , S ( H ) + B ). One has obviously ( j E κ 1 ) ∗ ( m ) − ( j E κ 0 ) ∗ ( m ) = m ◦ ( { κ 1 } − { κ 0 } ) = m ◦ ∂ H ( d ( κ 0 , κ 1 )) . 40 CHRISTIAN OKONEK, ANDREI TELEMAN W e know that ∂ H ( d ( κ 0 , κ 1 )) is re pr esented by c H ◦ ∆ and the co nnecting op erator ∂ in the exa c t sequence (4.2 .1) acts by rig h t comp osition with the sa me contraction c H . Therefo r e ( j E κ 1 ) ∗ ( m ) − ( j E κ 0 ) ∗ ( m ) = m ◦ ( c H ◦ ∆) = ∂ ( m ) ◦ d ( κ 0 , κ 1 ) = ∂ ( m ) ◦ ( d ( κ 0 , κ 1 ) · { id B + B } ) = ( d ( κ 0 , κ 1 ) · ∂ ( m )) ◦ { id B + B } = d ( κ 0 , κ 1 ) · ∂ ( m ) . Here we hav e used the fact that the comp osition multiplication ◦ is S 1 α ∗ ( B )- bilinear. This lemma has an imp ortant analo gue for the groups α ∗ ( x ) a sso ciated with a K-theory element x . F o r a compa c t space P we put α ∗ ( P ; x ) = lim − → ( E ,F ) ∈ x S 1 α ∗ B ( S ( E ) + B ∧ B P + B , F + B ) . where the inductiv e limit is taken with resp ect to the categ ory T ( x ). Using the metho ds used in sec tio n 2.3 for the deﬁnition of the gro ups α ∗ ( x ), and the results in section 5.1, we s ee that this inductive limit exists; it can b e construc ted b y taking ﬁrst the limit of S 1 α ∗ B ( S ( E ⊕ C n ) + B ∧ B P + B , [ F ⊕ C n ] + B ) ov er n , and factorizing the result b y the ac tion of ˜ J ( I [ K − 1 ( B )]) ⊂ S 1 α 0 ( B ). The gr aded group α ∗ ( P ; x ) comes with a n obvious homomor phism α ∗ ( P ; x ) → α ∗ (p ∗ B ( x )), where p B : B × P → B is the pro jection on the ﬁr st summand. T aking the inductive limit o f the co nnection morphis ms ∂ = ∂ E ,F in (4.2.1) with resp ect to the category T ( x ), o ne gets a morphism ∂ x := lim − → ( E ,F ) ∈ x ∂ E ,F : α b − 1 ( S ( H ); x ) − → α 0 ( x ) . (23) which is int rinsically asso cia ted with x . Let κ : B → S ( H ) b e a ﬁxed map. The system of morphisms ( j E κ ) ∗ : S 1 α ∗ B ([ S ( E ) × S ( H )] + B , F + B ) → S 1 α ∗ B ( S ( E ) + B , F + B ) induces a mor phism j ∗ κ : α ∗ ( S ( H ); x ) → α ∗ ( x ). Corollary 4. 10. L et m ∈ α b − 1 ( S ( H ); x ) , and let κ 0 , κ 1 : B → S ( H ) b e two maps. One has the identity ( j κ 1 ) ∗ ( m ) − ( j κ 0 ) ∗ ( m ) = d ( κ 0 , κ 1 ) · ∂ x ( m ) . 4.2.2. The universal p erturb ation and the invariant jump formulae. Let E , F be Hermitian v ector bundles o ver a co mpa ct basis B , let V , W b e Euclidea n v ector spaces, and let µ : E × V → [ F × W ] + B be an S 1 -equiv aria n t map ov er B satisfying the pro p erties P1 and P2 (1 ) with h = 0 . In o ther words, µ (0 E y , v ) = l ( v ) , ∀ y ∈ B ∀ v ∈ V , where l : V ≃ − → W 0 ⊂ W is a linear em bedding . The cylinder constr uctio n cannot be applied to such a map, beca use µ has v anishing po in ts on the co re 0 E × D R ( V ) of any cy linder D R ( E ) × D R ( V ). W e orient the orthog onal co mplemen t H of W 0 in W , and we denote b y b its dimension. Let ǫ > 0. F o r every ma p κ : B → S ǫ ( H ) we deﬁne the p erturbation µ κ : E × V → [ F × W ] + B by putting µ κ ( e, v ) := T κ ( y ) ( µ ( e, v )) for e ∈ E y . Here T κ ( y ) denotes the automor- phism of [ F × ( H ⊕ W 0 )] + B which extends the tra ns lation ( f , w ) 7→ ( f , w + κ ( y )) . COHOMOTOPY INV ARIANTS 41 Remark 4.11. If ǫ > 0 is suﬃciently smal l, the map µ κ satisﬁes t he pr op erties P1 , P2 of se ction 3.1, s o the cylinder c onstruction applies and yields a stable class { µ κ } ∈ S 1 α b − 1 B ( S ( E ) + B , F + B ) . Pro of: Supp ose that µ s atisﬁes the proper t y P1 with consta n ts C , c . Cho ose ǫ < c 2 . The map µ κ satisﬁes P1 with constants C , c ′ := c 2 , and P2 with co nstant ε 0 = ǫ . Another w ay to construct a ma p satisfying prope r ties P1 , P2 is to let κ v ar y in the spher e S ǫ ( H ) and consider the universal p ertu r b ation ˜ µ : ˜ E × V − → ˜ F × W ov er the basis ˜ B := B × S ǫ ( H ) (where ˜ E := p ∗ B ( E ), ˜ F := p ∗ B ( F )) which acts as µ κ ov er B × { κ } . This map also satisﬁes pr op e rties P1 , P2 with the sa me constants as any µ κ , so that the cylinder construction applies and yields a class { ˜ µ } ∈ S 1 α b − 1 ˜ B ( S ( ˜ E ) ˜ B , ˜ F + ˜ B ). Our nex t g oal is to understand this class { ˜ µ } . The essential po int is to iden tify the imag e of { ˜ µ } ∈ S 1 α b − 1 ˜ B ( S ( ˜ E ) + ˜ B , ˜ F + ˜ B ) } under the connecting mor phis m ∂ . Recall fro m sec tion 2.6 that { o ( E ,F ) } ∈ S 1 α 0 B ( S ( E ) + B , F + B ) is the class of the obvious pointed ma p S ( E ) + B → F + B ov er B which maps + B to the inﬁnit y section, and S ( E ) to the trivial section. Prop osition 4.12. (The ∂ -image of t he invariant of the universal p ertu rb ation) Via the identiﬁc ation S 1 α 0 B ( S ( E ) + B ∧ B H + B , [ F ⊕ H ] + B ) = S 1 α 0 B ( S ( E ) + B , F + B ) one has ∂ ( { ˜ µ } ) = −{ o ( E ,F ) } . Pro of: A s in section 3 .1 ﬁx R > C and ε < min( ε 0 , c ′ ) = min( ǫ, c 2 ). Let τ 0 < R be suﬃciently small such that µ ( e, v ) rema ins ﬁnite for every ( e, v ) ∈ D τ 0 ( E ) × D R ( V ). Step 1. W e r eplace ˜ µ D R ( ˜ E ) × D R ( V ) by a map ˜ µ τ which r epresents the sa me clas s { µ } a nd coincides with the κ -indep endent map µ outside the smaller cylinder D τ ( E ) × D R ( V ). Deﬁne ˜ µ τ : D R ( ˜ E ) × D R ( V ) − → [ ˜ F × W ] + ˜ B by the formula ˜ µ τ ( e, κ, v ) :=   1 − 1 τ k e k  ( κ + l ( v )) + 1 τ k e k µ ( e, v ) for 0 ≤ k e k ≤ τ µ ( e, v ) for k e k ≥ τ . The maps ˜ µ τ and ˜ µ c oincide on the cor e 0 ˜ E × D R ( V ) of the cylinder D R ( ˜ E ) × D R ( V ) a nd they diﬀer by the tr anslation T κ outside D τ ( ˜ E ) × D R ( V ). W e deﬁne a homotopy b etw een ˜ µ τ and ˜ µ D R ( ˜ E ) × D R ( V ) by putting ˜ µ t τ ( e, κ, v ) :=  (1 − t ) ˜ µ τ ( e, κ, v ) + t ˜ µ ( e, κ, v ) for k e k ≤ τ T tκ ◦ µ ( e, v ) for k e k ≥ τ . Claim: If τ is suﬃciently small, then k ˜ µ t τ k ≥ c ′ on ∂ h D R ( ˜ E ) × D R ( V ) i for e very t ∈ [0 , 1]. 42 CHRISTIAN OKONEK, ANDREI TELEMAN The claim is not obvious only for p oints ( e, v ) ∈ D τ ( ˜ E ) × S R ( V ). One has the ident it y ˜ µ t τ ( e, κ, v ) = (1 − t )  1 − 1 τ k e k  κ + l ( v ) + 1 τ k e k [ µ ( e, v ) − l ( v )]  + tµ ( e, v ) + tκ = = l ( v ) +  1 − 1 − t τ k e k  κ +  t + (1 − t ) τ k e k  [ µ ( e, v ) − l ( v )] . The ﬁrs t tw o terms b elong to orthog onal complements, so for e ∈ D τ ( E ) o ne has k ˜ µ t τ ( e, κ, v ) k ≥ k l ( v ) k − k µ ( e, v ) − l ( v ) k . Since µ (0 E y , v ) = l ( v ), and µ is ﬁb erwis e diﬀere ntiable with g lobally contin uo us deriv atives o n E × V , it holds lim τ → 0  sup { ( µ ( e, v ) − l ( v )) 0 ≤ k e k ≤ τ , k v k ≤ R }  = 0 . On the other hand, for k v k = R one has k l ( v ) k = k µ (0 E y , v ) k > c . This prov es the claim. Using the Claim and k ˜ µ t τ ( e, κ, v ) k = k κ k = ǫ > 0 we see tha t ( ˜ µ t τ ) t ∈ [0 , 1] deﬁnes a homotopy b et ween ˜ µ τ and ˜ µ D R ( ˜ E ) × D R ( V ) in the space of maps for which the cylinder constr uction applies. Therefo r e { ˜ µ } = { ˜ µ τ } ∈ S 1 α b − 1 B ([ S ( E ) × S ( H )] + B , F + B ) for a ll suﬃciently small τ > 0 . (24) Step 2. W e compute the class − ∂ ( { ˜ µ τ } ). Regard { ˜ µ τ } as a n element in the gr oup S 1 α − 1 B ([ S ( E ) × S ( H )] + B , [ F ⊕ H ] + B ) = S 1 α 0 B ([ S ( E ) + B ∧ B S ( H ) + B ∧ B S 1 , [ F ⊕ H ] + B ) . As e x plained at the beginning of this sec tio n the morphism ∂ is giv en by c ompo sition with the cont raction map c H : H + = S ǫ ( H ) × [0 , R ]  ∼ H → S ( H ) + ∧ S 1 = S ǫ ( H ) × [0 , R ]  S ( H ) × { 0 , R } induced by the iden tit y of S ǫ ( H ) × [0 , R ]. The morphism − ∂ is deﬁned by co mpo - sition with c ′ , where c ′ is induced by the map ( κ, ρ ) → ( κ, R − ρ ). The clas s { ˜ µ τ } is repr esented by the map ˜ m τ : S ( E ) × S ǫ ( H ) × [0 , R ] × D R ( V ) − → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) given by ˜ m τ ( e, κ, ρ, v ) = [ ˜ µ τ ( ρe, κ, v )] . As we hav e seen in section 3.1, this map induces a map S ( E ) + B ∧ B S ( H ) + B ∧ B S 1 ∧ B V + B − → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) bec ause it ha s the following prop erties (1) ˜ m τ ( e, κ, 0 , v ) a nd m τ ( e, κ, R , v ) belo ng alwa y s to the inﬁnit y section of the right ha nd space, (2) ˜ m τ ( e, κ, ρ, v ) belongs to the inﬁnity section of the righ t hand space when k v k = R . COHOMOTOPY INV ARIANTS 43 The class − ∂ ( { ˜ µ τ } ) is deﬁned by the map ˜ m ′ τ : S ( E ) × S ǫ ( H ) × [0 , R ] × D R ( V ) → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) given by ˜ m ′ τ ( e, κ, ρ, v ) = ˜ m τ ( e, κ, R − ρ, v ) . This map descends to a map S ( E ) + B ∧ B H + B ∧ B V + B → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) bec ause it ha s the following prop erties: (1) ˜ m ′ τ ( e, κ, 0 , v ) and ˜ m ′ τ ( e, κ, R , v ) are indep endent of κ . (2) ˜ m ′ τ ( e, κ, R , v ) b elongs always to the inﬁnity section o f the right hand space. (3) ˜ m ′ τ ( e, κ, ρ, v ) belongs to the inﬁnity section of the righ t hand space when k v k = R . These three conditions characterize the maps of p ointed spaces ov er B deﬁned o n S ( E ) × S ǫ ( H ) × [0 , R ] × D R ( V ) which descend to S ( E ) + B ∧ B H + B ∧ B V + B . Step 2 (a). W e deform the map ˜ m ′ τ in the space of maps satisfying the three prop erties ab ov e, by comp osing it with a 1-pa rameter family of con tractions in the ρ -direction. F or t ∈ [0 , 1 ] deﬁne the map [ ˜ m ′ τ ] t : S ( E ) × S ǫ ( H ) × [0 , R ] × D R ( V ) → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) by [ ˜ m ′ τ ] t ( e, κ, ρ, v ) = ˜ m τ  e, κ, (1 − t + t τ R )( R − ρ ) , v  . The family ([ ˜ m ′ τ ] t ) t ∈ [0 , 1] deﬁnes a homotopy in the space of maps satisfying prop- erties (1), (2 ), (3) ab ov e. The main p o in t in checking (1) is the fact that the map ˜ m τ is constant with resp ect to κ for ρ ∈ [ τ , R ]. Therefo r e it holds − ∂ ( { ˜ µ τ } ) = { [ ˜ m ′ τ ] 0 } = { [ ˜ m ′ τ ] 1 } . Putting ˜ m ′′ τ := [ ˜ m ′ τ ] 1 , one has ˜ m ′′ τ ( e, κ, ρ, v ) = ˜ m τ ( e, κ, τ R ( R − ρ ) , v ) =  1 − R − ρ R  ( κ + l ( v ))+ R − ρ R µ ( τ R − ρ R e, v ) = ρ R κ + l ( v ) + R − ρ R  µ ( τ R − ρ R e, v ) − l ( v )  . Step 2 (b). W e remark that the family of maps ˜ m ′′ τ has a uniform limit as τ → 0 and we co mpute this limit explicitly . Using a rguments as in the pro of o f the claim ab ov e, we s ee that lim τ → 0 R − ρ R  µ ( τ R − ρ R e, v ) − l ( v )  = 0 uniformly . Therefor e ˜ m ′′ := lim τ → 0 ˜ m ′′ τ op erates by ˜ m ′′ ( e, κ, ρ, v ) = ρ R κ + l ( v ). It is now easy to see that the ma p S ( E ) + B ∧ B H + B ∧ B V + B → [ F × W ] + B  B [ F × W ] + B \ D ε ( F × W ) = F + B ∧ B H + B ∧ B [ W 0 ] + B 44 CHRISTIAN OKONEK, ANDREI TELEMAN induced by ˜ m ′′ is homotopic to the smash pro duct ov er B o f the obvious ma p S ( E ) + B → F + B (whic h repr esents o ( E , F )) with l + B : V + B → [ W 0 ] + B , and id : H + B → H + B . F or a map κ : B → S ǫ ( H ) one has { µ κ } = ( j E κ ) ∗ ( { ˜ µ } ) . (25) This form ula shows that the individual in v ariant { µ κ } asso ciated with a map κ : B → S ǫ ( H ) is determined by the in v ar iant asso ciated with the universal p er- turbation ˜ µ and the homoto p y class of κ . Using Co rollary 4.10 we obtain Corollary 4.13. (Cohomo topy invaria nt jump formula) On e has { µ κ 0 } − { µ κ 1 } = o ( E ,F ) · d ( κ 0 , κ 1 ) , wher e d ( κ 0 , κ 1 ) ∈ S 1 α − 1 B ( B + B , H + B ) is the diﬀer en c e class of the maps κ 0 , κ 1 r e- gar de d as s e ctions in the spher e bu nd le S ǫ ( H ) . Suppo se no w that b = 1. In this case S ǫ ( H ) has tw o elements κ 0 , κ 1 , and the diﬀerence class d ( κ 0 , κ 1 ) is just the unit element of S 1 α 0 B ( B + B , B + B ). Therefor e, in this ca s e, our res ult gives Corollary 4. 14. (Cohomotopy wal l cr ossing) Supp ose b = 1 . Then the two classes { µ κ 0 } , { µ κ 1 } asso ciate d with the two p erturb ations µ κ 0 , µ κ 1 of µ ar e r elate d by the formula { µ κ 0 } − { µ κ 1 } = { o ( E , F ) } . W e can now extend our r esults to the inﬁnite dimensiona l case. Let B be an oriented co mpact manifo ld, E , F complex Hilbert bundles ov er B , V , W real Hilbert spaces, and µ : E × V → F × W an S 1 -equiv aria n t, ﬁb erwise diﬀerent iable map ov er B s atisfying prop erties P 1 , P 3 a nd P 2 (1) with h = 0. Then we have an orthogo nal decompo s ition W = H ⊕ W 0 , and µ (0 E y , v ) = l ( v ) for every v ∈ V , wher e l : V → W 0 is a linear isometry . W e ﬁx an or ien tation of the ﬁnite dimensional summand H . Deﬁning in the same wa y as in the ﬁnite dimensiona l framework the universal p erturba tion ˜ µ , one g ets a stable cla ss { ˜ µ } ∈ α ∗ ( S ǫ ( H ); x ) , where x ∈ K ( B ) is the index of the complex part of the ﬁberwis e lineariza tio n of µ at the zero section. Recall that the E uler class γ ( x ) ∈ α 0 ( x ) is deﬁned b y the system of stable classe s −{ o ( E ,F ) } ∈ S 1 α 0 B ( S ( E ) + B , F + B ) deﬁned by the o b vious maps S ( E ) + B → F + B (see sec tio n 2.6). Using the results obta ined ab ov e and taking inductive limit ov e r T ( x ), w e obtain Corollary 4.15. (1) The image of { ˜ µ } u nder the morphism ∂ x : α b − 1 ( S ǫ ( H ); x ) → α 0 ( x ) is given by ∂ x ( { ˜ µ } ) = γ ( x ) . (2) L et κ 0 , κ 1 : B → S ( H ) two maps. Then { µ κ 1 } − { µ κ 0 } = d ( κ 0 , κ 1 ) · γ ( x ) . (3) Supp ose b = 1 and wri te S ǫ ( H ) = { κ 0 , κ 1 } . Then { µ κ 1 } − { µ κ 0 } = γ ( x ) . COHOMOTOPY INV ARIANTS 45 4.3. A pro duct formula and a v anishing theorem. In this section we give the inﬁnite dimensional analogue of the pro duct formula prov en in section 3.2 .3. Let V i , W i be rea l Hilb ert spaces , E i , F i complex Hilber t bundles over a compact base B ( i = 1 , 2), and let µ i : E i × V i → [ F i × W i ] + B be S 1 -equiv aria n t maps ov e r B , satisfying the prop erties P 1, P 2 (1 ) and P 3 o f section 3.3 with constants C , c . Let W i = H i ⊕W 0 ,i be the corr esp onding orthogo nal sum deco mpos itions, l i : V i ≃ → W 0 ,i isometries, x i ∈ K ( B ) the K -theory elements deﬁned by the corr espo nding families δ i of F redholm o per ators, and h i : B → H i the maps g iv en by P 2 (1). W e intro duce the notatio ns : V := V 1 ⊕ V 2 , W := W 1 ⊕ W 2 , H := H 1 ⊕ H 2 , W 0 := W 0 , 1 ⊕ W 0 , 2 , l := l 1 ⊕ l 2 , and consider the Hilb ert bundles E := E 1 ⊕ E 2 , F := F 1 ⊕ F 2 . The pro duct map µ : E × V = [ E 1 × V 1 ] ⊕ [ E 2 × V 2 ] − → [ F × W ] + B = [ F 1 × W 1 ] + B ∧ B [ F 2 × W 2 ] + B also sa tisﬁes pro perties P 1 P 2 (1) (with asso ciated ma p h = ( h 1 , h 2 ) : B → H ) and P 3; it sa tisﬁes P 2 (2) as s o o n as one o f the tw o maps µ 1 , µ 2 do es. Suppo se that µ 1 satisﬁes prop erty P 2 (2). In this case the construction of section 3.3 applies and yields an inv ar ia n t { µ 1 } ∈ α b 1 − 1 ( x 1 ) . The ﬁnite dimensiona l approximations of the map µ 2 deﬁne clas ses { ( µ 2 ) + c,π 2 } ∈ S 1 α b 2 B ([ E 2 ] + B , [ F 2 ] + B ) . It can b e s hown that a compatibility result similar to P rop osition 3.13 holds, so that one obtains an inv ariant { µ + 2 } ∈ α b 2 ( x + 2 ) := lim − → ( E 2 ,F 2 ) ∈ x 2 S 1 α b 2 B ([ E 2 ] + B , [ F 2 ] + B ) . Here the inductiv e limit on the rig h t is ta ken ov er the categor y T ( x 2 ) and is con- structed using the same metho ds as in the deﬁnition of the groups α ∗ ( x ) (see sectio n 2.3). The direct limit of the obvious pro ducts S 1 α b 1 − 1 B ( S ( E 1 ) + B , [ F 1 ] + B ) × S 1 α b 2 B ([ E 2 ] + B , [ F 2 ] + B ) → S 1 α b 1 + b 2 − 1 B ( S ( E 1 ) + B ∧ B [ E 2 ] + B 2 , [ F 1 ⊕ F 2 ] + B ) 1 c ∗ → S 1 α b 1 + b 2 − 1 B ( S ( E 1 ⊕ E 2 ) + B , [ F 1 ⊕ F 2 ] + B ) gives a well deﬁned pro duct · : S 1 α b 1 − 1 ( x 1 ) × α b 2 ( x + 2 ) → S 1 α b 1 + b 2 − 1 ( x 1 + x 2 ) . Using ﬁnite dimensio nal approximations of µ o f the form µ c,π 1 × π 2 = ( µ 1 ) c,π 1 × ( µ 2 ) c,π 2 and applying P rop osition 3.3 we obtain Remark 4.16. Under the assumptions and with the notations ab ove, the invariant of the pr o duct m ap µ = µ 1 × µ 2 is given by the formula { µ 1 × µ 2 } = { µ 1 } · { µ + 2 } . Note that in this formula the map µ 2 is allowed to hav e S 1 -inv ar iant z e ro es. In the case when b oth maps µ i satisfy P 2 (2) (so they are nowhere zero o n their S 1 - ﬁxed p oint lo ci) o ne ha s the following impor tant v anishing result for the Hure w ic z image of the inv aria n t a sso ciated with a pro duct map: 46 CHRISTIAN OKONEK, ANDREI TELEMAN Prop osition 4.1 7. Put x := x 1 + x 2 ∈ K ( B ) and let h x : α ∗ ( x ) → H ∗ ( x ; Z ) b e the Hur ewicz morphism asso ciate d with x . Supp ose that b oth maps µ i satisfy pr op erties P 1 , P 2 (1), P 2 (2) and P 3 , and t hat B is a ﬁn ite CW c omplex. Then h x ( { µ 1 × µ 2 } ) = 0 . Pro of: Let m i := ( µ i ) c,π i be ﬁnite dimensional approximations of µ i and put m := m 1 × m 2 . Applying the cylinder construction to this ma ps we get a represe n- tative m R : S ( E 1 ⊕ E 2 ) + B ∧ B [ R ⊕ V 1 ⊕ V 2 ] + B → [ F 1 ⊕ F 2 ⊕ W 1 ⊕ W 2 ] + B of the class { µ 1 × µ 2 } . Put E := E 1 ⊕ E 2 , F := F 1 ⊕ F 2 , V := V 1 ⊕ V 2 , W := W 1 ⊕ W 2 , and b = b 1 + b 2 . Let ¯ m R : [ R ⊕ V ] + P ( E ) → [ ˜ F ⊕ W ] + P ( E ) be the asso ciated sphere bundle map, constructed as in section 4.1.1. W e denote by p : [ R ⊕ V ] + P ( E ) → P ( E ) , q : [ ˜ F ⊕ W ] + P ( E ) → P ( E ) the t wo bundle pr o jections, a nd b y h := h ¯ m R ∈ H 2 f + b 1 + b 2 − 1 ( P ( E ); Z ) the corre- sp onding Hurewicz cla s s, which is deﬁned by the equality ( ¯ m R ) ∗ (t ˜ F ⊕ W ) = p ∗ ( h ) ∪ t R ⊕ V (26) in H ∗ ([ R ⊕ V ] + P ( E ) , ∞ R ⊕ V ; Z ). Since both ma ps µ i satisfy pro per t y P2 , it follows that, for a suﬃciently small neig h bo r ho o d P o f P ( E 1 ) ∪ P ( E 2 ) in P ( E ), the map ¯ m R maps p − 1 ( P ) to the inﬁnity sectio n of the right hand bundle. W e can suppo se that P is a standar d compact neighbor ho od of this union, i.e. it has the fo rm P = P ( E ) \  [ e 1 , e 2 ] ∈ P ( E ) | e i 6 = 0 , ln k e 1 k k e 2 k ∈ ( − s s )  for s uﬃcie ntly lar ge s > 0. The pull-back cla ss ( ¯ m R ) ∗ (t ˜ F ⊕ W ) c a n b e regarded as an elemen t in H ∗ ([ R ⊕ V ] + P ( E ) , ∞ R ⊕ V ∪ p − 1 ( P ); Z ), which can b e identiﬁed with H ∗− (dim( V )+1) ( P ( E ) , P ; Z ) via the relative Thom iso morphism o ver the pair ( P ( E ) , P ). Ther efore, the equality ( ¯ m R ) ∗ (t ˜ F ⊕ W ) = p ∗ ( h ′ ) ∪ t R ⊕ V (27) in H ∗ (([ R ⊕ V ] + P ( E ) , ∞ R ⊕ V ∪ p − 1 ( P ); Z ) deﬁnes a cla ss h ′ ∈ H ∗ ( P ( E ) , P ; Z ), and h is just the image of h ′ via the morphism C ∗ : H ∗ ( P ( E ) , P ; Z ) → H ∗ ( P ( E ); Z ) asso ciated with the map C : ( P ( E ) , ∅ ) → ( P ( E ) , P ). P ut now P 0 := P ( E ) \ ( P ( E 1 ) ∪ P ( E 2 )) , P 0 := P \ ( P ( E 1 ) ∪ P ( E 2 )) , and denote b y h ′ 0 the image of h ′ via the morphism I ∗ : H ∗ ( P ( E ) , P ; Z ) → H ∗ ( P 0 , P 0 ; Z ) deﬁned by the map I : ( P 0 , P 0 ) → ( P ( E ) , P ). The ma in p oint in the pro of o f our prop osition is that the restriction ¯ m R P 0 : p − 1 ( P 0 ) → q − 1 ( P 0 ) . is eq uiv ariant with r espe ct to the fr e e S 1 -action ( ζ , [ e 1 , e 2 ]) 7→ [ ζ e 1 , e 2 ] o n P 0 and the obvious lift o f this action in the bundle ˜ F P 0 . This is just b ecause µ is the pro duct o f tw o S 1 -equiv aria n t maps µ i . Therefore, ¯ m R P 0 descends to a bundle map [ ¯ n R ] 0 : p − 1 ( P 0 )  S 1 − → q − 1 ( P 0 )  S 1 COHOMOTOPY INV ARIANTS 47 ov er Q 0 := P 0 /S 1 . The tw o spher e bundles ab ov e coincide with the ﬁbrewise compactiﬁcations [ R ⊕ V ] + Q 0 , [ ˜ F 0 ⊕ W ] + Q 0 , where ˜ F 0 is the S 1 -quotient of ˜ F , reg arded as a bundle ov er Q 0 . W e denote b y p 0 , q 0 the co rresp onding bundle pro jections on Q 0 . Put Q := P /S 1 , Q 0 := Q ∩ Q 0 . Using the r elative Thom isomor phism ov er the pair ( Q 0 , Q 0 ), it follows that the equality [ ¯ n R ] ∗ 0 (t ˜ F 0 ⊕ W ) = p ∗ 0 ( k 0 ) ∪ t R ⊕ V deﬁnes a class k 0 ∈ H ∗ ( Q 0 , Q 0 ; Z ). T aking the pull-back of this equality via the pro jection Π 0 : ( P 0 , P 0 ) → ( Q 0 , Q 0 ), (and comparing the o btained formula with a similar equality satisﬁed by h ′ 0 ), we see that Π ∗ 0 ( k 0 ) = h ′ 0 . Therefore h = C ∗ ◦ I ∗ − 1 ◦ Π ∗ 0 ( k 0 ) = C ∗ ◦ Π ∗ ◦ [ J ∗ ] − 1 ( k 0 ) , (28) where Π : ( P ( E ) , P ) →  P ( E )  S 1 , Q  , J : ( Q 0 , Q 0 ) →  P ( E )  S 1 , Q  denote the obvious maps. In this formula we used the ident it y J ◦ Π 0 = Π ◦ I , and that the maps I , J induce isomorphisms in cohomology , by the excis ion theorem. The result follows now directly from Lemma 4.18 b elow. Lemma 4 .18. The morphism U ∗ : H ∗  P ( E )  S 1 , Q ; Z  − → H ∗ ( P ( E ); Z ) induc e d by t he m ap U := Π ◦ C : ( P ( E ) , ∅ ) →  P ( E )  S 1 , Q  , vanishes. Pro of: By the excision a nd homotopy in v ariance theorem o ne has H ∗  P ( E )  S 1 , Q ; Z  = H ∗  P ( E )  S 1 \ ˚ Q , Q \ ˚ Q ; Z  , where ˚ Q is the interior of Q . One has a natural homeomo r phism P ( E )  S 1 \ ˚ Q ∼ = [ P ( E 1 ) × B P ( E 2 )] × [ − s, s ] , [ e 1 , e 2 ] 7→  [ e 1 ] , [ e 2 ] , ln k e 1 k k e 2 k  , and this homeo morphism identiﬁes Q \ ˚ Q with [ P ( E 1 ) × P ( E 2 )] × {− s, s } . Multi- plication with the Thom clas s of the trivial bundle P ( E 1 ) × B P ( E 2 ) × ( − s, s ) → P ( E 1 ) × B P ( E 2 ) deﬁnes an isomorphism H i ( P ( E 1 ) × B P ( E 2 ); Z ) ∼ = → H i +1  P ( E )  S 1 \ ˚ Q , Q \ ˚ Q ; Z  = H i +1  P ( E )  S 1 , Q ; Z  . Step 1. When B is a po int, the statement of the Lemma is obvious b ecause in this ca se b oth s paces P ( E 1 ) × B P ( E 2 ) and P ( E ) hav e trivial cohomolog y in o dd dimensions. Step 2. F or a general bas is, note that U induces a mor phism of the Leray sp ectra l sequences asso ciated with the pr o jections P ( E ) − → B ,  P ( E )  S 1 , Q  − → B . 48 CHRISTIAN OKONEK, ANDREI TELEMAN But the Leray s pectr al s equence for the r elative cohomology of the pair  P ( E )  S 1 , Q  can b e identiﬁed with the sp ectral sequence fo r the c ohomology w ith compact sup- po rts of P ( E )  S 1 \Q . It s uﬃces to note that the induced spectra l se quence mo rphism v anishes at the E p,q 1 -level, by Step 1 . 5. Appendix 5.1. Inductiv e lim its of functors. W e r ecall the following impo rtant Deﬁnition 5.1. ( [AM] p. 148) A ﬁ ltering c ate gory is c ate gory C with the pr op erties F1. F or every p air ( O, O ′ ) of obje cts, ther e exist s an ob je ct O ′′ and morphisms O → O ′′ , O ′ → O ′′ . F2. F or every two morphisms u , v : O → O ′ ther e exists an obje ct O ′′ and a morphism w : O ′ → O ′′ such that w ◦ u = w ◦ v . F or sm al l ﬁltering catego ries one has the following basic fact: Prop osition 5.2. ( [AM] , p. 149-150) L et A b e one of t he c ate gories S ets , A b or G r , and let C b e a ﬁltering smal l c ate gory. Then any functor F : C → A has an inductive limit, which c an c onstructe d in the classic al way: one factorizes the disjoint union ` O ∈O b ( C ) F ( O ) by the e quivalenc e r elation ( O, x ) ∼ ( O ′ , x ′ ) if ∃ u : O → O ′′ , u ′ : O ′ → O ′′ with F ( u )( x ) = F ( u ′ )( x ′ ) . (29) When A = A b or G r , one endows the obtaine d set of e quivalenc e classes with the op er ation induc e d by the gr oup op er ations on the su mmands F ( O ) of the disjoint union. W e will say that C is we akly ﬁ ltering if it satisﬁes F1 and the fo llowing weak form of the ax iom F 2. ˜ F2. F or every tw o morphisms u , v : O → O ′ there exists an ob ject O ′′ and morphisms w , z : O ′ → O ′′ such that w ◦ u = z ◦ v . Lemma 5.3. Supp ose that C is we akly ﬁltering and smal l. Then the r elation ∼ deﬁne d in (29) is s t il l an e quivalenc e r elation, and the c onclus ion of Pr op osition 5.2 holds for A = S ets . Pro of: It suﬃces to chec k that ∼ is tra ns itiv e. Let x ∈ F ( O ), x ′ ∈ F ( O ′ ), x ′′ ∈ F ( O ′′ ) with x ∼ x ′ , x ′ ∼ x ′′ . Ther efore there exists mo r phisms u : O → ˆ O , u ′ : O ′ → ˆ O , v ′ : O ′ → ˜ O , v ′′ : O ′′ → ˜ O such that F ( u )( x ) = F ( u ′ )( x ′ ) a nd F ( v ′ )( x ′ ) = F ( v ′′ )( x ′′ ). By F1 there exists morphis ms ˆ w : ˆ O → O 0 , ˜ w : ˜ O → O 0 . W e a pply ˜ F2 to the morphisms ˆ w u ′ , ˜ w v ′ : O ′ → O 0 . W e obtain morphisms ˆ z , ˜ z : O 0 → O 1 such that ˆ z ˆ wu ′ = ˜ z ˜ wv ′ . Therefo r e F ( ˆ z ˆ w u )( x ) = F ( ˆ z ˆ w )( F ( u )( x )) = F ( ˆ z ˆ w )( F ( u ′ )( x ′ )) = F ( ˆ z ˆ w u ′ )( x ′ ) = = F ( ˜ z ˜ wv ′ )( x ′ ) = F ( ˜ z ˜ w )( F ( v ′ )( x ′ )) = F ( ˜ z ˜ w )( F ( v ′′ )( x ′′ )) = F ( ˜ z ˜ w v ′′ )( x ′′ ) , hence x ∼ x ′′ . F or A = A b o r G one canno t endow the quo tien t of the disjoint union by this equiv alence relation with a coherent gro up structure using only the weakly ﬁltering condition. Unfortunately , w e will need inductiv e limits of functors deﬁned on index c a te- gories whic h are not small. In this case the disjoint unio n considered in Remark COHOMOTOPY INV ARIANTS 49 5.2 might not b e a set. How ever, there exists a simple situation when the existence of an inductive limit is guara n teed: Lemma 5.4. L et C b e a we akly ﬁltering c ate gory, Q ∈ O b ( C ) a ﬁ xe d obje ct and F : C → A a fun ctor su ch that F ( u ) is surje ct ive for every morph ism u : Q → O . (1) Supp ose A = S ets . (a) The r elation on F ( Q ) deﬁne d by y ≈ y ′ if ∃ u, v : Q → O such that F ( u )( y ) = F ( v )( y ′ ) (30) is an e quivalenc e re lation. Put L := F ( Q ) / ≈ . (b) F or any O ∈ O b ( C ) ther e exists a u n ique map f O : F ( O ) → L deﬁne d by f O ( x ) = [ y ] for any p air ( x, y ) ∈ F ( O ) × F ( Q ) for which ther e ex ist morphisms u : O → ˆ O , v : Q → ˆ O with F ( u )( x ) = F ( v )( y ) . The system ( f O ) O ∈O b ( C ) is F -c omp atible (i.e. it holds f O ′ ◦ F ( w ) = f O for any morphism w : O → O ′ ). (c) The system ( f O ) O ∈O b ( C ) satisﬁes the universal pr op erty of the inductive limit, so the inductive limit of F exists and c an b e identiﬁe d with L . (2) Supp ose A = A b or G r . (a) L et H b e a smallest normal sub gr oup of F ( Q ) which c ontains the ele- ments x ′ x − 1 with x ≈ x ′ . Put L := F ( Q ) /H . (b) The system of morphism ( f O : F ( O ) → L ) O ∈O b ( C ) deﬁne d in a similar way as in (1) is F - c omp atible and satisﬁes t he universal pr op erty of the inductive limit. Ther efor e the inductive limit of F exists and c an b e identiﬁe d with L . Pro of: (1) (a) is clear . F or (b) we have to prov e that the map f O is well deﬁned. Let y ∈ F ( Q ), y ′ ∈ F ( Q ), u : O → ˆ O , v : Q → ˆ O , u ′ : O → ˆ O ′ , and v ′ : Q → ˆ O ′ such that F ( u )( x ) = F ( v )( y ) and F ( u ′ )( x ) = F ( v ′ )( y ). W e can ﬁnd an ob ject ˜ O and morphis ms w : ˆ O → ˜ O , w ′ : ˆ O ′ → ˜ O . Since C is weakly ﬁltering, the ex is t morphisms z : ˜ O → O 0 , z ′ : ˜ O → O 0 such that z w u = z ′ w ′ u ′ . This implies F ( z wv )( y ) = F ( z w )( F ( u )( x )) = F ( z ′ w ′ )( F ( u ′ )( x )) = F ( z wv ′ )( y ′ ) , so y ≈ y ′ . The F -compatibility of the system ( f O ) O ∈O b ( C ) and the fact that this system satisﬁes the universal pro per t y o f the inductive limit are easily veriﬁed. (2) F ollows easily fr om (1). Deﬁnition 5.5. ( [AM] p. 149) L et N , C b e c ate gories. A funct or Θ : N → C is c al le d (1) c oﬁnal, if C1. F or any O ∈ O b ( C ) ther e exists n ∈ O b ( N ) and u : O → Θ( n ) . C2. F or every n ∈ O b ( N ) , O ∈ O b ( C ) , and u : Θ ( n ) → O , ther e exists m ∈ O b ( N ) , ν : n → m and v : O → Θ( m ) such that v u = Θ( ν ) . (2) c oﬁnal in the sense of Artin-Mazur ( [AM] p. 149) , if C1. holds, ˜ C2. F or every O ∈ O b ( C ) , n ∈ O b ( N ) and u , v : O → Θ( n ) , ther e exists a morphism µ : n → m in N such that Θ ( µ ) u = Θ ( µ ) v . Lemma 5 .6. (1) If N is ﬁltering and Θ is c oﬁnal in t he sense of Artin-Mazur, then Θ is c oﬁn al and C is ﬁltering. 50 CHRISTIAN OKONEK, ANDREI TELEMAN (2) If C is ﬁltering and Θ is c oﬁnal, then Θ is c oﬁnal in the sense of Artin- Mazur. (3) Supp ose Θ : N → C is c oﬁn al, a nd N , C ar e b oth smal l and ﬁltering. F or any functor F : C → A (with A = S ets , A b or G r ) the c anonic al morphism lim − → n ∈O b ( N ) F (Θ ( n )) → lim − → O ∈O b ( C ) F ( O ) is an isomorphism. Pro of: 1. Let u : Θ ( n ) → O b e a morphism. Using C1, we ca n ﬁnd a morphism w : O → Θ( m ); since N is ﬁltering, we can ﬁnd morphisms η : n → k , κ : m → k . Therefore, we get t w o morphis ms Θ( η ), Θ( κ ) wu : Θ( n ) → Θ( k ). By ˜ C2, there e x ists µ : k → l such that Θ( µ )Θ( η ) = Θ( µ )Θ( κ ) w u . This shows [Θ( µκ ) w ] u = Θ( µη ), so C2 holds with v = Θ( µκ ) w and ν = µη . The fact that C is ﬁltering is s ta ted in [AM] p. 1 49. 2. Let u , v : O → Θ( n ) b e tw o morphis ms . Since C is ﬁltering, there exists w : Θ ( n ) → O ′ with w u = w v . By C2, we can ﬁnd m ∈ O b ( N ), ν : n → m and v ′ : O ′ → Θ( n ), suc h tha t v ′ w = Θ( ν ). W e will ha ve Θ( ν ) u = v ′ wu = v ′ wv = Θ( ν ) v , which proves ˜ C2. 3. See Prop osition 1.8 in [AM ] p. 15 0. Example 1. Let B be a co mpact space and let U B be the category of complex vector bundles over B . A morphism U → U ′ is a pa ir u = ( i, U 1 ) co nsisting o f a bundle embedding i : U → U ′ and a complemen t U 1 of i ( U ) in U ′ (see section 2.3). The categor y U B satisﬁes F1 but not F2, so it is not ﬁltering. Let N be category asso ciated with the or dered set ( N , ≤ ). Then the functor Θ : N → U B which asso cia tes to n the trivial bundle C n and to an inequalit y n ≤ m the standard morphism C n → C m is coﬁnal. This follows from the fact that a n y v ector bundle on B p ossesses a complement . Note ho wever that Θ is not coﬁnal in the sense of Artin-Mazur. Example 2. F or a category C and an ob ject Q ∈ O b ( C ) we will deno te by C Q the category whose o b jects are morphism u : Q → O a nd whose morphisms are Hom( Q u → O , Q v → O ′ ) := { w : O → O ′ | w ◦ u = v } . A morphism u : Q → Q ′ induces in an obvious wa y a pull-back functor u ∗ : C Q ′ → C Q . If C is ﬁltering then C Q is ﬁltering and the target functor functor T : C Q → C is b oth coﬁna l and coﬁnal in the sense of Artin-Maz ur . Deﬁnition 5. 7. A c ate gory with automorphism push-forwar d is a p air ( U , A ) , wher e U is a c ate gory and A : U → G r a functor, su ch that F1. holds in U . S1. A ( O ) = Aut( O ) for every O ∈ O b ( C ) . S2. F or any u : O → O ′ and a ∈ Aut( O ) one has A ( u )( a ) ◦ u = u ◦ a S3. F or every two morphisms u , v : O → O ′ in U ther e exists an obje ct O ′′ , a morphism w : O ′ → O ′′ and an automorphism a ∈ A ( O ′′ ) su ch that a ◦ w ◦ u = w ◦ v . Note that when ( U , A ) is a ca tegory with automorphism push- fo rward, then U is weakly ﬁltering (use S3). COHOMOTOPY INV ARIANTS 51 Example 3. Deﬁning the automorphism push-forward functors in the obvious wa y , the catego ries U B , C B , T ( x ) in tro duced in this article b ecome categor ies with automorphism push-forward. Let ( U , A ) b e a catego ry with automorphism push-forward, Q ∈ O b ( U ) a ﬁxed ob ject, and F : U → A b a functor suc h that F ( u ) is a iso mo rphism for any morphism u : Q → O . W e know by Lemma 5.4 that the inductiv e limit o f F exists and is a quotient of F ( Q ). W e need an explicit description of this quotient. F o r ev ery ob ject u : Q → O in the catego r y U Q the group A ( T ( u )) a cts on F ( Q ) v ia the isomorphism F ( u ) : F ( Q ) → F ( T ( u )). A mo r phism w : T ( u ) → T ( v ) ca n be reg arded as an element in Hom U Q ( u, v ) and deﬁnes a group morphism A ( w ) : A ( T ( u )) → A ( T ( v )) which intert wines the actio ns of these g roups on G ( Q ). Prop osition 5. 8. Le t ( U , A ) b e a c ate gory with automorphism push-forwar d, Q ∈ O b ( U ) a ﬁ xe d obje ct, and F : U → A b a functor such that F ( u ) is a isomorphism for any u : Q → O . L et N b e a smal l ﬁltering c ate gory and Θ : N → U Q a functor satisfying the c oﬁnality axiom C1. Put A := lim − → n ∈O b ( N ) A ( T (Θ( n ))) . Then A acts on F ( Q ) in a natur al way, the inductive limit lim − → O ∈O b ( U ) F ( O ) exists and c an b e identiﬁe d with t he qu otient F ( Q ) /I [ A ] F ( Q ) . Pro of: By Lemma 5.4 the inductive limit of F exists and can b e identiﬁ ed with a quotien t F ( Q ) /H . Here H is the group generated b y the elements of the form x − x ′ where x , x ′ ∈ F ( Q ) are such that ther e exis ts u , u ′ : Q → O with F ( u )( x ) = F ( u ′ )( x ′ ). W e cla im that the s et o f suc h pa irs ( x, x ′ ) coincides with the set of pa irs of the form ( a x ′ , x ′ ) with x ′ ∈ F ( Q ), a ∈ A . Indeed, if F ( u )( x ) = F ( u ′ )( x ′ ), choo se v : O → ˆ O and a ∈ A ( ˆ O ) s uc h that v u ′ = av u . The mor phism v u can b e re g arded as an ob ject in the categor y U Q . Since Θ s atisﬁes the axiom C 1 , there exists n ∈ O b ( N ) and a morphism v u → Θ( n ) in U Q , i.e. a morphism w : ˆ O → T (Θ( n )) such that wv u = Θ( n ). W e o btain F (Θ ( n ))( x ) = F ( wv u )( x ) = F ( wv u ′ )( x ′ ) = F ( wav u )( x ′ ) = F ( A ( w )( a ) wv u )( x ′ ) = = A ( w )( a )( F ( w v u )( x ′ )) = A ( w )( a )( F (Θ( n ))( x ′ )) , which s hows that x = a x ′ , wher e a is the class of A ( w )( a ) ∈ A ( T (Θ( n )) in A . Conv er sely let a = [ a ] ∈ A b e represented by a ∈ A ( T (Θ( n )) and supp ose that x = a x ′ . This means F (Θ( n ))( x ) = a ( F (Θ( n ))( x ′ )) so, putting u := Θ( n ), u ′ := a Θ ( n ) one has F ( u )( x ) = F ( u ′ )( x ′ ). Let ( U , A ) b e a category with automorphism push-forward, and let G : C → A be a functor, where A is one of the categ ories S ets , G r or A b . Deﬁnition 5 .9. We say that the stabilize d automorphisms act trivial ly on G if TSA. F or every O ∈ O b ( C ) , x ∈ G ( O ) and a ∈ A ( O ) t her e exists a morphism u : O → O ′ such that G ( u )( G ( a )( x )) = G ( u )( x ) . In t he pr esenc e of funct or Θ : N → U , we say that the Θ -stabilize d automorphisms act trivial ly on G if ΘSA. F or every n ∈ O b ( N ) , x ∈ G (Θ( n )) and a ∈ A (Θ( n )) ther e exists a mor- phism ν : n → m such that G (Θ( ν ))( G ( a )( x )) = G (Θ( ν ))( x ) . 52 CHRISTIAN OKONEK, ANDREI TELEMAN Remark 5.10 . If Θ is c oﬁnal and G satisﬁes Θ SA , then it al so satisﬁes TSA. If C is ﬁ ltering, then any functor G : C → A satisﬁ es TSA. If, mor e over, Θ is c oﬁnal, then G also satisﬁes Θ SA. Let ( U , A ) b e a catego ry with automorphism pus h-forward, and let G : U → A be a functor. Let N be a smal l ﬁltering catego ry a nd Θ : N → U a coﬁnal functor such that Θ SA holds. Co nsider the clas s ical inductive limit L Θ := lim − → n ∈O b ( N ) G (Θ( n )). F or every O ∈ O b ( U ) we deﬁne a morphism f O : G ( O ) → L Θ by f O ( x ) := [ G ( v )( x )] where v : O → Θ ( n ) is a mo r phism (whose existence is guar anteed by C1 ). Prop osition 5.11. Under the assu mptions and with the n otations ab ove it holds (1) F or any O ∈ O b ( U ) the map f O is wel l deﬁne d. The system of m aps ( f O ) O ∈O b ( U ) is G -c omp atible i.e. for any u : O → O ′ one has f O ′ ◦ G ( u ) = f O . When A = A b or G r , the map f O is a gr oup morphism. (2) The system ( f O ) O ∈O b ( U ) satisﬁes t he universal pr op erty of the inductive limit, t her efor e the functor G admits an induct ive limit in A which c an b e identiﬁe d with L Θ . W e agree to write u ( x ), v ( x ) . . . , instead of G ( u )( x ), G ( v )( x ) . . . , to sav e o n notations. Pro of: 1. Le t v : O → Θ( n ), v ′ : O → Θ ( n ′ ) be t w o morphis ms . Since N is ﬁltering , there exists morphisms ν : n → m , ν ′ : n ′ → m . Applying a x iom S3 to the morphisms Θ( ν ) v , Θ( ν ′ ) v ′ , we get a mo rphism w : Θ( m ) → ˆ O and an automorphism a ∈ A ( ˆ O ) such that w Θ( ν ′ ) v ′ = aw Θ( ν ) v . Now we a pply the axiom C2 to w and w e get mor phisms u : ˆ O → Θ( k ), µ : m → k such that uw = Θ( µ ). W e have Θ( µν ′ ) v ′ = uw Θ( ν ′ ) v ′ = u aw Θ( ν ) v = A ( u )( a ) uw Θ( ν ) v = A ( u )( a )Θ( µν ) v . Using the a xiom ΘSA we obtain a morphism η : k → l such that Θ( η ) [ A ( u )( a )Θ( µν ) v ( x )] = Θ( η ) [ Θ( µν ) v ( x )] . Therefore Θ ( η µν ′ )( v ′ ( x )) = Θ( η µν )( v ( x )), which shows that v ( x ) and v ′ ( x ′ ) deﬁne the same elemen t in L Θ . The sec o nd and the third cla im are obvious. 2. Let Λ ∈ O b ( A ) a nd ( g O ) O ∈O b ( U ) , g O : G ( O ) → Λ a system of G -compatible morphisms. Using the system ( g Θ( n ) ) n ∈O b ( N ) (whic h is G ◦ Θ-compatible) we get a unique morphism g : L Θ → Λ s uch that g ◦ c n = g Θ( n ) for every n ∈ O b ( N ), where c n : G (Θ( n )) → L Θ is the canonical morphism. It remains to prove tha t g ◦ f O = g O for every O ∈ O b ( U ). Let x ∈ G ( O ) and choose v : G ( O ) → Θ( n ). One has g ◦ f O ( x ) = g ( c n ( v ( x ))) = g Θ( n ) ( v ( x )) = g O ( x ) . 5.2. Bundle m aps b et ween p ointed sphere bundles . Let X b e a CW com- plex a nd Y ⊂ X a subcomplex. F o r tw o s ections s ′ , s ′′ in a an orient ed r -sphere bundle over X which coincide over Y , we denote b y o Y ( s ′ , s ′′ ) ∈ H r ( X, Y ; Z ) the primary obstruction to the existence of a homo to p y be tw een s ′ and s ′′ in the space of sections which coincide with s ′ Y = s ′′ Y on Y [S]. COHOMOTOPY INV ARIANTS 53 Let π ζ : ζ → B b e an or ien ted r eal bundle o f ra nk r over a CW co mplex B . Denote by π + ζ : ζ + B =: ˆ B → B the bundle pro jection of the asso ciated sphere bundle, and consider the pull-back bundle ˆ ζ := [ π + ζ ] ∗ ( ζ ) on ˆ B . The sphere bundle ˆ ζ + ˆ B = [ π + ζ ] ∗ ( ζ + B ) c omes with a tautologica l section θ ζ and a n “inﬁnite” section s ∞ ˆ ζ . These sections co incide on the subs pa ce ∞ ζ ⊂ ˆ B . W e endow the space ˆ B with a CW structure in the following way: First, o n the subspace ∞ ζ we copy the CW structure from the ba se B via s ∞ ζ . Second, for every k -cell e ⊂ B we put ˆ e := π − 1 ζ ( e ). The a ttac hing map corre spo nding to ˆ e is deﬁned in the following wa y: let u : D k → ¯ e ⊂ B the attaching ma p of e . The pullback bundle u ∗ ( ζ ) is trivia l, so it can b e identiﬁed with D k × R r = D k × ˚ D r . The induced map D k × ˚ D r → π − 1 ζ ( ¯ e ) ⊂ ζ can b e extended to map ˆ u : D k × D r → [ π + ζ ] − 1 ( ¯ e ) ⊂ ζ + in an obvious w ay . Let t ζ be the Tho m class of the bundle ζ . W e c laim Lemma 5.12. With r esp e ct to such a c el lular str u ctur e on ˆ B one has o ∞ ζ ( s ∞ ˆ ζ , θ ζ ) = t ζ in H r ( ˆ B , ∞ ζ ; Z ) . Pro of: Let P : E → B := B S O ( r ) b e the univ er sal v ector bundle with struc- ture gro up S O ( r ) and a ﬁxed C W structure on the classifying space B . Since H r ( ˆ B , ∞ E ; Z ) ≃ H 0 ( B ; Z ) ≃ Z , there exists an integer N such that o ∞ E ( s ∞ ˆ E , θ E ) = N t E . Let f : B → B a cellular map which induces the bundle ζ . This map is cov er ed by a bundle map ˆ f : ˆ B → ˆ B , which is obviously cellula r and maps the sub- complex ∞ ζ of ˆ B int o the s ubcomplex ∞ E of ˆ B . Using the functorial pro per ties of the rela tive obs tr uction class and o f the Thom class , we obtain o ∞ ζ ( s ∞ ˆ ζ , θ ζ ) = N t ζ . The in teger N can b e computed using any bundle ζ , so we will choose the bundle R r → { ∗ } . The tautologica l section is just the identit y of [ R r ] + . It ’s easy to s ee that b oth c la sses can b e identiﬁed with the generato r o f H r ([ R r ] + , ∞ ; Z ). Corollary 5.13. L et ζ b e an oriente d r -bu nd le over a CW c omplex B , and let s b e a se ction in ζ + B which c oincides with s ∞ ζ on a sub c omplex A ⊂ B . Then o A ( s ∞ ζ , s ) = s ∗ (t ζ ) in H r ( B , A ; Z ) . Pro of: Note that, with resp ect to the cellula r decompos ition of ˆ B considered ab ov e, the section s : B → ˆ B is a cellular map a nd maps to sub complex A into the s ubcomplex ∞ ζ . It suﬃces to a pply the functorial pro per t y of the rela tiv e obstruction classes with resp ect to cellular maps. Corollary 5.14. L et ζ b e an oriente d r -bund le over a ﬁnite CW c omplex B of dimension n ≤ r , and let A ⊂ B b e a sub c omplex. Then the map o A : s 7→ s ∗ (t ζ ) deﬁnes a bije ction b etwe en the set Γ A ( ζ + B ) of homotopy classes of se ctions in ζ + B which c oincide with s ∞ ζ on A , and H r ( B , A ; Z ) . Pro of: Injectivity: Since dim( B ) ≤ r , for a section s ∈ Γ A ( ζ + B ) the only obstruc- tion to the existence o f a homoto p y betw een s ∞ ζ and s is the primary obstr uction o A ( s ∞ ζ , s ). T o prov e sur jectivit y , consider, for any r -cell e ⊂ B \ A , a section s e which coincides with s ∞ ζ on ¯ e \ e and has a sing le v a nishing p oint , which is non-degenerate. The pull-back s ∗ e (t ζ ) is a generator of H r ( B , B \ e ; Z ) ∼ = Z . Corollary 5.15. L et ζ 0 , ζ 1 b e two oriente d bund les of r anks r 0 , r 1 over an n - dimensional c omplex B . 54 CHRISTIAN OKONEK, ANDREI TELEMAN (1) If n + r 0 < r 1 , any p ointe d bund le map f : [ ζ 0 ] + B → [ ζ 1 ] + B over B is homotopic (in the sp ac e of p ointe d bund le maps over B ) t o the ﬁb erwise c onstant map f ∞ which maps [ ζ 0 ] + into ∞ ζ 1 . (2) If n + r 0 = r 1 , then a p ointe d bund le m ap f : [ ζ 0 ] + B → [ ζ 1 ] + B over B is homotopic t o f ∞ if and only if t he class h f ∈ H n ( B ; Z ) , deﬁne d by the c ondition f ∗ (t ζ 1 ) = [ π + ζ 0 ] ∗ ( h f ) ∪ t ζ 0 , vanishes. Mor e over, the assignment f 7→ h f deﬁnes a bije ction b etwe en the set of homotopy classes of p ointe d bund le maps [ ζ 0 ] + B → [ ζ 1 ] + B and H n ( B ; Z ) . Pro of: It suﬃces to apply Corollary 5.1 4 to the pull- back bundle ˜ ζ 1 := [ π + ζ 0 ] ∗ ( ζ 1 ) ov er ˜ B := [ ζ 0 ] + B and to identify the space of p ointed bundle maps [ ζ 0 ] + B → [ ζ 1 ] + B with the space of thos e sections in [ ˜ ζ 1 ] + ˜ B which coincide with s ∞ ˜ ζ 1 on ∞ ζ 0 ⊂ ˜ B . Then use the Thom iso morphism · ∪ t ζ 0 : H n ( B ; Z ) → H r 1 ( ˜ B , ∞ ζ 0 ; Z ). References [AM] Artin, M. ; Mazur, B.: Etale homotopy , Lect ure Notes in Math. 100, Springer V erlag, 1969. [B] Bader, M .: Cohomotopy inv arian ts in gaug e the oretic Gromov-Witten theory, PhD Thesis, Z¨ urich , 2007, in preparation. [B1] Bauer, S.: A stable c ohomotopy r eﬁnement of Sei ber g-Witten invariants: II , Inv ent. math 155, 21-40, 2004. [B2] Bauer, S.: R eﬁne d Seiber g -Witten invariants , Diﬀeren t faces of Geometry , 1-46, In t. Math. Ser. (N. Y. ), Kluw er/Plen um, New Y ork, 2004. [B3] Bauer, S.: O n co nne cte d sums of four dimensional manifold s , Preprintreihe 2000, Univ. Bielefeld, http://w ww.mathem atik.uni- bielefeld.de/sfb343/preprints/pr00001.ps.gz . [BF] Bauer, S.; F uruta, M.: A stable c ohomotopy r eﬁnement of Seib er g-Wit ten invariants: I , In v en t. math 155, 1-19, 2004. [Br] Brussee, R: The c anonic al class and the C ∞ -pr op erties of Kahler surfac es , New Y ork J. Math. 2, 103-146, 1996. [C] Crabb, M.: The ﬁbr ewise L er ay-Schauder index , J. ﬁxed p oint theory appl. 1, 3-30, 2007. [CJ] Cr abb, M.; James, I.: F ib erwise homotopy the ory , Springer V erlag, 1998. [CK] Crabb, M.; Knapp, K.: On t he c o de gr e e of negative multiplies of t he Hopf bund le , Pr oc. of the Roy al So c. of Edinburgh, 107A, 87-107, 1987. [CS] Crabb, M.; Sutherland.: The sp ac e of se ct ions of a spher e bund le I , Pro c. Edinburgh Math. Soc. 29, 383-403, 1986. [tD] tom Di ec k, T . : T r ansformation Gr oups , De Gruyter, 1987. [F u1] F uruta, M.: Monop ole e quations and the 11/ 8 c onje ctur e , Math. Res. Lett. 8, 279-291, 2001. [F u2] F uruta, M.: Stable homotopy version of Seib er g-Witt en invariant , preprint , M PI Bonn, 1997, http://www.mpi m- bonn.mpg.de . [F u3] F uruta, M .: The Pontrjagin-Thom c onstructi on and non-line ar F r ed holm t he ory , plenary talk given at the Postnik ov M emorial Conference, June 2007. [H] Hausc hild, H.: Zersp altung ¨ aquivarianter Homotopiemengen , M ath. Ann. 230, 279-292, 1977. [LL] Li, T. J.; Liu, A.: Gener al wal l cr ossing formula , Math. Res. Lett. 2 no. 6, 797–810, 1995. [OO] Ohta, H.; Ono, K.: Notes on symple ctic 4 -manifolds with b + 2 = 1, II., Inte rnat. J. Math. 7 no. 6, 755–770, 1996. [OT] Ok onek, Ch.; T eleman, A. : Seib e r g-Witten invariants for manifolds with b + = 1 and the universal wal l cr ossing formula , Internat. J. Math. 7, no. 6, 811-832, 1996. [S] Steenrod, N .: The top olo gy of Fibr e Bund les , Pr i nceton Universit y Press, 1951. COHOMOTOPY INV ARIANTS 55 Christian Okonek: Institut f ¨ ur Mathematik, U nivers it¨ at Z ¨ u ric h , Winterth urerstrasse 150, CH-8057 Z ¨ urich, e-mail: okonek@math.unizh.ch Andrei T eleman: CMI, Universit ´ e de Pro vence, 39 Rue F. Joliot-Curie, F- 13453 Marseille Cedex 13, e-mail: teleman@cmi.univ-mrs.fr

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