aa r X i v : . [ m a t h . K T ] M a y SMOOTH K-THEORY by Ulrich Bunke & Thomas Schick
Dedicated to Jean-Michel Bismut on the occasion of his 60th birthday
Abstract . —
In this paper we consider smooth extensions of cohomology theo-ries. In particular we construct an analytic multiplicative model of smooth K -theory.We further introduce the notion of a smooth K -orientation of a proper submersion p : W → B and define the associated push-forward ˆ p ! : ˆ K ( W ) → ˆ K ( B ). We showthat the push-forward has the expected properties as functoriality, compatibility withpull-back diagrams, projection formula and a bordism formula.We construct a multiplicative lift of the Chern character ˆ ch : ˆ K ( B ) → ˆ H ( B, Q ),where ˆ H ( B, Q ) denotes the smooth extension of rational cohomology, and we showthat ˆ ch induces a rational isomorphism.If p : W → B is a proper submersion with a smooth K -orientation, then we definea class A ( p ) ∈ ˆ H ev ( W, Q ) (see Lemma 6.17) and the modified push-forward ˆ p A ! :=ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B, Q ) . One of our main results lifts the cohomologicalversion of the Atiyah-Singer index theorem to smooth cohomology. It states thatˆ p A ! ◦ ˆ ch = ˆ ch ◦ ˆ p ! . Mathematics Subject Classification . —
Key words and phrases . —
Deligne cohomology, smooth K-theory, Chern character, families ofelliptic operators, Atiyah-Singer index theorem.Thomas Schick was funded by Courant Research Center G¨ottingen “Higher order structures inmathematics” via the German Initiative of Excellence.
ULRICH BUNKE & THOMAS SCHICK
R´esum´e (K-theorie differentiable). —
Nous considerons les extensions differ-entiables des theories de cohomology. En particulier, nous construisons un mod`eleanalytique et avec multiplication de la K-theorie differentiable. Nous introduisonsle concept d’une K-orientation differentiable d’une submersion propre p : W → B .Nous contruisons une application d’integration associ´e ˆ p ! : ˆ K ( W ) → ˆ K ( B ); et nousdemontrons les propri´et´es attendues comme functorialit´e, compatibilit´e avec pull-back, formules de projection et de bordism.Nous construisons une version differentiable du charact`ere de Chern ˆ ch : ˆ K ( B ) → ˆ H ( B, Q ), o`u ˆ H ( B, Q ) est une extension differentiable de la cohomologie rationelle, etnous demontrons que ˆ ch induit un isomorphisme rationel.Si p : W → B est une submersion propre avec une K -orientation differentiable,nous definissons une classe A ( p ) ∈ ˆ H ev ( W, Q ) (compare Lemma 6.17) et une ap-plication d’integration modifi´e ˆ p A ! := ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B, Q ) . Un denos resultats principales est une version en cohomologie differentiable du theor`emed’indice de Atiyah-Singer. Cette version dits que ˆ p A ! ◦ ˆ ch = ˆ ch ◦ ˆ p ! . Contents
1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1. The main results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. A short introduction to smooth cohomology theories. . . . . . . . 61.3. Related constructions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Definition of smooth K-theory via cycles and relations . . . . . . . . . . 142.1. Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2. Relations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3. Smooth K -theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4. Natural transformations and exact sequences. . . . . . . . . . . . . . . 222.5. Comparison with the Hopkins-Singer theory and the flattheory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263. Push-forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.1. K -orientation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2. Definition of the Push-forward. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3. Functoriality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414. The cup product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.1. Definition of the product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2. Projection formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3. Suspension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505. Constructions of natural smooth K -theory classes. . . . . . . . . . . . . . . 515.1. Calculations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515.2. The smooth K -theory class of a mapping torus. . . . . . . . . . . . . 525.3. The smooth K -theory class of a geometric family with kernelbundle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.4. A canonical ˆ K -class on S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.5. The product of S -valued maps and line-bundles. . . . . . . . . . . . 565.6. A bi-invariant ˆ K - class on SU (2). . . . . . . . . . . . . . . . . . . . . . . . . . 57 MOOTH K-THEORY Z /k Z -invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.10. Spin c -bordism invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.11. The e -invariant. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656. The Chern character and a smooth Grothendieck-Riemann-Rochtheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.1. Smooth rational cohomology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 686.2. Construction of the Chern character . . . . . . . . . . . . . . . . . . . . . . . 706.3. The Chern character is a rational isomorphism and multiplicative 766.4. Riemann Roch theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
1. Introduction1.1. The main results. —
In this paper we construct a model of a smooth extension of the generalizedcohomology theory K , complex K -theory. Historically, the concept of smooth exten-sions of a cohomology theory started with smooth integral cohomology [ CS85 ], alsocalled real Deligne cohomology, see [
Bry93 ]. A second, geometric model of smoothintegral cohomology is given in [
CS85 ], where the smooth integral cohomology classeswere called differential characters. One important motivation of its definition was thatone can associate natural differential characters to hermitean vector bundles with con-nection which refine the Chern classes. The differential character in degree two evenclassifies hermitean line bundles with connection up to isomorphism. The multiplica-tive structure of smooth integral cohomology also encodes cohomology operations, see[
Gom ].The holomorphic counterpart of the theory became an important ingredient ofarithmetic geometry.
Motivated by the problem of setting up lagrangians for quantum field the-ories with differential form field strength it was argued in [
FH00 ], [
Fre00 ] that onemay need smooth extensions of other generalized cohomology theories. The choice ofthe generalized cohomology theory is here dictated by a charge quantization condi-tion, which mathematically is reflected by a lattice in real cohomology. Let N be agraded real vector space such that the field strength lives in Ω d =0 ( B ) ⊗ N , the closedforms on the manifold B with coefficients in N . Let L ( B ) ⊂ H ( B, N ) be the latticegiven by the charge quantization condition on B . Then one looks for a generalizedcohomology theory h and a natural transformation c : h ( B ) → H ( B, N ) such that c ( h ( B )) = L ( B ). It was argued in [ FH00 ], [
Fre00 ] that the fields of the theory
ULRICH BUNKE & THOMAS SCHICK should be considered as cycles for a smooth extension ˆ h of the pair ( h, c ). For exam-ple, if N = R and the charge quantization leads to L ( B ) = im ( H ( B, Z ) → H ( B, R )),then the relevant smooth extension could be the smooth integral cohomology theoryof [ CS85 ].In Subsection 1.2 we will introduce the notion of a smooth extension in an axiomaticway. [ Fre00 ] proposes in particular to consider smooth extensions of complexand real versions of K -theory. In that paper it was furthermore indicated how cyclemodels of such smooth extensions could look like. The goal of the present paper is tocarry through this program in the case of complex K -theory. In the remainder of the present subsection we describe, expanding the ab-stract, our main results. The main ingredient is a construction of an analytic modelof smooth K -theory (1) using cycles and relations. Our philosophy for the construction of smooth K -theory is that a vectorbundle with connection or a family of Dirac operators with some additional geome-try should represent a smooth K -theory class tautologically. In this way we followthe outline in [ Fre00 ]. Our class of cycles is quite big. This makes the construc-tion of smooth K -theory classes or transformations to smooth K -theory easy, but itcomplicates the verification that certain cycle level constructions out of smooth K -theory are well-defined. The great advantage of our choice is that the constructionsof the product and the push-forward on the level of cycles are of differential geometricnature.More precisely we use the notion of a geometric family which was introducedin [ Bun ] in order to subsume all geometric data needed to define a Bismut super-connection in one notion. A cycle of the smooth K -theory ˆ K ( B ) of a compact manifold B is a pair ( E , ρ ) of a geometric family E and an element ρ ∈ Ω( B ) / im ( d ), see Section2. Therefore, cycles are differential geometric objects. Secondary spectral invariantsfrom local index theory, namely η -forms, enter the definition of the relations (seeDefinition 2.10). The first main result is that our construction really yields a smoothextension in the sense of Definition 1.1. Our smooth K -theory ˆ K ( B ) is a contravariant functor on the category ofcompact smooth manifolds (possibly with boundary) with values in the category of Z / Z -graded rings. This multiplicative structure is expected since K -theory is a mul-tiplicative generalized cohomology theory, and the Chern character is multiplicative,too. As said above, the construction of the product on the level of cycles (Defini-tion 4.1) is of differential-geometric nature. Analysis enters the verification of well-definedness. The main result is here that our construction produces a multiplicativesmooth extension in the sense of Definition 1.2. (1) or differentiable K -theory in the language of other authors MOOTH K-THEORY Let us consider a proper submersion p : W → B with closed fibres whichhas a topological K -orientation. Then we have a push-forward p ! : K ( W ) → K ( B ),and it is an important part of the theory to extend this push-forward to the smoothextension.For this purpose one needs a smooth refinement of the notion of a K -orientationwhich we introduce in 3.5. We then define the associated push-forward ˆ p ! : ˆ K ( W ) → ˆ K ( B ), again by a differential-geometric construction on the level of cycles (17). Weshow that the push-forward has the expected properties: functoriality, compatibilitywith pull-back diagrams, projection formula, bordism formula. Let V = ( V, h V , ∇ V ) be a hermitean vector bundle with connection. In[ CS85 ] a smooth refinement ˆ ch ( V ) ∈ ˆ H ( B, Q ) of the Chern character was con-structed. In the present paper we construct a lift of the Chern character ch : K ( B ) → H ( B, Q ) to a multiplicative natural transformation of smooth cohomology theories(see (30)) ˆ ch : ˆ K ( B ) → ˆ H ( B, Q )such that ˆ ch ( V ) = ˆ ch ([ V , V is the geometric family determined by V . TheChern character induces a natural isomorphism of Z / Z -graded ringsˆ K ( B ) ⊗ Q ∼ → ˆ H ( B, Q )(Proposition 6.12). If p : W → B is a proper submersion with a smooth K -orientation, then wedefine a class (see Lemma 6.17) A ( p ) ∈ ˆ H ev ( W, Q ) and the modified push-forwardˆ p A ! := ˆ p ! ( A ( p ) ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B, Q ) . Our index theorem 6.19 lifts the characteristic class version of the Atiyah-Singer indextheorem to smooth cohomology. It states that the diagramˆ K ( W ) ˆ p ! (cid:15) (cid:15) ˆ ch / / ˆ H ( W, Q ) ˆ p A ! (cid:15) (cid:15) ˆ K ( B ) ˆ ch / / ˆ H ( B, Q )commutes. In Subsection 1.2 we present a short introduction to the theory of smoothextensions of generalized cohomology theories. In Subsection 1.3 we review in somedetail the literature about variants of smooth K -theory and associated index theo-rems. In Section 2 we present the cycle model of smooth K -theory. The main resultis the verification that our construction satisfies the axioms given below. Section 3is devoted to the push-forward. We introduce the notion of a smooth K -orientation,and we construct the push-forward on the cycle level. The main results are thatthe push-forward descends to smooth K -theory, and the verification of its functorialproperties. In Section 4 we discuss the ring structure in smooth K -theory and its ULRICH BUNKE & THOMAS SCHICK compatibility with the push-forward. Section 5 presents a collection of natural con-structions of smooth K -theory classes. In Section 6 we construct the Chern characterand prove the smooth index theorem. The first example of a smooth cohomology theory appeared under the nameCheeger-Simons differential characters in [
CS85 ]. Given a discrete subring R ⊂ R wehave a functor (2) B ˆ H ( B, R ) from smooth manifolds to Z -graded rings. It comeswith natural transformations1. R : ˆ H ( B, R ) → Ω d =0 ( B ) (curvature)2. I : ˆ H ( B, R ) → H ( B, R ) (forget smooth data)3. a : Ω( B ) / im ( d ) → ˆ H ( B, R ) (action of forms).Here Ω( B ) and Ω d =0 ( B ) denote the space of smooth real differential forms and itssubspace of closed forms. The map a is of degree 1. Furthermore, one has the followingproperties, all shown in [ CS85 ].1. The following diagram commutesˆ H ( B, R ) R (cid:15) (cid:15) I / / H ( B, R ) R → R (cid:15) (cid:15) Ω d =0 ( B ) dR / / H ( B, R ) , where dR is the de Rham homomorphism.2. R and I are ring homomorphisms.3. R ◦ a = d ,4. a ( ω ) ∪ x = a ( ω ∧ R ( x )), ∀ x ∈ ˆ H ( B, R ), ∀ ω ∈ Ω( B ) / im ( d ),5. The sequence H ( B, R ) → Ω( B ) / im ( d ) a → ˆ H ( B, R ) I → H ( B, R ) → Cheeger-Simons differential characters are the first example of a more gen-eral structure which is described for instance in the first section of [
Fre00 ]. In viewof our constructions of examples for this structure in the case of bordism theories and K -theory, and the presence of completely different pictures like [ HS05 ] we think thatan axiomatic description of smooth cohomology theories is useful.Let N be a Z -graded vector space over R . We consider a generalized cohomologytheory h with a natural transformation of cohomology theories c : h ( B ) → H ( B, N ).The natural universal example is given by N := h ∗ ⊗ R , where c is the canonical (2) In the literature, this group is sometimes denoted by ˆ H ( B, R / R ), possibly with a degree-shift byone. MOOTH K-THEORY transformation. Let Ω( B, N ) := Ω( B ) ⊗ R N . To a pair ( h, c ) we associate the notion ofa smooth extension ˆ h . Note that manifolds in the present paper may have boundaries. Definition 1.1 . —
A smooth extension of the pair ( h, c ) is a functor B → ˆ h ( B ) fromthe category of compact smooth manifolds to Z -graded groups together with naturaltransformations R : ˆ h ( B ) → Ω d =0 ( B, N ) (curvature) I : ˆ h ( B ) → h ( B ) (forget smooth data) a : Ω( B, N ) / im ( d ) → ˆ h ( B ) (action of forms) .These transformations are required to satisfy the following axioms: The following diagram commutes ˆ h ( B ) R (cid:15) (cid:15) I / / h ( B ) c (cid:15) (cid:15) Ω d =0 ( B, N ) dR / / H ( B, N ) . R ◦ a = d . (2)3. a is of degree . The sequence h ( B ) c → Ω( B, N ) / im ( d ) a → ˆ h ( B ) I → h ( B ) → . (3) is exact. The Cheeger-Simons smooth cohomology B ˆ H ( B, R ) considered in 1.2.1 is thesmooth extension of the pair ( H ( . . . , R ) , i ), where i : H ( B, R ) → H ( B, R ) is inducedby the inclusion R → R . The main object of the present paper, smooth K -theory,is a smooth extension of the pair ( K, ch R ), and we actually work with the obvious Z / Z -graded version of these axioms. If h is a multiplicative cohomology theory, then one can consider a Z -gradedring N over R and a multiplicative transformation c : h ( B ) → H ( B, N ). In this caseis makes sense to talk about a multiplicative smooth extension ˆ h of ( h, c ). Definition 1.2 . —
A smooth extension ˆ h of ( h, c ) is called multiplicative, if ˆ h to-gether with the transformations R, I, a is a smooth extension of ( h, c ) , and in addition
1. ˆ h is a functor to Z -graded rings, R and I are multiplicative, a ( ω ) ∪ x = a ( ω ∧ R ( x )) for x ∈ ˆ h ( B ) and ω ∈ Ω( B, N ) / im ( d ) . The smooth extension ˆ H ( . . . , R ) of ordinary cohomology H ( . . . , R ) with coefficientsin a subring R ⊂ R considered in 1.2.1 is multiplicative. The smooth extension ˆ K of K -theory which we construct in the present paper is multiplicative, too. ULRICH BUNKE & THOMAS SCHICK
Consider two pairs ( h i , c i ), i = 0 , u : h → h such that c ◦ h = c . Then we definethe notion of a natural transformation of smooth cohomology theories which refines u . Definition 1.3 . —
A natural transformation of smooth extensions ˆ u : ˆ h → ˆ h which refines u is a natural transformation ˆ u : ˆ h ( B ) → ˆ h ( B ) such that the followingdiagram commutes: Ω( B, N ) / im ( d ) a / / ˆ h ( B ) R I / / ˆ u (cid:15) (cid:15) h ( B ) u (cid:15) (cid:15) Ω d =0 ( B, N )Ω(
B, N ) / im ( d ) a / / ˆ h ( B ) I / / R ; ; h ( B ) Ω d =0 ( B, N ) . Our main example is the Chern characterˆ ch : ˆ K ( B ) → ˆ H ( B, Q )which refines the ordinary Chern character ch : K ( B ) → H ( B, Q ). The Chern char-acter and its smooth refinements are actually multiplicative. One can show that two smooth extensions of ( H ( . . . , R ) , i ) are canonicallyisomorphic (see [ SS ] and [ BS09 , Section 4]). There is no uniqueness result for ar-bitrary pairs ( h, c ). Appropriate examples in the case of K -theory are presented in[ BS09 , Section 6]. In order to fix the uniqueness problem one has to require moreconditions, which are all quite natural.The projection pr : S × B → B has a canonical smooth K -orientation (see 4.3.2for details). Hence we have a push-forward ( ˆ pr ) ! : ˆ K ( S × B ) → ˆ K ( B ) (see Definition3.18). This map plays the role of the suspension for the smooth extension. It is naturalin B , and the following diagram commutes (see Proposition 3.19)Ω( S × B ) / im ( d ) R S × B/B (cid:15) (cid:15) a / / ˆ K ( S × B ) R ( ˆ pr ) ! (cid:15) (cid:15) I / / K ( B ) ( pr ) ! (cid:15) (cid:15) Ω( S × B ) R S × B/B (cid:15) (cid:15) Ω( B ) / im ( d ) a / / ˆ K ( B ) R ; ; I / / K ( B ) Ω( B ) . (4) MOOTH K-THEORY Furthermore, it satisfies (see 4.6) ( ˆ pr ) ! ◦ pr ∗ = 0 . (5)We have the following theorem, also discovered by Wiethaup. Theorem 1.4 ( [ BS09 , Section 3, Section 4] ) . — There is a unique (up to isomor-phism) smooth extension of the pair ( K, ch R ) for which in addition the push-forwardalong pr : S × B → B is defined, is natural in B , satisfies (5), and is such that (4)commutes. If we require the isomorphism to preserve ( ˆ pr ) ! , then it is also unique.1.2.6. — The theory of [
HS05 ] gives the following general existence result.
Theorem 1.5 ( [ HS05 ] ) . — For every pair ( h, c ) of a generalized cohomology theoryand a natural transformation h → HN there exists a smooth extension ˆ h in the senseof Definition 1.1. A similar general result about multiplicative extensions is not known. Besidessmooth extensions of ordinary cohomology and K -theory we have a collection ofmultiplicative extensions of bordism theories, again by an an explicit construction ina cycle model. The details can be found in [ BSSW07 ]. Let us now assume that ( h, c ) is multiplicative, and that ˆ h is a multiplicativesmooth extension of the pair ( h, c ). Let p : W → B be a proper submersion with closedfibres. An h -orientation of p is given by a collection of compatible choices of h -Thomclasses on representatives of the stable normal bundle of p . Equivalently, we can fixa Thom class on the vertical tangent bundle, and we will adopt this point of view inthe present paper. If p is h -oriented, then we have a push-forward p ! : h ( W ) → h ( B ) . It is an inportant question for applications and calculations how one can lift thepush-forward to the smooth extensions.In the case of smooth ordinary cohomology with coefficients in R it turns out thatan ordinary orientation of p suffices in order to define ˆ p ! : ˆ H ( W, R ) → ˆ H ( B, R ). Thispush-forward has been considered e.g. in [
Bry93 ], [
DL05 ], [
K¨o7 ]. We refer to 6.1.1for more details.A push-forward for more general pairs ( h, c ) has been considered in [
HS05 ] withouta discussion of functorial properties.
The philosophy in the present paper is that the push-forward in K -theoryis realized analytically using families of fibre-wise Dirac operators. Therefore, in thepresent paper a smooth K -orientation is given by a collection of geometric data whichallows to define the push-forward on the level of cycles, which are given by familiesof Dirac type operators. We add a differential form to the data in order to capturethe behaviour under deformations. ULRICH BUNKE & THOMAS SCHICK
We have cycle models of multiplicative smooth extensions of bordism the-ories Ω G , where G in particular can be SO, Spin, U, Spin c , see [ BSSW07 ]. In theseexamples the natural transformation c is the genus associated to a formal power series φ ( x ) = 1 + a x + . . . with coefficients in some graded ring. These bordism theoriesadmit a theory of orientations and push-forward which is very similar to the case of K -theory. Concerning the product and the integration bordism theories turn out tobe much simpler than ordinary cohomology. Motivated by this fact, in a joint projectwith M. Kreck we develop a bordism like version of the smooth extension of integralcohomology based on the notion of orientifolds.We also have an equivariant version of the theory of the present paper for finitegroups which will be presented in a future publication. Recall that [
HS05 ] provides a topological construction of smooth K -theory.In this subsection we review the literature about analytic variants of smooth K -theoryand related index theorems. Note that we will completely ignore the development ofholomorphic variants which are more related to arithmetic questions than to topology.This subsection will use the language which is set up later in the paper. It should beread in detail only after obtaining some familiarity with the main definitions (thoughwe tried to give sufficiently many forward references). Let p : W → B be a proper submersion with closed fibres. To give a K -orientation of p is equivalent to give a Spin c -structure on its vertical bundle T v p .The K -orientation of p yields, by a stable homotopy construction, a push-forward p ! : K ( W ) → K ( B ). Let ˆ A ( T v p ) denote the ˆ A -class of the vertical bundle, and let c ( L ) ∈ H ( W, Z ) be the cohomology class determined by the Spin c -structure (see3.1.6). The ”index theorem for families” in the characteristic class version states that ch ( p ! ( x )) = Z W/B ˆ A ( T v p ) ∪ e c ( L ) ∪ ch ( x ) , ∀ x ∈ K ( W ) . If one realizes the push-forward in an analytic model, then this statement is indeedan index theorem for families of Dirac operators.
The cofibre of the map of spectra K → H R induced by the Chern char-acter represents a generalized cohomology theory K R / Z , called R / Z - K -theory. Itis a module theory over K -theory and therefore also admits a push-forward for K -oriented proper submersions. This push-forward is again defined by constructionsin stable homotopy theory. An analytic/geometric model of R / Z - K -theory was pro-posed in [ Kar87 ], [
Kar97 ]. This led to the natural question whether there is ananalytic description of the push-forward in R / Z - K -theory. This question was solvedin [ Lot94 ]. The solution gives a topological interpretation of ρ -invariants.Furthermore, in [ Lot94 ] a Chern character from R / Z - K -theory to cohomologywith R / Q -coefficients has been constructed, and an index theorem has been proved. MOOTH K-THEORY Let us now explain the relation of these constructions and results with the presentpaper. In the present paper we define the flat theory ˆ K flat ( B ) as the kernel of the cur-vature R : ˆ K ( B ) → Ω d =0 ( B ). It turns out that ˆ K flat ( B ) is isomorphic to K R / Z ( B )up to a degree-shift by one (Proposition 2.25). One can actually represent all classesof K flat ( B ) by pairs ( E , ρ ), where E is a geometric family with zero-dimensional fibre(see 2.1.4). If one restricts to these special cycles, then our model of K flat ( B ) andthe model of K R / Z − ( B ) of [ Lot94 ] coincide.By an inspection of the constructions one can further check that the restrictionof our cycle level push-forward (17) to these particular flat cycles is the same as theone in [
Lot94 ]. At a first glance our push-forward of flat classes seems to depend ona smooth refinement of the topological K -orientation of the map p , but it is in factindependent of these geometric choices as can be seen using the homotopy invarianceof the flat theory. The comparison with [ Lot94 ] shows that the restriction of ourpush-forward to flat classes coincides with the homotopy theorists’ one.The restriction of our smooth lift of the Chern character ˆ ch : ˆ K ( B ) → ˆ H ( B, Q )(see Theorem 6.2) to the flat theories exactly gives the Chern character of [ Lot94 ]ˆ ch : ˆ K flat ( B ) → ˆ H flat ( B, Q )(using our notation and the isomorphism of ˆ H ∗ flat ( B ) ∼ = H ∗− ( B, R / Q )). If we restrictour index theorem 6.19 to flat classes, then it specializes toˆ ch (ˆ p ! ( x )) = Z W/B ˆ A ( T v p ) ∪ e c ( L ) ∪ ˆ ch ( x ) , ∀ x ∈ ˆ K ( W ) , and this is exactly the index theorem of [ Lot94 ].In this sense the present paper is a direct generalization of [
Lot94 ] from the flatto the general case.
The analytic model of R / Z - K -theory and the analytic construction of thepush-forward in [ Lot94 ] fits into a series of constructions of homotopy invariant func-tors with a push-forward which encodes secondary spectral invariants. Let us mentionthe two examples in [
Lot00 ] which are based on flat bundles or flat bundles with dual-ity, respectively. The spectral geometric invariants in these examples are the analytictorsion forms of [
BL95 ] and the η -forms introduced e.g. in [ BC90a ]. The functori-ality of the push-fowards under compositions is discussed in [
Bun02 ] and [
BM04 ].But these construction do not fit (at least at the moment) into the world of smoothcohomology theory, and it is still an open problem to interpret the push-forward intopological terms.Let us also mention the paper [
Pek93 ] devoted to smooth lifts of Chern classes.
In [
Berb ], [
Bera ] several variants of functors derived from K -theory areconsidered. In the following we recall the names of these groups used in that referenceand explain, if possible, their relation with the present paper. ULRICH BUNKE & THOMAS SCHICK
1. relative K -theory K rel : the cycles are triples ( V, ∇ V , f ) of Z / Z -graded flatvector bundles and an odd selfadjoint bundle automorphism f (which need notbe parallel).2. free multiplicative K -theory K ch (also called transgressive in [ Bera ]): it isessentially (3) a model of ˆ K based on cycles of the form ( E , ρ ), where E is ageometric family with zero-dimensional fibre coming from a geometric vectorbundle (see 2.1.4).3. multiplicative K -theory M K : it is the same model of K flat as in [ Lot94 ], see1.3.3.4. flat K -theory K flat : it is the Grothendieck group of flat vector bundles.Besides the definition of these groups and the investigation of their interrelation themain topic of [ Berb ], [
Bera ] is the construction of push-forward operations. In thefollowing we will only discuss multiplicative and transgressive K -theory since they arerelated to the present paper. The difference to the constructions of [ Lot94 ] and thepresent paper is that Berthomiau’s analytic push-forward (which we denote here by p B ! ) does not use the Spin c -Dirac operator but the fibre-wise de Rham complex. Fromthe point of view of analysis the difference is essentially that the class ˆ A ( T v p ) ∪ e c ( L ) or the corresponding differential form has to be replaced by the Euler class E ( T v p )or the Euler form of the vertical bundle.The advantage of working with the de Rham complex is that in order to define thepush-forward p B ! one does not need a Spin c -structure. If there is one, then one canactually express p B ! in terms of ˆ p ! as p B ! ( x ) = ˆ p ! ( x ∪ s ∗ ) , where s ∗ ∈ K ( W ) is the class of the dual of the spinor bundle S c ( T v p ), or the ˆ K ( W )-class represented by the geometric version of this bundle in the case of transgressive K -theory, respectively. The point here is that the Dirac operator induced by the deRham complex is the Spin c -Dirac operator twisted by S c ( T v p ) ∗ .As said above, the homotopy theorists’ p ! is the push-forward associated to a K -orientation of p . In contrast, the homotopy theorists’ version of p B ! is the Gottlieb-Becker transfer.The motivation of [ Berb ] , [
Bera ] to define the push-forward with the de Rhamcomplex is that it is compatible with the push-forward for flat K -theory. The push-forward of a flat vector bundle is expressed in terms of fibre-wise cohomology whichforms again a flat vector bundle on the base. This additional structure also plays acrucial role in [ Lot00 ], [
BL95 ], [
Bun02 ], and [
BM04 ]. If one interprets the push-forward using the
Spin c -calculus, then the flat connection is lost. Let us mention that (3) The connections are not assumed to be hermitean and the corresponding differential forms havecomplex coefficients.
MOOTH K-THEORY the first circulated version of the present paper predates the papers [ Berb ] , [
Bera ]which actually adapt some of our ideas.
The topics of [
Bis05 ] are two index theorems involving ˆ H ( B, Q )-valuedcharacteristic classes. Here we only review the first one, since the second is relatedto flat vector bundles. (Compare also [ MZ04 ] for a “flat version”). Let us formulatethe result of [
Bis05 ] in the language of the present paper.Let p : W → B be a proper submersion with closed fibres with a fibre-wise spin -structure over a compact base B . The spin structure induces a Spin c -structure,and we choose a representative of a smooth K -orientation o := ( g T v p , T h p, ˜ ∇ , ∇ is indced from the Levi-Civita connection on T v p (see 3.1.9 for details). Let V = ( V, h V , ∇ V ) be a geometric vector bundle over W with associated geometricfamily V (compare 2.1.4). Then we can form the geometric family E := p ! V (see 3.7)over B .The family of Dirac operators D ( E ) acts on sections of a bundle of Hilbert spaces H ( E ) → B . The geometric structures of the K -orientation o and V induce a con-nection ∇ H ( E ) (it is the connection part of the Bismut superconnection [ BGV04 ,Prop. 10.15] associated to this situation). We assume that the family of Dirac op-erators of D ( E ) has a kernel bundle K := ker( D ( E )). This bundle has an inducedmetric h K . The projection of ∇ H ( E ) to K gives a hermitean connection ∇ K . Wethus get a geometric bundle K := ( K, h K , ∇ K ), and an associated geometric family K (see 5.3.1). The index theorem in [ Bis05 ] calculates the smooth Chern characterˆ ch ( K ) ∈ ˆ H ( B, Q ) of [ CS85 ] and states:ˆ ch ( K ) = ˆ p ! ( ˆˆ A ( T v p ) ∪ ˆ ch ( V )) + a ( η BC ( E )) , where we refer to (33) and 5.3.3 for notation.Note that this theorem could also be derived from our index Theorem 6.19. ByCorollary 5.5, (17) , our special choice of o , and Theorem 6.19 (the marked step) wehave ˆ ch ( K ) − a ( η BC ( E )) = ˆ ch [ K , η BC ( E )]= ˆ ch [ E , ch ([ p ! V , ch ( p ! ([ V , ! = ˆ p K ! ( ˆ ch ( V ))= p ! ( ˆˆ A ( T v p ) ∪ ˆ ch ( V )) . Acknowledgement: We thank Moritz Wiethaup for explaining to us his insights andresult. We further thank Mike Hopkins and Dan Freed for their interest in this workand many helpful remarks. We thank the referee for many helpful comments whichlead to considerable improvements of the exposition. ULRICH BUNKE & THOMAS SCHICK
2. Definition of smooth K-theory via cycles and relations2.1. Cycles. —
One goal of the present paper is to construct a multiplicative smooth ex-tension of the pair ( K, ch R ) of the multiplicative generalized cohomology theory K ,complex K -theory, and the composition ch R : K ch → H Q → H R of the Chern charac-ter with the natural map from ordinary cohomology with rational to real coefficientsinduced by the inclusion Q → R . In this section we define the smooth K -theorygroup ˆ K ( B ) of a smooth compact manifold, possibly with boundary, and constructthe natural transformations R, I, a . The main result of the present section is thatour construction really yields a smooth extension in the sense of Definition 1.1. Widiscuss the multiplicative structure in Section 4.Our restriction to compact manifolds with boundary is due to the fact that wework with absolute K -groups. One could in fact modify the constructions in orderto produce compactly supported smooth K -theory or relative smooth K -theory. Butin the present paper, for simplicity, we will not discuss relative smooth cohomologytheories. We define the smooth K -theory ˆ K ( B ) as the group completion of a quotientof a semigroup of isomorphism classes of cycles by an equivalence relation. We startwith the description of the cycles. Definition 2.1 . —
Let B be a compact manifold, possibly with boundary. A cyclefor a smooth K -theory class over B is a pair ( E , ρ ) , where E is a geometric family,and ρ ∈ Ω( B ) / im ( d ) is a class of differential forms.2.1.3. — The notion of a geometric family has been introduced in [
Bun ] in order tohave a short name for the data needed to define a Bismut super-connection [
BGV04 ,Prop. 10.15]. For the convenience of the reader we are going to explain this notionin some detail.
Definition 2.2 . —
A geometric family over B consists of the following data: a proper submersion with closed fibres π : E → B , a vertical Riemannian metric g T v π , i.e. a metric on the vertical bundle T v π ⊂ T E , defined as T v π := ker( dπ : T E → π ∗ T B ) . a horizontal distribution T h π , i.e. a bundle T h π ⊆ T E such that T h π ⊕ T v π = T E . a family of Dirac bundles V → E , an orientation of T v π . Here, a family of Dirac bundles consists of1. a hermitean vector bundle with connection ( V, ∇ V , h V ) on E ,2. a Clifford multiplication c : T v π ⊗ V → V ,3. on the components where dim( T v π ) has even dimension a Z / Z -grading z . MOOTH K-THEORY We require that the restrictions of the family Dirac bundles to the fibres E b := π − ( b ), b ∈ B , give Dirac bundles in the usual sense (see [ Bun , Def. 3.1]):1. The vertical metric induces the Riemannian structure on E b ,2. The Clifford multiplication turns V | E b into a Clifford module (see [ BGV04 ,Def.3.32]) which is graded if dim( E b ) is even.3. The restriction of the connection ∇ V to E b is a Clifford connection (see[ BGV04 , Def.3.39]).A geometric family is called even or odd, if dim( T v π ) is even-dimensional or odd-dimensional, respectively. Here is a simple example of a geometric family with zero-dimensional fibres.Let V → B be a complex Z / Z -graded vector bundle. Assume that V comes with ahermitean metric h V and a hermitean connection ∇ V which are compatible with the Z / Z -grading. The geometric bundle ( V, h V , ∇ V ) will usually be denoted by V .We consider the submersion π := id B : B → B . In this case the vertical bundleis the zero-dimensional bundle which has a canonical vertical Riemannian metric g T v π := 0, and for the horizontal bundle we must take T h π := T B . Furthermore,there is a canonical orientation of p . The geometric bundle V can naturally beinterpreted as a family of Dirac bundles on B → B . In this way V gives rise to ageometric family over B which we will usually denote by V . In order to define a representative of the negative of the smooth K -theoryclass represented by a cycle ( E , ρ ) we introduce the notion of the opposite geometricfamily. Definition 2.3 . —
The opposite E op of a geometric family E is obtained by revers-ing the signs of the Clifford multiplication and the grading (in the even case) of theunderlying family of Clifford bundles, and of the orientation of the vertical bundle.2.1.6. — Our smooth K -theory groups will be Z / Z -graded. On the level of cyclesthe grading is reflected by the notions of even and odd cycles. Definition 2.4 . —
A cycle ( E , ρ ) is called even (or odd, resp.), if E is even (or odd,resp.) and ρ ∈ Ω odd ( B ) / im ( d ) ( or ρ ∈ Ω ev ( B ) / im ( d ) , resp.).2.1.7. — Let E and E ′ be two geometric families over B . An isomorphism E ∼ → E ′ consists of the following data: V (cid:15) (cid:15) F / / V ′ (cid:15) (cid:15) E π (cid:31) (cid:31) @@@@@@@ f / / E ′ π ′ ~ ~ }}}}}}} B ULRICH BUNKE & THOMAS SCHICK where1. f is a diffeomorphism over B ,2. F is a bundle isomorphism over f ,3. f preserves the horizontal distribution, the vertical metric and the orientation.4. F preserves the connection, Clifford multiplication and the grading. Definition 2.5 . —
Two cycles ( E , ρ ) and ( E ′ , ρ ′ ) are called isomorphic if E and E ′ are isomorphic and ρ = ρ ′ . We let G ∗ ( B ) denote the set of isomorphism classes ofcycles over B of parity ∗ ∈ { ev, odd } .2.1.8. — Given two geometric families E and E ′ we can form their sum E ⊔ B E ′ over B .The underlying proper submersion with closed fibres of the sum is π ⊔ π ′ : E ⊔ E ′ → B .The remaining structures of E ⊔ B E ′ are induced in the obvious way. Definition 2.6 . —
The sum of two cycles ( E , ρ ) and ( E ′ , ρ ′ ) is defined by ( E , ρ ) + ( E ′ , ρ ′ ) := ( E ⊔ B E ′ , ρ + ρ ′ ) . The sum of cycles induces on G ∗ ( B ) the structure of a graded abelian semigroup.The identity element of G ∗ ( B ) is the cycle 0 := ( ∅ , ∅ is the empty geometricfamily. In this subsection we introduce an equivalence relation ∼ on G ∗ ( B ). Weshow that it is compatible with the semigroup structure so that we get a semigroup G ∗ ( B ) / ∼ . We then define the smooth K -theory ˆ K ∗ ( B ) as the group completion ofthis quotient.In order to define ∼ we first introduce a simpler relation ”paired” which has anice local index-theoretic meaning. The relation ∼ will be the equivalence relationgenerated by ”paired”. The main ingredients of our definition of ”paired” are the notions of ataming of a geometric family E introduced in [ Bun , Def. 4.4], and the η -form of atamed family [ Bun , Def. 4.16].In this paragraph we shortly review the notion of a taming. For the definition ofeta-forms we refer to [
Bun , Sec. 4.4]. In the present paper we will use η -forms as ablack box with a few important properties which we explicitly state at the appropriateplaces below.If E is a geometric family over B , then we can form a family of Hilbert spaces( H b ) b ∈ B , where H b := L ( E b , V | E b ). If E is even, then this family is in addition Z / Z -graded. The geometric family E gives rise to a family of Dirac operators ( D ( E b )) b ∈ B ,where D ( E b ) is an unbounded selfadjoint operator on H b , which is odd in the evencase.A pre-taming of E is a family ( Q b ) b ∈ B of selfadjoint operators Q b ∈ B ( H b ) givenby a smooth fibrewise integral kernel Q ∈ C ∞ ( E × B E, V ⊠ V ∗ ). In the even case we MOOTH K-THEORY assume in addition that Q b is odd, i.e. that it anticommutes with the grading z . Thepre-taming is called a taming if D ( E b ) + Q b is invertible for all b ∈ B .The family of Dirac operators ( D ( E b )) b ∈ B has a K -theoretic index which we denoteby index ( E ) ∈ K ( B ) . If the geometric family E admits a taming, then the associated family of Dirac oper-ators operators admits an invertible compact perturbation, and hence index ( E ) = 0.Vice versa, if index ( E ) = 0 and the even part is empty or has a component withdim( T v π ) >
0, then by [
Bun , Lemma. 4.6] the geometric family admits a taming.If the even part of E has zero-dimensional fibres, then the existence of a tamingmay require some stabilization. This means that we must add a geometric family V ⊔ B V op (see 2.1.4 and Definition 2.3), where V is the bundle B × C n → B forsufficiently large n . Definition 2.7 . —
A geometric family E together with a taming will be denoted by E t and called a tamed geometric family. Let E t be a taming of the geometric family E by the family ( Q b ) b ∈ B . Definition 2.8 . —
The opposite tamed family E opt is given by the taming ( − Q b ) b ∈ B of E op .2.2.4. — The local index form Ω( E ) ∈ Ω( B ) is a differential form canonically asso-ciated to a geometric family. For a detailed definition we refer to [ Bun , Def..4.8],but we can briefly formulate its construction as follows. The vertical metric T v π andthe horizontal distribution T h π together induce a connection ∇ T v π on T v π (see 3.1.3for more details). Locally on E we can assume that T v π has a spin structure. Welet S ( T v π ) be the associated spinor bundle. Then we can write the family of Diracbundles V as V = S ⊗ W for a twisting bundle ( W, h W , ∇ W , z W ) with metric, metricconnection, and Z / Z -grading which is determined uniquely up to isomorphism. Theform ˆ A ( ∇ T v π ) ∧ ch ( ∇ W ) ∈ Ω( E ) is globally defined, and we get the local index formby applying the integration over the fibre R E/B : Ω( E ) → Ω( B ):Ω( E ) := Z E/B ˆ A ( ∇ T v π ) ∧ ch ( ∇ W ) . The local index form is closed and represents a cohomology class [Ω( E )] ∈ H dR ( B ).We let ch dR : K ( B ) → H dR ( B ) be the composition ch dR : K ( B ) ch → H ( B ; Q ) can → H dR ( B ) . The characteristic class version of the index theorem for families is ULRICH BUNKE & THOMAS SCHICK
Theorem 2.9 ( [ AS71 ] ) . — ch dR ( index ( E )) = [Ω( E )] . A proof using methods of local index theory has been given by [
Bis85 ]. For apresentation of the proof we refer to [
BGV04 ]. An alternative proof can be obtainedfrom [
Bun , Thm.4.18] by specializing to the case of a family of closed manifolds.
If a geometric family E admits a taming E t (see Definition 2.7), then wehave index ( E ) = 0. In particular, the local index form Ω( E ) is exact. The importantfeature of local index theory in this case is that it provides an explicit form whoseboundary is Ω( E ) (see equation (6) below).Let E t be a tamed geometric family over B . In [ Bun , Def. 4.16] we have definedthe η -form η ( E t ) ∈ Ω( B ). By [ Bun , Theorem 4.13]) it satisfies dη ( E t ) = Ω( E ) . (6)The first construction of η -forms has been given in [ BC90a ], [
BC90b ], [
BC91 ] underthe assumption that ker( D ( E b )) vanishes or has constant dimension. The variantwhich we use here has also been considered in [ Lot94 ], [
MP97b ], [
MP97a ].Since the analytic details of the definition of the η -form η ( E t ) are quite complicatedwe will not repeat them here but refer to [ Bun , Def. 4.16]. For most of the presentpaper we can use the construction of the η -form as a black box refering to [ Bun ] fordetails of the construction and the proofs of properties. Exceptions are argumentsinvolving adiabatic limits for which we use [
BM04 ] as the reference.
Now we can introduce the relations ”paired” and ∼ . Definition 2.10 . —
We call two cycles ( E , ρ ) and ( E ′ , ρ ′ ) paired if there exists ataming ( E ⊔ B E ′ op ) t such that ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ) . We let ∼ denote the equivalence relation generated by the relation ”paired”. Lemma 2.11 . —
The relation ”paired” is symmetric and reflexive.Proof . — In order to show that ”paired” is reflexive and symmetric we are goingemploy the relation [
Bun , Lemma 4.12] η ( E opt ) = − η ( E t ) . (7)Let E be a geometric family over B , and let H b denote the Hilbert space of sectionsof the Dirac bundle along the fibre over b ∈ B . The family E ⊔ B E op has an involution τ which flips the components, the signs of the Clifford multiplications, the gradingand the orientations. We use the same symbol τ in order to denote the action of τ on the Hilbert space of sections of the Dirac bundle of E b ⊔ B E opb . The latter can be MOOTH K-THEORY identified with H b ⊕ H opb , and in this picture τ = (cid:18) (cid:19) . Note that τ anticommutes with D b := D ( E b ⊔ B E opb ) = (cid:18) D ( E b ) 00 − D ( E b ) (cid:19) . We choose an even, compactly supported smooth function χ : R → [0 , ∞ ) such that χ (0) = 1 and form Q b := τ χ ( D b ) . This operator also anticommutes with D b , and ( D b + Q b ) = D b + χ ( D b ) is positiveand therefore invertible for all b ∈ B . The family ( Q b ) b ∈ B thus defines a taming( E ⊔ B E op ) t .The involution σ := (cid:18) i − i (cid:19) on the Hilbert space H b ⊕ H opb is induced by an isomorphism( E ⊔ B E op ) t ∼ = ( E ⊔ B E op ) opt . Because of the relation (7) we have η (( E ⊔ B E op ) t ) = 0. It follows that ( E , ρ ) is pairedwith ( E , ρ ).Assume now that ( E , ρ ) is paired with ( E ′ , ρ ′ ) via the taming ( E ⊔ B E ′ op ) t so that ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ). Then ( E ⊔ B E ′ op ) opt is a taming of E ′ ⊔ B E op such that ρ ′ − ρ = η (( E ⊔ B E ′ op ) opt ), again by (7). It follows that ( E ′ , ρ ′ ) is paired with ( E , ρ ). Lemma 2.12 . —
The relations ”paired” and ∼ are compatible with the semigroupstructure on G ∗ ( B ) .Proof . — In fact, if ( E i , ρ i ) are paired with ( E ′ i , ρ ′ i ) via tamings ( E i ⊔ B E ′ opi ) t for i = 0 ,
1, then ( E , ρ ) + ( E ′ , ρ ′ ) is paired with ( E , ρ ) + ( E ′ , ρ ′ ) via the taming( E ⊔ B E ⊔ B ( E ′ ⊔ B E ′ ) op ) t := ( E ⊔ B E ′ op ) t ⊔ B ( E ⊔ B E ′ op ) t . In this calculation we use the additivity of the η -form [ Bun , Lemma 4.12] η ( E t ⊔ B F t ) = η ( E t ) + η ( F t ) . The compatibilty of ∼ with the sum follows from the compatibility of ”paired”.We get an induced semigroup structure on G ∗ ( B ) / ∼ . Lemma 2.13 . — If ( E , ρ ) ∼ ( E , ρ ) , then there exists a cycle ( E ′ , ρ ′ ) such that ( E , ρ ) + ( E ′ , ρ ′ ) is paired with ( E , ρ ) + ( E ′ , ρ ′ ) . ULRICH BUNKE & THOMAS SCHICK
Proof . — Let ( E , ρ ) be paired with ( E , ρ ) via a taming ( E ⊔ B E op ) t , and ( E , ρ ) bepaired with ( E , ρ ) via ( E ⊔ B E op ) t . Then ( E , ρ ) + ( E , ρ ) is paired with ( E , ρ ) +( E , ρ ) via the taming(( E ⊔ B E ) ⊔ B ( E ⊔ B E ) op ) t := ( E ⊔ B E op ) t ⊔ B ( E ⊔ B E op ) t . If ( E , ρ ) ∼ ( E , ρ ), then there is a chain ( E ,α , ρ ,α ), α = 1 , . . . , r with ( E , , ρ , ) =( E , ρ ), ( E ,r , ρ ,r ) = ( E , ρ ), such that ( E ,α , ρ ,α ) is paired with ( E ,α +1 , ρ ,α +1 ).The assertion of the Lemma follows from an ( r − K -theory. — In this subsection we define the contravariant functor B → ˆ K ( B ) fromcompact smooth manifolds to Z / Z -graded abelian groups. Recall the definition 2.6of the semigroup of isomorphism classes of cycles. By Lemma 2.12 we can form thesemigroup G ∗ ( B ) / ∼ . Definition 2.14 . —
We define the smooth K -theory ˆ K ∗ ( B ) of B to be the groupcompletion of the abelian semigroup G ∗ ( B ) / ∼ . If ( E , ρ ) is a cycle, then let [ E , ρ ] ∈ ˆ K ∗ ( B ) denote the corresponding class in smooth K -theory.We now collect some simple facts which are helpful for computations in ˆ K ( B ) onthe level of cycles. Lemma 2.15 . —
We have [ E , ρ ] + [ E op , − ρ ] = 0 .Proof . — We show that ( E , ρ ) + ( E op , − ρ ) = ( E ⊔ B E op ,
0) is paired with 0 = ( ∅ , E ⊔ B E op ) ⊔ B ∅ op ) t = ( E ⊔ E op ) t introducedin the proof of Lemma 2.11 with η (( E ⊔ B E op ) t ) = 0. Lemma 2.16 . —
Every element of ˆ K ∗ ( B ) can be represented in the form [ E , ρ ] .Proof . — An element of ˆ K ∗ ( B ) can be represented by a difference [ E , ρ ] − [ E , ρ ].Using Lemma 2.15 we get [ E , ρ ] − [ E , ρ ] = [ E , ρ ] + [ E op , − ρ ] = [ E ⊔ B E op , ρ − ρ ]. Lemma 2.17 . — If [ E , ρ ] = [ E , ρ ] , then there exists a cycle ( E ′ , ρ ′ ) such that ( E , ρ ) + ( E ′ , ρ ′ ) is paired with ( E , ρ ) + ( E ′ , ρ ′ ) .Proof . — The relation [ E , ρ ] = [ E , ρ ] implies that there exists a cycle ( ˜ E , ˜ ρ ) suchthat ( E , ρ ) + ( ˜ E , ρ ) ∼ ( E , ρ ) + ( ˜ E , ˜ ρ ). The assertion now follows from Lemma2.13. MOOTH K-THEORY In this paragraph we extend B ˆ K ∗ ( B ) to a contravariant functor fromsmooth manifolds to Z / Z -graded groups. Let f : B → B be a smooth map. Thenwe have to define a map f ∗ : ˆ K ∗ ( B ) → ˆ K ( B ). We will first define a map of abeliansemigroups f ∗ : G ∗ ( B ) → G ∗ ( B ), and then we show that it passes to ˆ K .If E is a geometric family over B , then we can define an induced geometric family f ∗ E over B . The underlying submersion and vector bundle of f ∗ E are given by thecartesian diagram f ∗ V (cid:15) (cid:15) / / V (cid:15) (cid:15) f ∗ E f ∗ π (cid:15) (cid:15) F / / E π (cid:15) (cid:15) B f / / B . The metric g T v f ∗ π and the orientation of T v f ∗ π are defined such that dF : T v f ∗ π → F ∗ T v π is an isometry and orientation preserving. The horizontal distribution T h f ∗ π is given by the condition that dF ( T h f ∗ π ) ⊆ F ∗ T h π . Finally, the Dirac bundlestructure of f ∗ V is induced from the Dirac bundle structure on V in the usual way.For b ∈ B let H b be the Hilbert space of sections of V along the fibre E b . If b ∈ B satisfies f ( b ) = b , then we can identify the Hilbert space of sections of f ∗ V along the fibre f ∗ E b canonically with H b . If ( Q b ) b ∈ B defines a taming E t of E ,then the family ( Q f ( b ) ) b ∈ B is a taming f ∗ E t of f ∗ E . We have the following relationof η -forms: η ( f ∗ E t ) = f ∗ η ( E t ) . (8)In order to see this note the following facts. The geometric family E gives rise to abundle of Hilbert spaces H ( E ) → B with fibres H ( E ) b = H b , using the notationintroduced above. We have a natural isomorphism H ( f ∗ E ) ∼ = f ∗ H ( E ). The geometryof E together with the taming induces a family of super-connections A s ( E t ) on H parametrized by s ∈ (0 , ∞ ) (see [ Bun , 4.4.4] for explicit formulas). By constructionwe have f ∗ A s ( E t ) = A s ( f ∗ E t ). The η -form η ( E t ) is defined as an integral of the trace ofa family of operators on H ( E ) (with differential form coefficients) build from ∂ s A s ( E t )and A s ( E ) [ Bun , Definition 4.16]. Equation (8) now follows from f ∗ ∂ s A s ( E t ) = ∂ s A s ( f ∗ E t ) and f ∗ A s ( E ) = A s ( f ∗ E t ) .If ( E , ρ ) ∈ G ( B ), then we define f ∗ ( E , ρ ) := ( f ∗ E , f ∗ ρ ) ∈ G ( B ). The pull-backpreserves the disjoint union and opposites of geometric families. In particular, f ∗ isa semigroup homomorphism. Assume now that ( E , ρ ) is paired with ( E ′ , ρ ′ ) via thetaming ( E ⊔ B E ′ op ) t . Then we can pull back the taming as well and get a taming f ∗ ( E ⊔ B E ′ op ) t of f ∗ E ⊔ B f ∗ E ′ op . Equation (8) now implies that f ∗ ( E , ρ ) is pairedwith f ∗ ( E ′ , ρ ′ ) via the taming f ∗ ( E ⊔ B E ′ op ) t . ULRICH BUNKE & THOMAS SCHICK
Hence, the pull-back f ∗ passes to G ∗ ( B ) / ∼ , and being a semigroup homomor-phism, it induces a map of group completions f ∗ : ˆ K ∗ ( B ) → ˆ K ∗ ( B ) . Evidently, ( id B ) ∗ = ˆ id ˆ K ∗ ( B ) . Let f ′ : B → B be another smooth map. If E is ageometric family over B , then ( f ◦ f ′ ) ∗ E is isomorphic to f ′∗ f ∗ E . This observationimplies that f ′∗ f ∗ = ( f ◦ f ′ ) ∗ : ˆ K ∗ ( B ) → ˆ K ( B ) . This finishes the construction of the contravariant functor ˆ K ∗ on the level of mor-phisms. In this subsection we introduce the transformations
R, I, a , and we showthat they turn the functor ˆ K into a smooth extension of ( K, ch R ) in the sense ofDefinition 1.1. We first define the natural transformation I : ˆ K ( B ) → K ( B )by I [ E , ρ ] := index ( E ) . We must show that I is well-defined. Consider ˜ I : G ( B ) → K ( B ) defined by ˜ I ( E , ρ ) := index ( E ). If ( E , ρ ) is paired with ( E ′ , ρ ′ ), then the existence of a taming ( E ⊔ B E ′ op ) t implies that index ( E ) = index ( E ′ ). The relation index ( E ⊔ B E ′ ) = index ( E ) + index ( E ′ ) (9)together with Lemma 2.13 now implies that ˜ I descends to G ( B ) / ∼ . The additivity(9) and the definition of ˆ K ( B ) as the group completion of G ( B ) / ∼ implies that ˜ I further descends to the homomorphism I : ˆ K ( B ) → K ( B ).The relation index ( f ∗ E ) = f ∗ index ( E ) shows that I is a natural transformationof functors from smooth manifolds to Z / Z -graded abelian groups. Lemma 2.18 . —
For every compact manifold B , the transformation I : ˆ K ( B ) → K ( B ) is surjective.Proof . — We discuss even and odd degrees seperately. In the even case, a K-theoryclass ξ ∈ K ( B ) is represented by a Z / Z -graded vector bundle V on B . Simplychoose a hermitean metric and a connection on V . We obtain a resulting geometricfamily V on B , with underlying submersion id : B → B (i.e. 0-dimensional fibres) asin 2.1.4, and clearly I ( V ) = index ( V ) = [ V ] = ξ ∈ K ( B ).For odd degrees, the statement is proved in [ Bun , 3.1.6.7].
MOOTH K-THEORY We consider the functor B Ω ∗ ( B ) / im ( d ), ∗ ∈ { ev, odd } as a functor frommanifolds to Z / Z -graded abelian groups. We construct a parity-reversing naturaltransformation a : Ω ∗ ( B ) / im ( d ) → ˆ K ∗ ( B )by a ( ρ ) := [ ∅ , − ρ ] . Let Ω ∗ d =0 ( B ) be the group of closed forms of parity ∗ on B . Again weconsider B Ω ∗ d =0 ( B ) as a functor from smooth manifolds to Z / Z -graded abeliangroups. We define a natural transformation R : ˆ K ( B ) → Ω d =0 ( B )by R ([ E , ρ ]) = Ω( E ) − dρ . Again we must show that R is well-defined. We will use the relation (6) of the η -formand the local index form, and the obvious properties of local index formsΩ( E ⊔ B E ′ ) = Ω( E ) + Ω( E ′ ) , Ω( E op ) = − Ω( E ) . We start with ˜ R : G ( B ) → Ω( B ) , ˜ R ( E , ρ ) := Ω( E ) − dρ . Since Ω( E ) is closed, ˜ R ( E , ρ ) is closed. If ( E , ρ ) is paired with ( E ′ , ρ ′ ) via the taming( E ⊔ B E ′ op ) t , then ρ − ρ ′ = η (( E ⊔ B E ′ op ) t ). It follows R ( E , ρ ) = Ω( E ) − dρ = Ω( E ) − dρ ′ − dη (( E ⊔ B E ′ op ) t )= Ω( E ) − dρ ′ − Ω( E ) − Ω( E ′ op )= Ω( E ′ ) − dρ ′ = R ( E ′ , ρ ′ ) . Since ˜ R is additive it descends to G ( B ) / ∼ and finally to the map R : ˆ K ( B ) → Ω d =0 ( B ). It follows from Ω( f ∗ E ) = f ∗ Ω( E ) that R is a natural transformation. The natural transformations satisfy the following relations:
Lemma 2.19 . — 1. R ◦ a = d ch dR ◦ I = [ . . . ] ◦ R .Proof . — The first relation is an immediate consequence of the definition of R and a . The second relation is the local index theorem 2.9. ULRICH BUNKE & THOMAS SCHICK
Via the embedding H dR ( B ) ⊆ Ω( B ) / im ( d ), the Chern character ch dR : K ( B ) → H dR ( B ) can be considered as a natural transformation ch dR : K ( B ) → Ω( B ) / im ( d ) . Proposition 2.20 . —
The following sequence is exact: K ( B ) ch dR → Ω( B ) / im ( d ) a → ˆ K ( B ) I → K ( B ) → . We give the proof in the following couple of subsection.
We start with the surjectivity of I : ˆ K ( B ) → K ( B ). The main point isthe fact that every element x ∈ K ( B ) can be realized as the index of a family ofDirac operators by Lemma 2.18. So let x ∈ K ( B ) and E be a geometric family with index ( E ) = x . Then we have I ([ E , x . Next we show exactness at ˆ K ( B ). For ρ ∈ Ω( B ) / im ( d ) we have I ◦ a ( ρ ) = I ([ ∅ , − ρ ]) = index ( ∅ ) = 0, hence I ◦ a = 0. Consider a class [ E , ρ ] ∈ ˆ K ( B ) whichsatisfies I ([ E , ρ ]) = 0. We can assume that the fibres of the underlying submersion arenot zero-dimensional. Indeed, if necessary, we can replace E by E⊔ B ( ˜ E ⊔ B ˜ E op ) for someeven family with nonzero-dimensional fibres without changing the smooth K -theoryclass by Lemma 2.15. Since index ( E ) = 0 this family admits a taming E t (2.2.2).Therefore, ( E , ρ ) is paired with ( ∅ , ρ − η ( E t )). It follows that [ E , ρ ] = a ( η ( E t ) − ρ ). In order to prepare the proof of exactness at Ω( B ) / im ( d ) in 2.4.11 we needsome facts about the classification of tamings of a geometric family E . The mainidea is to measure the difference between tamings of E using a local index theoremfor E × [0 ,
1] (compare [
Bun , Cor. 2.2.19]). Let us assume that the underlyingsubmersion π : E → B decomposes as E = E ev ⊔ B E odd such that the restriction of π to the even and odd parts is surjective with nonzero- and even-dimensional andodd-dimensional fibres, and which is such that the Clifford bundle is nowhere zero-dimensional. If index ( E ) = 0, then there exists a taming E t (see 2.2.2). Assumethat E t ′ is a second taming. Both tamings together induce a boundary taming of thefamily with boundary ( E × [0 , bt . In [ Bun ] we have discussed in detail geometricfamilies with boundaries and the operation of taking a boundary of a geometric familywith boundary. In the present case
E × [0 ,
1] has two boundary faces labeled by theendpoints { , } of the interval. We have ∂ ( E × [0 , ∼ = E and ∂ ( E × [0 , ∼ = E op .A boundary taming ( E × [0 , bt is given by tamings of ∂ i ( E × [0 , i = 0 , Bun , Def. 2.1.48]). We use E t at E × { } and E opt ′ at E × { } .The boundary tamed family has an index index (( E × [0 , bt ) ∈ K ( B ) which isthe obstruction against extending the boundary taming to a taming [ Bun , Lemma2.2.6]. The construction of the local index form extends to geometric families withboundaries. Because of the geometric product structure of
E × [0 ,
1] we have Ω(
E × [0 , Bun , Theorem 2.2.18]gives ch dR ◦ index (( E × [0 , bt ) = [ η ( E t ) − η ( E t ′ )] . MOOTH K-THEORY On the other hand, given x ∈ K ( B ) and E t , since we have chosen our family E sufficiently big, there exists a taming E t ′ such that index (( E × [0 , bt ) = x .To prove this, we argue as follows. Given tamings E t and E t ′ we obtain a family D ( E t , E t ′ ) of perturbed Dirac operators over B × R which restricts to D ( E t ) on B ×{ β } for β <
0, and to D ( E t ′ ) for β ≥
1, and which interpolates these families for β ∈ [0 , D ( E t , E t ′ ) is invertible outside of a compact subset of B × R (note that B is compact) it gives rise to a class [ E t , E t ′ ] ∈ KK ( C , C ( B ) ⊗ C ( R )). TheDirac operator on R provides a class [ ∂ ] ∈ KK ( C ( R ) , C ), and one checks —using themethod of connections as in [ Bun95 , proof of Proposition 2.11] or directly workingwith the unbounded picture [
BJ83 ]— that D ( E × [0 , bt represents the Kasparovproduct [ E t , E t ′ ] ⊗ C ( R ) [ ∂ ] ∈ KK ( C , C ( B )) . The map K c ( B × R ) ∼ → KK ( C , C ( B ) ⊗ C ( R )) ·⊗ C R ) [ ∂ ] → KK ( C , C ( B )) ∼ → K ( B )is by [ Kas81 , Paragraph 5, Theorem 7] the inverse of the suspension isomorphism, soin particular surjective. It remains to see that one can exhaust KK ( C , C ( B ) ⊗ C ( R ))with classes of the form [ E t , E t ′ ] by varying the taming E t ′ .We sketch an argument in the even-dimensional case. The odd-dimensional case issimilar. For a separable infinite-dimensional Hilbert space H let GL ( H ) ⊂ GL ( H )be the group of invertible operators of the form 1 + K with K ∈ K ( H ) compact.The space GL ( H ) has the homotopy type of the classifying space for K . Thebundle of Hilbert spaces H ( E ) + → B gives rise to a (canonically trivial, up to ho-motopy) bundle of groups GL ( H ( E ) + ) → B by taking GL ( . . . ) fibrewise (it is herewhere we use that the family is sufficiently big so that H ( E ) + is infinite-dimensional).Let Γ( GL ( H ( E ) + )) be the topological group of sections. Then we have an isomor-phism π Γ( GL ( H ( E ) + )) ∼ = K ( B ). Let x ∈ K ( B ) be represented by a section s ∈ Γ( GL ( H ( E ) + )). We can approximate s − s − Bun ]) (4) .There is a bijection between tamings E t ′ and sections s ∈ Γ( GL ( H ( E ) + )) of thistype which maps E t ′ to s := D + ( E t ) − D + ( E t ′ ). The map which associates the KK -class [ E t , E t ′ ] to the section s is just one realization of the suspension isomorphism K ( B ) → K c ( B × R ) (using the Kasparov picture of the latter group). In particularwe see that all classes in K c ( B × R ) arise as [ E t , E t ′ ] for various tamings E t ′ . We now show exactness at Ω( B ) / im ( d ). Let x ∈ K ( B ). Then we have a ◦ ch dR ( x ) = [ ∅ , − ch dR ( x )]. We choose a geometric family E as in 2.4.10 and set˜ E := E ⊔ B E op . In the proof of Lemma 2.11 we have constructed a taming ˜ E t suchthat η ( ˜ E t ) = 0. Using the discussion 2.4.10 we choose a second taming ˜ E t ′ such that (4) Alternatively one can directly produce such a section using the setup described in [
MR07 ]. ULRICH BUNKE & THOMAS SCHICK index (( ˜
E × [0 , bt ) = − x , hence η ( ˜ E t ′ ) = ch dR ( x ). By the taming ˜ E t ′ we see thatthe cycle ( ˜ E ,
0) pairs with ( ∅ , − ch dR ( x )). On the other hand, via ˜ E t the cycle ( ˜ E , ∅ , − ch dR ( x )) ∼ a ◦ ch dR = 0.Let now ρ ∈ Ω( B ) / im ( d ) be such that a ( ρ ) = [ ∅ , − ρ ] = 0. Then by Lemma 2.17there exists a cycle ( ˆ E , ˆ ρ ) such that ( ˆ E , ˆ ρ − ρ ) pairs with ( ˆ E , ˆ ρ ). Therefore there existsa taming E t ′ of E := ˆ E ⊔ B ˆ E op such that η ( E t ′ ) = − ρ .Let E t be the taming with vanishing η -form constructed in the proof ofLemma 2.11. The two tamings induce a boundary taming ( E × [0 , bt suchthat ch dR ◦ index (( E × [0 , bt ) = − η ( E t ′ ) = ρ . This shows that ρ is in the image of ch dR . We now improve Lemma 2.13. This result will be very helpful in verifyingwell-definedness of maps out of smooth K -theory, e.g. the smooth Chern character. Lemma 2.21 . — If [ E , ρ ] = [ E , ρ ] and at least one of these families has a higher-dimensional component, then ( E , ρ ) is paired with ( E , ρ ) .Proof . — By Lemma 2.13 there exists [ E ′ , ρ ′ ] such that ( E , ρ ) + ( E ′ , ρ ′ ) is pairedwith ( E , ρ ) + ( E ′ , ρ ′ ) by a taming ( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t . We have ρ − ρ = η (( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t ) . Since index ( E ) = index ( E ) there exists a taming ( E ⊔ B E op ) t . Furthermore, thereexists a taming ( E ′ ⊔ B E ′ op ) t with vanishing η -invariant (see the proof of Lemma 2.11).These two tamings combine to a taming ( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t ′ . There exists ξ ∈ K ( B ) such that ch dR ( ξ ) = η (( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t ) − η (( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t ′ )= η (( E ⊔ B E ′ ⊔ B ( E ⊔ B E ′ ) op ) t ) − η (( E ⊔ B E op ) t ) . We can now adjust (using 2.4.10) the taming ( E ⊔ B E op ) t such that we can choose ξ = 0. It follows that ρ − ρ = η (( E ⊔ B E op ) t ). An important consequence of the axioms 1.1 for a smooth generalized co-homology theory is the homotopy formula. Let ˆ h be a smooth extension of a pair( h, c ). Let x ∈ ˆ h ([0 , × B ), and let i k : B → { k } × B ⊂ [0 , × B , k = 0 ,
1, be theinclusions.
Lemma 2.22 . — i ∗ ( x ) − i ∗ ( x ) = a Z [0 , × B/B R ( x ) ! . MOOTH K-THEORY Proof . — Let p : [0 , × B → B denote the projection. If x = p ∗ y , then on the onehand the left-hand side of the equation is zero. On the other hand, R ( x ) = p ∗ R ( y )so that R [0 , × B/B R ( x ) = 0, too.Since p is a homotopy equivalence there exists ¯ y ∈ h ( B ) such that I ( x ) = p ∗ (¯ y ).Because of the surjectivity of I we can choose y ∈ ˆ h ( B ) such that I ( y ) = ¯ y . It followsthat I ( x − p ∗ y ) = 0. By the exactness of (3) there exists a form ω ∈ Ω( I × B ) / im ( d )such that x − p ∗ y = a ( ω ). By Stokes’ theorem we have the equality i ∗ ω − i ∗ ω = R [0 , × B/B dω in Ω( B ) / im ( d ). By (2) we have dω = R ( a ( ω )). It follows that Z [0 , × B/B dω = Z [0 , × B/B R ( a ( ω )) = Z [0 , × B/B R ( x − p ∗ y ) = Z [0 , × B/B R ( x ) . This implies i ∗ x − i ∗ x = i ∗ a ( ω ) − i ∗ a ( ω ) = a i ∗ ω − i ∗ ω ) = a ( Z [0 , × B/B R ( x ) ! . Let ˆ h be a smooth extension of a pair ( h, c ). We use the notation introducedin 1.2.2. Definition 2.23 . —
The associated flat functor is defined by B ˆ h flat ( B ) := ker { R : ˆ h ( B ) → Ω d =0 ( B, N ) } . Recall that a functor F from smooth manifolds is homotopy invariant, if for thetwo embeddings i k : B → { k } × B → [0 , × B , k = 0 ,
1, we have F ( i ) = F ( i ). Asa consequence of the homotopy formula Lemma 2.22 the functor ˆ h flat is homotopyinvariant.In interesting cases it is part of a generalized cohomology theory. The map c : h → HN gives rise to a cofibre sequence in the stable homotopy category h c → HN → h N, R / Z which defines a spectrum h N, R / Z . Proposition 2.24 . — If ˆ h is the Hopkins-Singer extension of ( h, c ) , then we have anatural isomorphism ˆ h flat ( B ) ∼ = h N, R / Z ( B )[ − . In the special case that N = h ∗ ⊗ Z R this is [ HS05 , (4.57)]. ULRICH BUNKE & THOMAS SCHICK
In the case of K -theory and the Chern character ch R : K → H ( K ∗ ⊗ Z R )one usually writes K R / Z := h K ∗ ⊗ Z R , R / Z . The functor B K R / Z ( B ) is called R / Z - K -theory. Since R / Z is an injective abeliangroup we have a universal coefficient formula K R / Z ∗ ( B ) ∼ = Hom ( K ∗ ( B ) , R / Z ) , (10)where K ∗ ( B ) denotes the K -homology of B . A geometric interpretation of R / Z - K -theory was first proposed in [ Kar87 ], [
Kar97 ]. In these reference it was calledmultiplicative K -theory. The analytic construction of the push-forward has beengiven in [ Lot94 ]. Proposition 2.25 . —
There is a natural isomorphism of functors ˆ K flat ( B ) ∼ = K R / Z ( B )[ − .Proof . — In the following (the paragraphs 2.5.5, 2.5.6) we sketch two conceptuallyvery different arguments. For details we refer to [ BS09 , Section 5, Section 7].
In the first step one extends ˆ K flat to a reduced cohomology theory onsmooth manifolds. The reduced group of a pointed manifold is defined as the kernelof the restriction to the point. The missing structure is a suspension isomorphism.It is induced by the map ˆ K ( B ) → ˆ K ( S × B ) given by x pr ∗ x S ∪ pr ∗ x , where x S ∈ ˆ K ( S ) is defined in Definition 5.6, and the ∪ -product is defined below in4.1. The inverse is induced by the push-forward ( ˆ pr ) ! : ˆ K ( S × B ) → ˆ K ( B ) along pr : S × B → B introduced below in 3.18. Finally one verifies the exactness ofmapping cone sequences.In order to identify the resulting reduced cohomology theory with R / Z - K -theoryone constructs a pairing between ˆ K flat and K -homology, using an analytic modelas in [ Lot94 ]. This pairing, in view of the universal coefficient formula (10) gives amap of cohomology theories ˆ K flat ( B ) → K R / Z ( B )[ −
1] which is an isomorphism bycomparison of coefficients.
The second argument is based on the comparison with the Hopkins-Singertheory. We let B ˆ K HS ( B ) denote the version of the smooth K -theory functordefined by Hopkins-Singer [ HS05 ]. In [
BS09 , Section 5] we show that there is aunique natural isomorphism ˆ K ev ∼ → ˆ K evHS . In view of 2.24 we get the isomorphismˆ K evflat ( B ) ∼ → ˆ K evHS,flat ( B ) ∼ → K R / Z ev [ − B ) . In [
BS09 ] we furthermore show that using the integration for ˆ K and the suspensionisomorphism for K R / Z this isomorphism extends to the odd parts. MOOTH K-THEORY Many of the interesting examples given in Section 5 can be understood (atleast to a large extend) already at this stage. We recommend to look them up now,if one is less interseted in structural questions. This should also serve as a motivationfor the constructions in Sections 3 and 4.
3. Push-forward3.1. K -orientation. — The groups
Spin ( n ) and Spin c ( n ) fit into exact sequences1 −−−−→ Z / Z −−−−→ Spin ( n ) −−−−→ SO ( n ) −−−−→ y y y id −−−−→ U (1) i −−−−→ Spin c ( n ) π −−−−→ SO ( n ) −−−−→ → Z / Z → Spin c ( n ) ( λ,π ) → U (1) × SO ( n ) → λ ◦ i : U (1) → U (1) is a double covering. Let P → B be an SO ( n )-principalbundle. We let Spin c ( n ) act on P via the projection π . Definition 3.1 . — A Spin c -reduction of P is a diagram Q (cid:31) (cid:31) ??????? f / / P (cid:127) (cid:127) (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) B , where Q → B is a Spin c ( n ) -principal bundle and f is Spin c ( n ) -equivariant.3.1.2. — Let p : W → B be a proper submersion with vertical bundle T v p . Weassume that T v p is oriented. A choice of a vertical metric g T v p gives an SO -reduction SO ( T v p ) of the frame bundle Fr ( T v p ), the bundle of oriented orthonormal frames.Usually one calls a map between manifolds K -oriented if its stable normal bundleis equipped with a K -theory Thom class. It is a well-known fact [ ABS64 ] that thisis equivalent to the choice of a
Spin c -structure on the stable normal bundle. Finally,isomorphism classes of choices of Spin c -structures on T v p and the stable normalbundle of p are in bijective correspondence. So for the purpose of the present paperwe adopt the following definition. Definition 3.2 . —
A topological K -orientation of p is a Spin c -reduction of SO ( T v p ) . In the present paper we prefer to work with
Spin c -structures on the vertical bundlesince it directly gives rise to a family of Dirac operators along the fibres. The goal ofthis section is to introduce the notion of smooth K -orientation which refines a giventopological K -orientation. ULRICH BUNKE & THOMAS SCHICK
In order to define such a family of Dirac operators we must choose additionalgeometric data. If we choose a horizontal distribution T h p , then we get a connection ∇ T v p which restricts to the Levi-Civita connection along the fibres. Its constructiongoes as follows. First one chooses a metric g T B on B . It induces a horizontal metric g T h p via the isomorphism dp | T h p : T h p ∼ → p ∗ T B . We get a metric g T v p ⊕ g T h p on T W ∼ = T v p ⊕ T h p which gives rise to a Levi-Civita connection. Its projection to T v p is ∇ T v p . Finally one checks that this connection is independent of the choice of g T B . The connection ∇ T v p can be considered as an SO ( n )-principal bundle con-nection on the frame bundle SO ( T v p ). In order to define a family of Dirac operators,or better, the Bismut super-connection we must choose a Spin c -reduction ˜ ∇ of ∇ T v p ,i.e. a connection on the Spin c -principal bundle Q which reduces to ∇ T v p . If we thinkof the connections ∇ T v p and ˜ ∇ in terms of horizontal distributions T h SO ( T v p ) and T h Q , then we say that ˜ ∇ reduces to ∇ T v p if dπ ( T h Q ) = π ∗ ( T h SO ( T v p )). The
Spin c -reduction of Fr ( T v p ) determines a spinor bundle S c ( T v p ), andthe choice of ˜ ∇ turns S c ( T v p ) into a family of Dirac bundles.In this way the choices of the Spin c -structure and ( g T v p , T h p, ˜ ∇ ) turn p : W → B into a geometric family W . Locally on W we can choose a Spin -structure on T v p with associated spinorbundle S ( T v p ). Then we can write S c ( T v p ) = S ( T v p ) ⊗ L for a hermitean line bundle L with connection. The spin structure is given by a Spin -reduction q : R → SO ( T v p )(similar to 3.1) which can actually be considered as a subbundle of Q . Since q isa double covering and thus has discrete fibres, the connection ∇ T v p (in contrast tothe Spin c -case) has a unique lift to a Spin ( n )-connection on R . The spinor bundle S ( T v p ) is associated to R and has an induced connection. In view of the relationsof the groups 3.1.1 the square of the locally defined line bundle L is the globallydefined bundle L → W associated to the Spin c -bundle Q via the representation λ : Spin c ( n ) → U (1). The connection ˜ ∇ thus induces a connection on ∇ L , andhence a connection on the locally defined square root L . Note that vice versa, ∇ L and ∇ T v p determine ˜ ∇ uniquely. We introduce the form c ( ˜ ∇ ) := 14 πi R L (11)which would be the Chern form of the bundle L in case of a global Spin -structure.Let R ∇ Tvp ∈ Ω ( W, End ( T v p )) denote the curvature of ∇ T v p . The closed formˆ A ( ∇ T v p ) := det / R ∇ Tvp π sinh (cid:16) R ∇ Tvp π (cid:17) represents the ˆ A -class of T v p . MOOTH K-THEORY Definition 3.3 . —
The relevant differential form for local index theory in the
Spin c -case is ˆ A c ( ˜ ∇ ) := ˆ A ( ∇ T v p ) ∧ e c ( ˜ ∇ ) . If we consider p : W → B with the geometry ( g T v p , T h p, ˜ ∇ ) and the Dirac bundle S c ( T v p ) as a geometric family W over B , then by comparison with the description2.2.4 of the local index form Ω( W ) we see that Z W/B ˆ A c ( ˜ ∇ ) = Ω( W ) . The dependence of the form ˆ A c ( ˜ ∇ ) on the data is described in terms of thetransgression form. Let ( g T v pi , T hi p, ˜ ∇ i ), i = 0 ,
1, be two choices of geometric data.Then we can choose geometric data ( g T v p , T h p, ˜ ∇ ) on p = id [0 , × p : [0 , × W → [0 , × B (with the induced Spin c -structure on T v p ) which restricts to ( g T v pi , T hi p, ˜ ∇ i )on { i } × B . The class˜ˆ A c ( ˜ ∇ , ˜ ∇ ) := Z [0 , × W/W ˆ A c ( ˜ ∇ ) ∈ Ω( W ) / im ( d )is independent of the extension and satisfies d ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) = ˆ A c ( ˜ ∇ ) − ˆ A c ( ˜ ∇ ) . (12) Definition 3.4 . —
The form ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) is called the transgression form. Note that we have the identity˜ˆ A c ( ˜ ∇ , ˜ ∇ ) + ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) = ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) . (13)As a consequence we get the identities˜ˆ A c ( ˜ ∇ , ˜ ∇ ) = 0 , ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) = − ˆ A c ( ˜ ∇ , ˜ ∇ ) . (14) We can now introduce the notion of a smooth K -orientation of a propersubmersion p : W → B . We fix an underlying topological K -orientation of p (seeDefinition 3.2) which is given by a Spin c -reduction of SO ( T v p ). In order to makethis precise we must choose an orientation and a metric on T v p .We consider the set O of tuples ( g T v p , T h p, ˜ ∇ , σ ) where the first three entries havethe same meaning as above (see 3.1.3), and σ ∈ Ω odd ( W ) / im ( d ). We introduce arelation o ∼ o on O : Two tuples ( g T v pi , T hi p, ˜ ∇ i , σ i ), i = 0 , σ − σ = ˜ˆ A ( ˜ ∇ , ˜ ∇ ). We claim that ∼ is an equivalence relation. In fact,symmetry and reflexivity follow from (14), while transitivity is a consequence of (13). Definition 3.5 . —
The set of smooth K -orientations which refine a fixed underlyingtopological K -orientation of p : W → B is the set of equivalence classes O / ∼ . ULRICH BUNKE & THOMAS SCHICK
Note that Ω odd ( W ) / im ( d ) acts on the set of smooth K -orientations. If α ∈ Ω odd ( W ) / im ( d ) and ( g T v p , T h p, ˜ ∇ , σ ) represents a smooth K -orientation, thenthe translate of this orientation by α is represented by ( g T v p , T h p, ˜ ∇ , σ + α ). As aconsequence of (13) we get: Corollary 3.6 . —
The set of smooth K -orientations refining a fixed underlying topo-logical K -orientation is a torsor over Ω odd ( W ) / im ( d ) .3.1.11. — If o = ( g T v p , T h p, ˜ ∇ , σ ) ∈ O represents a smooth K -orientation, then wewill write ˆ A c ( o ) := ˆ A c ( ˜ ∇ ) , σ ( o ) := σ . We consider a proper submersion p : W → B with a choice of a topological K -orientation. Assume that p has closed fibres. Let o = ( g T v p , T h p, ˜ ∇ , σ ) representa smooth K -orientation which refines the given topological one. To every geometricfamily E over W we want to associate a geometric family p ! E over B .Let π : E → W denote the underlying proper submersion with closed fibres of E which comes with the geometric data g T v π , T h π and the family of Dirac bundles( V, h V , ∇ V ).The underlying proper submersion with closed fibres of p ! E is q := p ◦ π : E → B .
The horizontal bundle of π admits a decomposition T h π ∼ = π ∗ T v p ⊕ π ∗ T h p , where theisomorphism is induced by dπ . We define T h q ⊆ T h π such that dπ : T h q ∼ = π ∗ T h p .Furthermore we have an identification T v q = T v π ⊕ π ∗ T v p . Using this decompositionwe define the vertical metric g T v q := g T v π ⊕ π ∗ g T v p . The orientations of T v π and T v p induce an orientation of T v q . Finally we must construct the Dirac bundle p ! V → E .Locally on W we choose a Spin -structure on T v p and let S ( T v p ) be the spinor bundle.Then we can write S c ( T v p ) = S ( T v p ) ⊗ L for a hermitean line bundle with connection.Locally on E we can choose a Spin -structure on T v π with spinor bundle S ( T v π ). Thenwe can write V = S ( T v π ) ⊗ Z , where Z is the twisting bundle of V , a hermitean vectorbundle with connection ( Z / Z -graded in the even case). The local spin structures on T v π and π ∗ T v p induce a local Spin -structure on T v q = T v π ⊕ π ∗ T v p . Thereforelocally we can define the family of Dirac bundles p ! V := S ( T v q ) ⊗ π ∗ L ⊗ Z . It iseasy to see that this bundle is well-defined independent of the choices of local Spin -structures and therefore is a globally defined family of Dirac bundles.
Definition 3.7 . —
Let p ! E denote the geometric family given by q : E → B and p ! V → E with the geometric structures defined above. It immediately follows from the definitions, that p ! ( E op ) ∼ = ( p ! E ) op . MOOTH K-THEORY Let p : W → B be a proper submersion with a smooth K -orientation rep-resented by o . In 3.2.1 we have constructed for each geometric family E over W apush-forward p ! E . Now we introduce a parameter λ ∈ (0 , ∞ ) into this construction. Definition 3.8 . —
For λ ∈ (0 , ∞ ) we define the geometric family p λ ! E as in 3.2.1with the only difference that the metric on T v q = T v π ⊕ π ∗ T v p is given by g T v qλ = λ g T v π ⊕ π ∗ g T v p . More specifically, we use scaling invariance of the spinor bundle to canonicallyidentify the Dirac bundle for the metric g λ locally with p ! V := S ( T v q ) ⊗ π ∗ L ⊗ Z (for g ). This uses the description of S ( T v p ) in terms of tensor products of S ( T v π )and π ∗ S ( T v p ) (compare [ Bun , Section 2.1.2]) and the scaling invariance of S ( T v π ).However, with this identification the Clifford multiplication by vectors in T v q = T v π ⊕ π ∗ T v p is rescaled on the summand T v π by λ . The connection is slightly morecomplicated, but converges for λ → p λ ! E is called the adiabatic deformation of p ! E .There is a natural way to define a geometric family F on (0 , ∞ ) × B such that itsrestriction to { λ } × B is p λ ! E . In fact, we define F := ( id (0 , ∞ ) × p ) ! ((0 , ∞ ) × E )with the exception that we take the appropriate vertical metric. Note again that theunderlying bundle can be canonically identified with (0 , ∞ ) × p ! V . In the following,we work with this identifications throughout.Although the vertical metrics of F and p λ ! E collapse as λ → T v q converge and simplify inthis limit. This fact is heavily used in local index theory, and we refer to [ BGV04 ,Sec 10.2] for details. In particular, the integral˜Ω( λ, E ) := Z (0 ,λ ) × B/B Ω( F ) (15)converges, and we have lim λ → Ω( p λ ! E ) = Z W/B ˆ A c ( o ) ∧ Ω( E ) , Ω( p λ ! E ) − Z W/B ˆ A c ( o ) ∧ Ω( E ) = d ˜Ω( λ, E ) . (16) Let p : W → B be a proper submersion with closed fibres with a smooth K -orientation represented by o . We now start with the construction of the push-forward p ! : ˆ K ( W ) → ˆ K ( B ). For λ ∈ (0 , ∞ ) and a cycle ( E , ρ ) we defineˆ p λ ! ( E , ρ ) := [ p λ ! E , Z W/B ˆ A c ( o ) ∧ ρ + ˜Ω( λ, E ) + Z W/B σ ( o ) ∧ R ([ E , ρ ])] ∈ ˆ K ( B ) . (17)Since ˆ A c ( o ) and R ([ E , ρ ]) are closed, the mapsΩ( W ) / im ( d ) ∋ ρ Z W/B ˆ A c ( o ) ∧ ρ ∈ Ω( B ) / im ( d ) , ULRICH BUNKE & THOMAS SCHICK Ω( W ) / im ( d ) ∋ σ ( o ) Z W/B σ ( o ) ∧ R ([ E , ρ ]) ∈ Ω( B ) / im ( d )are well-defined. It immediately follows from the definition that ˆ p λ ! : G ( W ) → ˆ K ( B )is a homomorphism of semigroups. The homomorphism ˆ p λ ! : G ( W ) → ˆ K ( B ) commutes with pull-back. Moreprecisely, let f : B ′ → B be a smooth map. Then we define the submersion p ′ : W ′ → B ′ by the cartesian diagram W ′ p ′ (cid:15) (cid:15) F / / W p (cid:15) (cid:15) B ′ f / / B .
The differential dF : T W ′ → F ∗ T W induces an isomorphism dF : T v W ′ ∼ → F ∗ T v W .Therefore the metric, the orientation, and the Spin c -structure of T v p induce by pull-back corresponding structures on T v p ′ . We define the horizontal distribution T h p ′ such that dF ( T h p ′ ) ⊆ F ∗ T h p . Finally we set σ ′ := F ∗ σ . The representative of asmooth K -orientation given by these structures will be denoted by o ′ := f ∗ o . Aninspection of the definitions shows: Lemma 3.9 . —
The pull-back of representatives of smooth K -orientations preservesequivalence and hence induces a pull-back of smooth K -orientations. Recall from 3.1.5 that the representatives o and o ′ of the smooth K -orientationsenhance p and p ′ to geometric families W and W ′ . We have f ∗ W ∼ = W ′ .Note that we have F ∗ ˆ A c ( o ) = ˆ A c ( o ′ ). If E is a geometric family over W , then aninspection of the definitions shows that f ∗ p ! ( E ) ∼ = p ′ ! ( F ∗ E ). The following lemma nowfollows immediately from the definitions Lemma 3.10 . —
We have f ∗ ◦ ˆ p λ ! = ˆ p ′ λ ! ◦ F ∗ : G ( W ) → ˆ K ( B ′ ) .3.2.5. — Lemma 3.11 . —
The class ˆ p λ ! ( E , ρ ) does not depend on λ ∈ (0 , ∞ ) .Proof . — Consider λ < λ . Note thatˆ p λ ! ( E , ρ ) − ˆ p λ ! ( E , ρ ) = [ p λ ! E , ˜Ω( λ , E )] − [ p λ ! E , ˜Ω( λ , E )] . Consider the inclusion i λ : B → { λ } × B ⊂ [ λ , λ ] × B and let F be the family over[ λ , λ ] × B as in 3.2.2 such that p λ ! E = i ∗ λ F . We apply the homotopy formula Lemma2.22 to x = [ F , i ∗ λ ( x ) − i ∗ λ ( x ) = a Z [ λ ,λ ] × B/B R ( x ) ! = a Z [ λ ,λ ] × B/B Ω( F ) ! = a (cid:16) ˜Ω( λ , E ) − ˜Ω( λ , E ) (cid:17) , MOOTH K-THEORY where the last equality follows directly from the definition of ˜Ω. This equality isequivalent to [ p λ ! E , ˜Ω( λ , E )] = [ p λ ! E , ˜Ω( λ , E )] . In view of this Lemma we can omit the superscript λ and write ˆ p ! ( E , ρ ) for ˆ p λ ! ( E , ρ ). Let E be a geometric family over W which admits a taming E t . Recall thatthe taming is given by a family of smoothing operators ( Q w ) w ∈ W .We have identified the Dirac bundle of p λ ! E with the Dirac bundle of p E in anatural way in 3.2.2. The λ -dependence of the Dirac operator takes the form D ( p λ ! E ) = λ − D ( E ) + ( D H + R ( λ )) , where D H is the horizontal Dirac operator, and R ( λ ) is of zero order and remainsbounded as λ →
0. We now replace D ( E ) by the invertible operator D ( E ) + Q . Thenfor small λ > λ − ( D ( E ) + Q ) + ( D H + R ( λ ))is invertible. To see this, we consider its square which has the structure λ − ( D ( E ) + Q ) + λ − { D ( E ) + Q, ( D H + R ( λ )) } + ( D H + R ( λ )) . The anticommutator { D ( E ) , D H + R ( λ ) } is a first-order vertical operator which is thusdominated by a multiple of the positive second order ( D ( E ) + Q ) . The remainingparts of the anticommutator are zero-order and therefore also dominated by multiplesof ( D ( E ) + Q ) . The last summand is a square of a selfadjoint operator and hencenon-negative.The family of operators along the fibres of p ! E induced by Q is not a taming sinceit is not given by a family of integral operators along the fibres of p ! E → B . Inorder to understand its structure note the following. For b ∈ B the fibre of ( p ! E ) b is the total space of the bundle E | W b → W b . The integral kernel Q induces a familyof smoothing operators on the bundle of Hilbert spaces H ( E | W b ) → W b . Using thenatural identification H ( p ! E ) b ∼ = L ( W, S ( T v p ) ⊗ H ( E | W b ))we get the induced operator on H ( p ! E ) b . We will call a family of operators with thisstructure a generalized taming.Now recall that the η -form η ( F t ) of a tamed or generalized tamed family F t isbuild from a family of superconnections A s ( F t ) parametrized by s ∈ (0 , ∞ ) (see[ Bun , 2.2.4.3]). For 0 < s < ∂ s A s ( F t ) e − A s ( F t ) for s →
0. In the interval s ∈ [1 ,
2] the family A s ( F t ) smoothly connects with the family of superconnections given by A s ( F t ) = sD ( F t ) + terms with higher form degree ULRICH BUNKE & THOMAS SCHICK for s ≥
2. In order to define the η -form η ( F t ) the main points are:1. For small s the family A s ( F t ) behaves like the Bismut superconnection. Theformula (6) dη ( F t ) = Ω( F )only depends on the behavior of A s ( F t ) for small s . Therefore this formulacontinues to hold for generalized tamings.2. ∂ s A s ( F t ) e − A s ( F t ) is given by a family of integral operators with smooth integralkernel. This holds true for tamed families as well as for familes which are tamedin the generalized sense explained above. A proof can be based on Duhamel’sprinciple.3. The integral kernel of ∂ s A s ( F t ) e − A s ( F t ) together with all derivatives vanishesexponentially as s → ∞ . This follows by spectral estimates from the invertibilityand selfadjointness of D ( F t ). Now the invertibility of D ( F t ) is exactly thedesired effect of a taming or generalized taming.Coming back to our iterated fibre bundle we see that we can use the generalizedtaming for sufficiently small λ > η -form whichwe will denote by η ( p λ ! E t ). To be precise this eta form is associated to the family ofoperators A s ( p λ ! E ) + χ ( sλ − ) sλ − Q , s ∈ (0 , ∞ ) , where χ vanishes near zero and is equal to 1 on [1 , ∞ ). This means that we switchon the taming at time s ∼ λ , and we rescale it in the same way as the vertical partof the Dirac operator.We can control the behaviour of η ( p λ ! E t ) in the adiabatic limit λ → Theorem 3.12 . — lim λ → η ( p λ ! E t ) = Z W/B ˆ A c ( o ) ∧ η ( E t ) . Proof . — To write out a formal proof of this theorem seems too long for the presentpaper, without giving fundamental new insights. Instead we point out the followingreferences. Adiabatic limits of η -forms of twisted signature operators were studiedin [ BM04 , Section 5]. The same methods apply in the present case. The L -form in[ BM04 , Section 5] is the local index form of the signature operator. In the presentcase it must be replaced by the form ˆ A c ( o ), the local index form of the Spin c -Diracoperator. The absence of small eigenvalues simplifies matters considerably.Since the geometric family p λ ! E admits a generalized taming it follows that index ( p λ ! E ) = 0. Hence we can also choose a taming ( p λ ! E ) t . The latter choicetogether with the generalized taming induce a generalized boundary taming of thefamily p λ ! E × [0 ,
1] over B . The index theorem [ Bun , Theorem 2.2.18] can be extendedto generalized boundary tamed families (by copying the proof) and gives:
MOOTH K-THEORY Lemma 3.13 . —
The difference of η -forms η (( p λ ! E ) t ) − η ( p λ ! E t ) is closed. Its deRham cohomology class satisfies [ η (( p λ ! E ) t ) − η ( p λ ! E t )] ∈ ch dR ( K ( B )) . We now show that ˆ p ! : G ( W ) → ˆ K ( B ) passes through the equivalence rela-tion ∼ . Since ˆ p ! is additive it suffices by Lemma 2.13 to show the following assertion. Lemma 3.14 . — If ( E , ρ ) is paired with ( ˜ E , ˜ ρ ) , then ˆ p ! ( E , ρ ) = ˆ p ! ( ˜ E , ˜ ρ ) .Proof . — Let ( E ⊔ W ˜ E op ) t be the taming which induces the relation between the twocycles, i.e. ρ − ˜ ρ = η (cid:16) ( E ⊔ W ˜ E op ) t (cid:17) . In view of the discussion in 3.2.6 we can choosea taming p λ ! ( E ⊔ ˜ E op ) t .[ p λ ! E , − [ p λ ! ˜ E ,
0] = [ p λ ! ( E ⊔ W ˜ E op ) , a (cid:16) η (cid:16) p λ ! ( E ⊔ W ˜ E op ) t (cid:17)(cid:17) . By Proposition 2.20 and Lemma 3.13 we can replace the taming by the generalizedtaming and still get[ p λ ! E , − [ p λ ! ˜ E ,
0] = a (cid:16) η (cid:16) p λ ! ( E ⊔ W ˜ E op ) t (cid:17)(cid:17) . For sufficiently small λ > p ! ( E , ρ ) − ˆ p ! ( ˜ E , ˜ ρ ) = a (cid:16) η (cid:16) p λ ! ( E ⊔ W ˜ E op ) t (cid:17)(cid:17) − Z W/B ˆ A c ( o ) ∧ ( ρ − ˜ ρ )+ ˜Ω( λ, E ) − ˜Ω( λ, ˜ E ))We now go to the limit λ → p ! ( E , ρ ) − ˆ p ! ( ˜ E , ˜ ρ ) = a Z W/B ˆ A c ( o ) ∧ η (cid:16) ( E ⊔ W ˜ E op ) t (cid:17)! = − Z W/B ˆ A c ( o ) ∧ ( ρ − ˜ ρ )= 0We let ˆ p ! : ˆ K ( W ) → ˆ K ( B )denote the map induced by the construction (17). Though not indicated in the nota-tion until now this map may depend on the choice of the representative of the smooth K -orientation o (later in Lemma 3.17 we see that it only depends on the smooth K -orientation). ULRICH BUNKE & THOMAS SCHICK
Let p : W → B be a proper submersion with closed fibres with a smooth K -orientation represented by o . We now have constructed a homomorphismˆ p ! : ˆ K ( W ) → ˆ K ( B ) . In the present paragraph we study the compatibilty of this construction with thecurvature map R : ˆ K → Ω d =0 . Definition 3.15 . —
We define the integration of forms p o ! : Ω( W ) → Ω( B ) by p o ! ( ω ) = Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ ω Since ˆ A c ( o ) − dσ ( o ) is closed we also have a factorization p o ! : Ω( W ) / im ( d ) → Ω( B ) / im ( d ) . Lemma 3.16 . —
For x ∈ ˆ K ( W ) we have R (ˆ p ! ( x )) = p o ! ( R ( x )) . Proof . — Let x = ( E , ρ ). We insert the definitions, R ( x ) = Ω( E ) − dρ , and (16) inthe marked step. R (ˆ p ! ( x )) = Ω( p λ ! E ) − d ( Z W/B ˆ A c ( o ) ∧ ρ + ˜Ω( λ, E ) + Z W/B σ ( o ) ∧ R ( x )) ! = Ω( p λ ! E ) − Z W/B ˆ A c ( o ) ∧ dρ + Z W/B ˆ A c ( o ) ∧ Ω( E ) − Ω( p λ ! E ) − Z W/B dσ ( o ) ∧ R ( x )= Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ R ( x )= p o ! ( R ( x )) Our constructions of the homomorphismsˆ p ! : ˆ K ( W ) → ˆ K ( B ) , p o ! : Ω( W ) → Ω( B )involve an explicit choice of a representative o = ( g T v p , T h p, ˜ ∇ , σ ) of the smooth K -orientation lifting the given topological K -orientation of p . In this paragraph weshow: Lemma 3.17 . —
The homomorphisms ˆ p ! : ˆ K ( W ) → ˆ K ( B ) and p o ! : Ω( W ) → Ω( B ) only depend on the smooth K -orientation represented by o .Proof . — Let o k := ( g T v pk , T hk p, ˜ ∇ k , σ k ), k = 0 , K -orientation. Then we have σ − σ = ˜ˆ A c ( ˜ ∇ , ˜ ∇ ). For the moment we indicate MOOTH K-THEORY by a superscript ˆ p k ! which representative of the smooth K -orientation is used in thedefinition. Let ω ∈ Ω( W ). Then using (12) we get p o ! ( ω ) − p o ! ( ω ) = Z W/B ( ˆ A c ( o ) − ˆ A c ( o ) − d ( σ − σ )) ∧ ω = Z W/B ( ˆ A c ( ˜ ∇ ) − ˆ A c ( ˜ ∇ ) − d ˜ˆ A c ( ˜ ∇ , ˜ ∇ )) ∧ ω = 0 . We now consider the projection p : [0 , × W → [0 , × B with the induced topological K -orientation. It can be refined to a smooth K -orientation o which restricts to o k at { k } × B . Let q : [0 , × W → W be the projection and x ∈ ˆ K ( W ). Furthermore let i k : B → { k } × B → [0 , × B be the embeddings. The following chain of equalitiesfollows from the homotopy formula Lemma 2.22, the curvature formula Lemma 3.16,Stokes’ theorem and the definition of ˜ˆ A c ( ˜ ∇ , ˜ ∇ ), and finally from the fact that o ∼ o .ˆ p ( x ) − ˆ p ( x ) = i ∗ ˆ p ! q ∗ ( x ) − i ∗ ˆ p ! q ∗ ( x )= a Z [0 , × B/B R (ˆ p ! q ∗ x ) ! = a Z [0 , × B/B p o ! R ( q ∗ ( x )) ! = a Z [0 , × B/B p o ! q ∗ ( R ( x )) ! = a Z [0 , × B/B Z [0 , × W/ [0 , × B ( ˆ A c ( o ) − dσ ( o )) ∧ q ∗ R ( x ) ! = a Z W/B [ Z [0 , × W/W ( ˆ A c ( o ) − dσ ( o ))] ∧ R ( x ) ! = a Z W/B [ ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) − ( σ ( o ) − σ ( o ))] ∧ R ( x ) ! = 0 . Let p : W → B be a proper submersion with closed fibres with a topological K -orientation. We choose a smooth K -orientation which refines the topological K -orientation. In this case we say that p is smoothly K -oriented. Definition 3.18 . —
We define the push-forward ˆ p ! : ˆ K ( W ) → ˆ K ( B ) to be the mapinduced by (17) for some choice of a representative of the smooth K -orientation ULRICH BUNKE & THOMAS SCHICK
We also have well-defined maps p o ! : Ω( W ) → Ω( B ) , p o ! : Ω( W ) / im ( d ) → Ω( B ) / im ( d )given by integration of forms along the fibres. Let us state the result about thecompatibility of ˆ p ! with the structure maps of smooth K -theory as follows. Proposition 3.19 . —
The following diagrams commute: K ( W ) ch dR −−−−→ Ω( W ) / im ( d ) a −−−−→ ˆ K ( W ) I −−−−→ K ( W ) y p ! y p o ! y ˆ p ! y p ! K ( B ) ch dR −−−−→ Ω( B ) / im ( d ) a −−−−→ ˆ K ( B ) I −−−−→ K ( B ) (18)ˆ K ( W ) R −−−−→ Ω d =0 ( W ) y ˆ p ! y p o ! ˆ K ( B ) R −−−−→ Ω d =0 ( B ) (19) Proof . — The maps between the topological K -groups are the usual push-forwardmaps defined by the K -orientation of p . The other two are defined above. The square(19) commutes by Lemma 3.16. The right square of (18) commutes because we havethe well-known fact from index theory index ( p ! ( E )) = p ! ( index ( E )) . Let ω ∈ Ω( W ) / im ( d ). Then we haveˆ p ! ( a ( ω )) = [ ∅ , Z W/B σ ( o ) ∧ dω − Z W/B ˆ A c ( o ) ∧ ω ]= [ ∅ , − Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ ω ]= a ( p ! ( ω )) . This shows that the middle square in (18) commutes. Finally, the commutativity ofthe left square in (18) is a consequence of the Chern character version of the familyindex theorem ch dR ( p ! ( x )) = Z W/B ˆ A c ( T v p ) ∧ ch dR ( x ) , x ∈ K ( W ) . If f : B ′ → B is a smooth map then we consider the cartesian diagram W ′ F −−−−→ W y p ′ y p B ′ f −−−−→ B .
We equip p ′ with the induced smooth K -orientation (see 3.2.4). MOOTH K-THEORY Lemma 3.20 . —
The following diagram commutes: ˆ K ( W ) F ∗ −−−−→ ˆ K ( W ′ ) y p ! y p ′ ! ˆ K ( B ) f ∗ −−−−→ ˆ K ( B ′ ) . Proof . — This follows from Lemma 3.10.
We now discuss the functoriality of the push-forward with respect to iter-ated fibre bundles. Let p : W → B be as before together with a representative of asmooth K -orientation o p = ( g T v p , T h p, ˜ ∇ p , σ ( o p )). Let r : B → A be another propersubmersion with closed fibres with a topological K -orientation which is refined by asmooth K -orientation represented by o r := ( g T v r , T h r, ˜ ∇ r , σ ( o r )).We can consider the geometric family W := ( W → B, g T v p , T h p, S c ( T v p )) andapply the construction 3.2.2 in order to define the geometric family r λ ! ( W ) over A .The underlying submersion of the family is q := r ◦ p : W → A . Its vertical bundle hasa metric g T v qλ and a horizontal distribution T h q . The topological Spin c -structures of T v p and T v r induce a topological Spin c -structure on T v q = T v p ⊕ p ∗ T v r . The familyof Clifford bundles of p ! W is the spinor bundle associated to this Spin c -structure.In order to understand how the connection ˜ ∇ λq behaves as λ → T v p and T v r . Then we write S c ( T v p ) ∼ = S ( T v p ) ⊗ L p and S c ( T v r ) ∼ = S ( T v r ) ⊗ L r for one-dimensional twisting bundles with connection L p , L r . The twolocal spin structures induce a local spin structure on T v q ∼ = T v p ⊕ p ∗ T v r . We get S c ( T v q ) ∼ = S ( T v q ) ⊗ L q with L q := L p ⊗ p ∗ L r . The connection ∇ λ,T v qq convergesas λ →
0. Moreover, the twisting connection on L q does not depend on λ at all.Since ∇ λ,T v qq and ∇ Lq determine ˜ ∇ λq (see 3.1.5) we conclude that the connection ˜ ∇ λq converges as λ →
0. We introduce the following notation for this adiabatic limit:˜ ∇ adia := lim λ → ˜ ∇ λq . We keep the situation described in 3.3.1.
Definition 3.21 . —
We define the composite o λq := o r ◦ λ o p of the representativesof smooth K -orientations of p and r by o λq := ( g T v qλ , T h q, ˜ ∇ λq , σ ( o λq )) , where σ ( o λq ) := σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ λq ) − dσ ( o p ) ∧ p ∗ σ ( o r ) . Lemma 3.22 . —
This composition of representatives of smooth ˆ K -orientations pre-serves equivalence and induces a well-defined composition of smooth K -orientationswhich is independent of λ . ULRICH BUNKE & THOMAS SCHICK
Proof . — We first show that o λq is independent of λ . In view of 3.1.9 for λ < λ wemust show that σ ( o λ q ) − σ ( o λ q ) = ˜ˆ A c ( ˜ ∇ λ q , ˜ ∇ λ q ). In fact, inserting the definitionsand using (13) and (14) we have σ ( o λ q ) − σ ( o λ q ) = − ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ λ q ) + ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ λ q ) = ˜ˆ A c ( ˜ ∇ λ q , ˜ ∇ λ q ) . Let us now take another representative o ′ p . The following equalities hold in thelimit λ → σ ( o q ) − σ ( o ′ q )= ( σ ( o p ) − σ ( o ′ p )) ∧ p ∗ ˆ A c ( o r ) + ( ˆ A c ( o p ) − ˆ A c ( o ′ p )) ∧ p ∗ σ ( o r ) − d ( σ ( o p ) − σ ( o ′ p )) ∧ p ∗ σ ( o r )= ˜ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p ) ∧ p ∗ ˆ A c ( o r ) + ( ˆ A c ( ˜ ∇ p ) − ˆ A c ( ˜ ∇ ′ p ) − d ˜ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p )) ∧ p ∗ σ ( o r )= ˜ˆ A c ( ˜ ∇ adiaq , ˜ ∇ ′ adiaq )The last equality uses (12) and that in the adiabatic limitˆ A c ( ˜ ∇ adiaq ) = ˆ A c ( ˜ ∇ p ) ∧ p ∗ ˆ A c ( ∇ r ) , (20)which implies a corresponding formula for the adiabatic limit of transgressions,˜ˆ A c ( ˜ ∇ adiaq , ˜ ∇ ′ adiaq ) = ˜ˆ A c ( ˜ ∇ p , ˜ ∇ ′ p ) ∧ p ∗ ˆ A c ( ∇ r ) . Next we consider the effect of changing the representative o r to the equivalent one o ′ r . We compute in the adiabatic limit σ ( o q ) − σ ( o ′ q ) = σ ( o p ) ∧ ( p ∗ ˆ A c ( o r ) − p ∗ ˆ A c ( o ′ r )) + ( ˆ A c ( o p ) − dσ ( o p )) ∧ p ∗ ( σ ( o r ) − σ ( o ′ r ))= σ ( o p ) ∧ dp ∗ ˜ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) + ( ˆ A c ( o p ) − dσ ( o p )) ∧ p ∗ ˜ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r )= ˆ A c ( o p ) ∧ p ∗ ˜ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r )= ˜ˆ A c ( ˜ ∇ adiaq , ˜ ∇ ′ adiaq ) . In the last equality we have used again (20) and the corresponding equality˜ˆ A c ( ˜ ∇ adiaq , ˜ ∇ ′ adiaq ) = ˆ A c ( o p ) ∧ p ∗ ˜ˆ A c ( ˜ ∇ r , ˜ ∇ ′ r ) . We consider the composition of proper K -oriented submersions W q p / / B r / / A with representatives of smooth K -orientations o p of p and o r of r . We let o q := o r ◦ o p be the composition. These choices define push-forwards ˆ p ! , ˆ r ! and ˆ q ! in smooth K -theory. Theorem 3.23 . —
We have the equality of homomorphisms ˆ K ( W ) → ˆ K ( A )ˆ q ! = ˆ r ! ◦ ˆ p ! . MOOTH K-THEORY Proof . — We calculate the push-forwards and the composition of the K -orientationsusing the parameter λ = 1 (though we do not indicate this in the notation). We takea class [ E , ρ ] ∈ ˆ K ( W ). The following equality holds since λ = 1: q ! E = r ! ( p ! E ) . So we must show that Z W/A ˆ A c ( o q ) ∧ ρ + ˜Ω( q, , E ) + Z W/A σ ( o q ) ∧ R ([ E , ρ ]) (21) ≡ Z B/A ˆ A c ( o r ) ∧ "Z W/B ˆ A c ( o p ) ∧ ρ + ˜Ω( p, , E ) + Z W/B σ ( o p ) ∧ R ([ E , ρ ]) + ˜Ω( r, , p ! E ) + Z B/A σ ( o r ) ∧ R ( p ! [ E , ρ ]) . where ≡ means equality modulo im ( d ) + ch dR ( K ( A )). The form Ω( q, , E ) is given by(15). Since in the present paragraph we consider these transgression forms for variousbundles we have included the projection q as an argument.By Proposition 3.19 we have R (ˆ p ! [ E , ρ ]) = Z W/B ( ˆ A c ( o p ) − dσ ( o p )) ∧ R ([ E , ρ ]) . Next we observe that˜Ω( q, , E ) ≡ ˜Ω( r, , p ! E )+ Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E )+ Z B/A ˆ A c ( o r ) ∧ ˜Ω( p, , E ) , (22)(where ≡ means equality up to im ( d )). To see this we consider the two-parameterfamily r λ ! ◦ p µ ! ( E ), λ, µ >
0, of geometric families. There is a natural geometric fam-ily F over (0 , × A which restricts to r λ ! ◦ p µ ! ( E ) on { ( λ, µ ) } × A (see 3.2.2 forthe one-parameter case). Note that the local index form Ω( F ) extends by continu-ity to [0 , × A . If P : [0 , ֒ → [0 , is a path, then one can form the integral R P × A/A Ω( F | P × A ), the transgression of the local index form of r λ ! ◦ p µ ! ( E ) along thepath P . The following square indicates four paths in the ( λ, µ )-plane. The arrowsare labeled by the evaluations of Ω( F ) (which follow from the adiabatic limit formula ULRICH BUNKE & THOMAS SCHICK , ˜Ω( r, ,p ! E )Ω( r λ ! ◦ p ! ( E )) / / (1 , , R B/A ˆ A c ( o r ) ∧ Ω( p µ ! E ) R B/A ˆ A c ( o r ) ∧ ˜Ω( p, , E ) O O R W/A ˆ A c ( o r ◦ λ o p ) ∧ Ω( E ) R W/A ˜ˆ A c ( ˜ ∇ q , ˜ ∇ adia ) ∧ Ω( E ) / / (1 , ˜Ω( q, , E )Ω( r ! ◦ p µ ! ( E )) O O . Note the equality r ! ◦ p µ ! ( E ) = q µ ! ( E ) which is relevant for the right vertical path. Alsonote that for the lower horizontal path that , as µ →
0, the fibres of E are scaled tozero, whereas the fibres of p are scaled by λ . The latter is exactly the effect of thescaled composition o r ◦ λ o p of orientations defined in 3.3.1, explaining its appearencein the above formula. The equation (22) follows since the transgression is additiveunder composition of paths, and since the transgression along a closed contractiblepath gives an exact form.We now insert Definition 3.21 of σ ( o q ) in order to get Z W/A σ ( o q ) ∧ R ([ E , ρ ])= Z W/A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) − ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) i ∧ R ([ E , ρ ])= Z W/A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o r ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ dρ = Z W/A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/A (cid:16) ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) − ˆ A c ( o q ) (cid:17) ∧ ρ (23)We insert (23) and (22) into the left-hand side of (21). MOOTH K-THEORY Z W/A ˆ A c ( o q ) ∧ ρ + ˜Ω( q, , E ) + Z W/A σ ( o q ) ∧ R ([ E , ρ ]) ≡ Z W/A ˆ A c ( o q ) ∧ ρ + ˜Ω( r, , p ! E ) + Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z B/A ˆ A c ( o r ) ∧ ˜Ω( p, , E )+ Z W/A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ]) − Z W/A ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ q ) ∧ Ω( E ) + Z W/A (cid:16) ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) − ˆ A c ( o q ) (cid:17) ∧ ρ = ˜Ω( r, , p ! E ) + Z B/A ˆ A c ( o r ) ∧ ˜Ω( p, , E )+ Z W/A h σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) i ∧ R ([ E , ρ ])+ Z W/A ˆ A c ( o p ) ∧ p ∗ ˆ A c ( o r ) ∧ ρ . An inspection shows that this is exactly the right-hand side of (21).
4. The cup product4.1. Definition of the product. —
In this section we define and study the cup product ∪ : ˆ K ( B ) ⊗ ˆ K ( B ) → ˆ K ( B ) . It turns smooth K -theory into a functor on manifolds with values in Z / Z -graded ringsand into a multiplicative extension of the pair ( K, ch R ) in the sense of Definition 1.2. Let E and F be geometric families over B . The formula for the productinvolves the product E × B F of geometric families over B . The detailed descriptionof the product is easy to guess, but let us employ the following trick in order to givean alternative definition.Let p : F → B be the proper submersion with closed fibres underlying F . Letus for the moment assume that the vertical metric, the horizontal distribution, andthe orientation of p are complemented by a topological Spin c -structure together witha Spin c -connection ˜ ∇ as in 3.2.1. The Dirac bundle V of F has the form V ∼ = W ⊗ S c ( T v p ) for a twisting bundle W with a hermitean metric and unitary connection(and Z / Z -grading in the even case), which is uniquely determined up to isomorphism.Let p ∗ E ⊗ W denote the geometric family which is obtained from p ∗ E by twistingits Dirac bundle with δ ∗ W , where δ : E × B F → F denotes the underlying proper ULRICH BUNKE & THOMAS SCHICK submersion with closed fibres of p ∗ E . Then we have E × B F ∼ = p ! ( p ∗ E ⊗ W ) . This description may help to understand the meaning of the adiabatic deformationwhich blows up F , which in this notation is given by p λ ! ( p ∗ E ⊗ W ).In the description of the product of geometric families we could interchange theroles of E and F .If the vertical bundle of E does not have a global Spin c -structure, then it has atleast a local one. In this case the description above again gives a complete descriptionof the local geometry of E × B F . We now proceed to the definition of the product in terms of cycles. Inorder to write down the formula we assume that the cycles ( E , ρ ) and ( F , θ ) arehomogeneous of degree e and f , respectively. Definition 4.1 . —
We define ( E , ρ ) ∪ ( F , θ ) := [ E × B F , ( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ θ ] . Proposition 4.2 . —
The product is well-defined. It turns B ˆ K ( B ) into a functorfrom smooth manifolds to unital graded-commutative rings.Proof . — We first show that this product is bilinear and compatible with the equiva-lence relation ∼ (2.10). The product is obviously biadditive and natural with respectto pull-backs along maps B ′ → B . We now show that the product preserves theequivalence relation in the first argument. Assume that E admits a taming E t . Thenwe have ( E , ρ ) ∼ ( ∅ , ρ − η ( E t )). Using the latter representative we get( ∅ , ρ − η ( E t )) ∪ ( F , θ ) = [ ∅ , ( ρ − η ( E t )) ∧ Ω( F ) − ( − e dρ ∧ θ + ( − e dη ( E t ) ∧ θ ]= [ ∅ , ρ ∧ Ω( F ) + ( − e Ω( E ) ∧ θ − ( − e dρ ∧ θ − η ( E t ) ∧ Ω( F )] . On the other hand, similar to in 3.2.6, the taming E t induces a generalized taming( E × B F ) t . Using Lemma 3.13 and argueing as in the proof of Lemma 3.14 we get[ E × B F , ( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ σ ]= [ ∅ , ( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ σ − η (( E × B F ) t )] . It suffices to show that η ( E t ) ∧ Ω( F ) − η (( E × B F ) t ) ∈ im ( ch dR ) . (24)We will actually show that this difference is exact.We first consider the adiabatic limit in which we blow up the metric of F . We getfrom Theorem 3.12 lim adia η (( E × B F ) t ) = η ( E t ) ∧ Ω( F ) . (25)In order to see this we use that E × B F ∼ = p ! ( p ∗ E ⊗ W ) (see 4.1.2), where p : F → B and W → F is the twisting bundle of this family. The taming E t induces a taming MOOTH K-THEORY p ∗ E t , and hence a taming ( p ∗ E ⊗ W ) t . It follows from standard properties of theinduced superconnection on a tensor product bundle (alternatively one can use thespecial case of Theorem 3.12 where the second fibration has zero-dimensional fibres)that η ( p ∗ E ⊗ W ) t = p ∗ η ( E t ) ∧ ch ( ∇ W ). From Theorem 3.12 we get ( ˜ ∇ is associatedto p ) lim adia η (( E × B F ) t ) = lim λ → η ( p λ ! ( p ∗ E ⊗ W ) t )= η ( E t ) ∧ Z F/B ˆ A c ( ˜ ∇ ) ∧ ch ( ∇ W ) ! = η ( E t ) ∧ Ω( F )As in 3.2.2 we now let G t be the tamed family over (0 , ∞ ) × B with underlyingprojection r : (0 , ∞ ) × E × B F → (0 , ∞ ) × B which restricts to p λ ! ( p ∗ E ⊗ W ) t on { λ } × B . Then we have dη ( G t ) = Ω( G ). Using the formulas for ∇ T v r given in[ BGV04 , Prop. 10.2] we observe that i ∂ Hλ R ∇ Tvr = 0, where ∂ Hλ is a horizontal lift of ∂ λ . This implies that i ∂ λ dη ( G t ) = i ∂ λ Ω( G ) = 0. We get η ( p λ ! ( p ∗ E ⊗ W ) t ) − η ( p ( p ∗ E ⊗ W ) t ) = d Z [ λ, × B/B η ( G t ) . The exactness of the difference (24) now follows by taking the limit λ → d is closed since lim λ → η ( p λ ! ( p ∗ E ⊗ W ) t ) = η ( E t ) ∧ Ω( F ) by(25) and η ( p ( p ∗ E ⊗ W ) t ) = η (( E × B F ) t ) by construction.In order to avoid repeating this argument for the second argument we show thatthe product is graded commutative. Note that E × B F ∼ = F × B E except if bothfamilies are odd, in which case E × B F ∼ = (
F × B E ) op [ E , ρ ] ∪ [ F , θ ] = [ E × B F , ( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ θ ]= [( − ef F × B E , ( − e + e ( f − θ ∧ Ω( E ) + ( − f ( e − Ω( F ) ∧ ρ − ρ ∧ dθ ]= [( − ef F × B E , ( − ef θ ∧ Ω( E ) + ( − ef ( − f Ω( F ) ∧ ρ − ( − ef ( − f dθ ∧ ρ ]= ( − ef [ F , θ ] ∪ [ E , ρ ] . We now have a well-defined Z / Z -graded commutative product ∪ : ˆ K ( B ) ⊗ ˆ K ( B ) → ˆ K ( B ) . We show next that it is associative. First of all observe that the fibre product ofgeometric families is associative. Let e, f, g be the parities of the homogeneous classes ULRICH BUNKE & THOMAS SCHICK [ E , ρ ], [ F , θ ], and [ G , κ ].([ E , ρ ] ∪ [ F , θ ]) ∪ [ G , κ ]= [ E × B F , ( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ θ ] ∪ [ G , κ ]= [ E × B F × B G , (( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ θ ) ∧ Ω( G )+( − e + f Ω( E × B F ) ∧ κ − ( − e + f d (( − e Ω( E ) ∧ θ + ρ ∧ Ω( F ) − ( − e dρ ∧ θ ) ∧ κ ]= [ E × B F × B G , ( − e Ω( E ) ∧ θ ∧ Ω( G ) + ρ ∧ Ω( F ) ∧ Ω( G ) − ( − e dρ ∧ θ ∧ Ω( G ) + ( − e + f Ω( E ) ∧ Ω( F ) ∧ κ − ( − e + f Ω( E ) ∧ dθ ∧ κ − ( − e + f dρ ∧ Ω( F ) ∧ κ + ( − e + f dρ ∧ dθ ∧ κ ]On the other hand[ E , ρ ] × ([ F , θ ] × [ G , κ ])= [ E , ρ ] × [ F × B G , ( − f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − f dθ ∧ κ ]= [ E × B ∧F × B G , ( − e Ω( E ) ∧ (( − f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − f dθ ∧ κ )+ ρ ∧ Ω( F × B G ) − ( − e dρ ∧ (( − f Ω( F ) ∧ κ + θ ∧ Ω( G ) − ( − f dθ ∧ κ )]= [ E × B F × B G , ( − e + f Ω( E ) ∧ Ω( F ) ∧ κ + ( − e Ω( E ) ∧ θ ∧ Ω( G ) − ( − e + f Ω( E ) ∧ dθ ∧ κ + ρ ∧ Ω( F ) ∧ Ω( G ) − ( − e + f dρ ∧ Ω( F ) ∧ κ − ( − e dρ ∧ θ ∧ Ω( G ) + ( − e + f dρ ∧ dθ ∧ κ ]By an inspection we see that the two right-hand sides agree. Let us observe that the unit 1 ∈ ˆ K ( B ) is simply given by ( B × C , C concentrated in evendegree, and with curvature form 1. The definition shows that this is actually a uniton the level of cycles. This finishes the proof of Proposition 4.2. In this paragraph we study the compatibility of the cup product in smooth K -theory with the cup product in topological K-theory and the wedge product ofdifferential forms. Lemma 4.3 . —
For x, y ∈ ˆ K ( B ) we have R ( x ∪ y ) = R ( x ) ∧ R ( y ) , I ( x ∪ y ) = I ( x ) ∪ I ( y ) . Furthermore, for α ∈ Ω( B ) / im ( d ) we have a ( α ) ∪ x = a ( α ∧ R ( x )) . Proof . — Straight forward calculation using the definitions.
Corollary 4.4 . —
With the ∪ -product smooth K -theory ˆ K is a multiplicative exten-sion of the pair ( K, ch R ) . MOOTH K-THEORY Let p : W → B be a proper submersion with closed fibres with a smooth K -orientation represented by o . In this case we have a well-defined push-forwardˆ p ! : ˆ K ( W ) → ˆ K ( B ). The explicit formula in terms of cycles is (17). The projectionformula states the compatibility of the push-forward with the ∪ -product. Proposition 4.5 . —
Let x ∈ ˆ K ( W ) and y ∈ ˆ K ( B ) . Then ˆ p ! ( p ∗ y ∪ x ) = y ∪ ˆ p ! ( x ) . Proof . — Let x = [ F , σ ] and y = [ E , ρ ]. By an inspection of the constructions weobserve that the projection formula holds true on the level of geometric families p ! ( p ∗ E × W F ) ∼ = E × B p ! F . This implies Ω( p λ ! ( p ∗ E × W F )) = Ω( E ) ∧ Ω( p λ ! ( F )) . Consequently we have ˜Ω( λ, p ∗ E × W F ) = ( − e Ω( E ) ∧ ˜Ω( λ, F ). Inserting the defini-tions of the product and the push-forward we get up to exact formsˆ p ! ( p ∗ y ∪ x )= ˆ p ! ([ p ∗ E × W F , ( − e p ∗ Ω( E ) ∧ σ + p ∗ ρ ∧ Ω( F ) − ( − e p ∗ dρ ∧ σ ])= [ p ! ( p ∗ E × W F ) , Z W/B ˆ A c ( o ) ∧ [( − e p ∗ Ω( E ) ∧ σ + p ∗ ρ ∧ Ω( F ) − ( − e p ∗ dρ ∧ σ ]+ Z W/B σ ( o ) ∧ R ( p ∗ y ∪ x ) + ˜Ω(1 , p ∗ E × W F )]= [ E × B p ! F , ρ ∧ Z W/B ˆ A c ( o ) ∧ Ω( F ) + ( − e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ +( − e Ω( E ) ∧ ˜Ω(1 , F ) − ρ ∧ Z W/B ˆ A c ( o ) ∧ dσ + ( − e R ( y ) ∧ Z W/B σ ( o ) ∧ R ( x )] . (26) ULRICH BUNKE & THOMAS SCHICK
Up to exact forms we have ρ ∧ Z W/B ˆ A c ( o ) ∧ Ω( F ) + ( − e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ +( − e Ω( E ) ∧ ˜Ω(1 , F ) − ρ ∧ Z W/B ˆ A c ( o ) ∧ dσ + ( − e R ( y ) ∧ Z W/B σ ( o ) ∧ R ( x )= ( − e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ Z W/B ˆ A c ( o ) ∧ (Ω( F ) − dσ )) − ( − e dρ ∧ Z W/B σ ( o ) ∧ R ( x )= ( − e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ Z W/B ( ˆ A c ( o ) − dσ ( o )) ∧ R ( x )= ( − e Ω( E ) ∧ Z W/B ˆ A c ( o ) ∧ σ + ˜Ω(1 , F ) + Z W/B σ ( o ) ∧ R ( x ) ! + ρ ∧ R (ˆ p ! x ) . Thus the form component of (26) is exactly the one needed for the product y ∪ p ! ( x ). We consider the projection pr : S × B → B . The goal of this subsectionis to verify the relation ( ˆ pr ) ! ◦ pr ∗ = 0which is an important ingredient in the uniqueness result Theorem 1.4. The projection pr fits into the cartesian diagram S × B pr / / pr (cid:15) (cid:15) S p (cid:15) (cid:15) B r / / ∗ . We choose the metric g T S of unit volume and the bounding spin structure on T S .This spin structure induces a Spin c structure on T S together with the connection˜ ∇ . In this way we get a representative o of a smooth K -orientation of p . By pull-backwe get the representative r ∗ o of a smooth K -orientation of pr which is used to define( ˆ pr ) ! . MOOTH K-THEORY Using the projection formula Proposition 4.5 we get for x ∈ ˆ K ( B )( ˆ pr ) ! ( pr ∗ ( x )) = ( ˆ pr ) ! ( pr ∗ ( x ) ∪
1) = x ∪ ( ˆ pr ) ! . Using the compatibility of the push-forward with cartesian diagrams Lemma 3.20 weget ( ˆ pr ) ! pr ) ! ( pr ∗ (1)) = r ∗ ˆ p ! (1) . We let S denote the geometric family over ∗ given by p : S → ∗ with the geometrydescribed above. Since S has the bounding Spin -structure the Dirac operator isinvertible and has a symmetric spectrum. The family S therefore has a canonicaltaming S t by the zero smoothing operator, and we have η ( S t ) = 0. This impliesˆ p ! (1) = [ S ,
0] = [ ∅ , η ( S t )] = [ ∅ ,
0] = 0 . Corollary 4.6 . —
We have ( ˆ pr ) ! ◦ pr ∗ = 0 .
5. Constructions of natural smooth K -theory classes5.1. Calculations. — Lemma 5.1 . —
We have ˆ K ∗ ( ∗ ) ∼ = (cid:26) Z ∗ = 0 R / Z ∗ = 1 . Proof . — We use the exact sequence given by Proposition 2.20. The assertion followsfrom the obvious identitiesˆ K ( ∗ ) ∼ = K ( ∗ ) ∼ = Z , ˆ K ( ∗ ) ∼ = Ω ev ( ∗ ) / ch dR ( K ( ∗ )) ∼ = R / Z . Lemma 5.2 . —
There are exact sequences → R / Z → ˆ K ( S ) → Z → → C ∞ ( S ) / Z → ˆ K ( S ) → Z → . Proof . — These assertions again follow from Proposition 2.20 and the identifications K ( S ) ∼ = Z , K ( S ) ∼ = Z , Ω ev ( S ) / ch dR ( K ( S )) ∼ = C ∞ ( S ) / Z . ULRICH BUNKE & THOMAS SCHICK
Let V := ( V, h V , ∇ V , z ) be a geometric Z / Z -graded bundle over S suchthat dim( V + ) = dim( V − ). Let V denote the corresponding geometric family. ByLemma 5.2 the class [ V , ∈ ˆ K ( S ) satisfies I ([ V , R / Z . This element is calculated in the following lemma. Let φ ± ∈ U ( n ) /conj denote the holonomies of V ± (well defined modulo conjugation in thegroup U ( n )). Lemma 5.3 . —
We have [ V ,
0] = a (cid:18) πi log det ( φ + ) det ( φ − ) (cid:19) . Proof . — We consider the map q : S → ∗ with the canonical K -orientation 4.3.2.By Proposition 3.19 we have a commutative diagram R / Z ∼ −−−−→ Ω ( S ) / ( im ( d ) + im ( ch dR )) a −−−−→ ˆ K ( S ) y = y q o ! y ˆ q ! R / Z ∼ −−−−→ Ω ( ∗ ) / im ( ch dR ) a −−−−→ ˆ K ( ∗ ) . In order to determine [ V ,
0] it therefore suffices to calculate ˆ q ! ([ V , q : S → ∗ is the boundary of p : D → ∗ . Since the underlying topological K -orientation of q is given by the bounding Spin -structure we can choose a smooth K -orientation of p with product structure which restricts to the smooth K -orientationof q . The bundle V is topologically trivial. Therefore we can find a geometric bundle W = ( W, h W , ∇ W , z ), again with product structure, on D which restricts to V onthe boundary. Let W denote the corresponding geometric family over D . Later weprove the bordism formula Proposition 5.18. It givesˆ q ! ([ V , ∅ , p ! R ([ W , − a Z D / ∗ Ω ( W ) ! . Note thatΩ ( W ) = ch ( ∇ W ) = ch ( ∇ det ( W + ) ) − ch ( ∇ det ( W − ) ) = − πi (cid:20) R ∇ det W + − R det ∇ W − (cid:21) . The holonomy det ( φ ± ) ∈ U (1) of det ( V ± ) is equal to the integral of the curvatureof det W ± : log det ( φ ± ) = Z D R ∇ det ( W ± ) . It follows that ˆ q ! ([ V , a (cid:18) πi log det ( φ + ) det ( φ − ) (cid:19) . K -theory class of a mapping torus. — MOOTH K-THEORY Let E be a geometric family over a point and consider an automorphism φ of E . Then we can form the mapping torus T ( E , φ ) := ( R × E ) / Z , where n ∈ Z actson R by x x + n , and by φ n on E . The product R × E is a Z -equivariant geometricfamily over R (the pull-back of E by the projection R → ∗ ). The geometric structuresdescend to the quotient and turn the mapping torus T ( E , φ ) into a geometric familyover S = R / Z . In the present subsection we study the class[ T ( E , φ ) , ∈ ˆ K ( S ) . In the following we will assume that the parity of E is even, and that index ( E ) = 0. Let dim : K ( S ) → Z be the dimension homomorphism, which in thiscase is an isomorphism. Since dim I ([ T ( E , φ ) , index ( E )) = 0 we have infact [ T ( E , φ ) , ∈ R / Z ⊂ ˆ K ( S ), where we consider R / Z as a subgroup of ˆ K ( S )according to Lemma 5.2.Let V := ker( D ( E )). This graded vector space is preserved by the action of φ . Weuse the same symbol in order to denote the induced action on V .We form the zero-dimensional family V := ( R × V ) / Z over S . This bundle isisomorphic to the kernel bundle of T ( E , φ ). The bundle of Hilbert spaces of thefamily T ( E , φ ) ⊔ S V op has a canonical subbundle of the form V ⊕ V op . We choose thetaming ( T ( E , φ ) ⊔ S V op ) t which is induced by the isomorphism (cid:18) (cid:19) on this subbundle. Note that [ T ( E , φ ) ,
0] = [ V , η (( T ( E , φ ) ⊔ S V op ) t )]. Since the pull-back of ( T ( E , φ ) ⊔ S V op ) t under R → R / Z is isomorphic to a tamed family pulledback under R → ∗ we see that the one-form η (( T ( E , φ ) ⊔ S V op ) t ) = 0. Thus it remains to evaluate [ T ( E , φ ) ,
0] = [ V , ∈ R / Z . By Lemma 5.3 thisnumber can be expressed in terms of the holonomy of the determinant bundle det ( V ).Let φ ± ∈ Aut ( V ± ) be the induced transformations. Proposition 5.4 . —
We have [ T ( E , φ ) ,
0] = [ πi log( det φ + det φ − )] R / Z . In particular, if D ( E ) is invertible, then [ T ( E , φ ) ,
0] = 0 . K -theory class of a geometric family with kernel bundle.— Let E be an even-dimensional geometric family over the base B . By ( D b ) b ∈ B we denote the associated family of Dirac operators on the family of Hilbert spaces( H b ) b ∈ B . The geometry of E induces a connection ∇ H on this family (the connec-tion part of the Bismut superconnection [ BGV04 , Prop. 10.15]). We assume thatdim(ker( D b )) is constant. In this case we can form a vector bundle K := ker( D ).The projection of ∇ H to K gives a connection ∇ K . Hence we get a geometric bundle K := ( K, h K , ∇ K ) and an associated geometric family K (see 2.1.4). ULRICH BUNKE & THOMAS SCHICK
The sum
E ⊔ B K op has a natural taming ( E ⊔ B K op ) t which is given by (cid:18) uu ∗ (cid:19) ∈ End ( H b ⊕ K opb ) , where u : K b → H b is the embedding. We thus have the following equality in ˆ K ( B ):[ E ,
0] = [ K , η (( E ⊔ B K op ) t )] . Under the standing assumption that dim(ker( D b )) is constant we also havethe η -form of Bismut-Cheeger η BC ( E ) ∈ Ω( B ) (see [ BC91 ], [
BC90b ], [
BC90a ]).Since other authors use η BC ( E ), in the following two paragraphs we shall analyse therelation between this and η (( E ⊔ B K op ) t ).We form the geometric family [0 , × ( E ⊔ B K op ) over B . The taming ( E ⊔ B K op ) t induces a boundary taming at { } × ( E ⊔ B K op ). In index theory the boundary tamingis used to construct a perturbation of the Dirac operator which is invertible at −∞ of( −∞ , × ( E ⊔ B K op ) (see [ Bun ] for details). On the other side { } × ( E ⊔ B K op ) weconsider APS-boundary conditions. We thus get a family of perurbed Dirac operatorson ( −∞ , × ( E ⊔ B K op ). The L -boundary condition at {−∞} × ( E ⊔ B K op ) and theAPS-boundary condition at { } × ( E ⊔ B K op ) together imply the Fredholm property(which can be checked locally for the various boundary components or ends). In thisway the family of Dirac operators on [0 , × ( E ⊔ B K op ) gives rise to a family ofFredholm operators. We will denote this structure by ([0 , × ( E ⊔ B K op )) bt,AP S .The Chern character of its index index (([0 , × ( E ⊔ B K op )) bt,AP S ) ∈ K ( B ) canbe calculated using the methods of local index theory. Using 2.4.10 we can choose a possibly different taming (
E ⊔ B K op ) t ′ such thatthe corresponding index index (([0 , × ( E ⊔ B K op )) bt ′ ,AP S ) ∈ K ( B ) vanishes. In thiscase we can extend the boundary taming to a taming index (([0 , × ( E ⊔ B K op )) t ′ ,AP S ).We set up the method of local index theory as usual by forming the family ofrescaled Bismut superconnections A s := A s (([0 , × ( E ⊔ B K op )) t ′ ,AP S ) which takethe tamings and boundary tamings into account as explained in [ Bun , 2.2.4.3], seealso 3.2.6. Invertibility of D (([0 , × ( E ⊔ B K op )) t ′ ,AP S ) ensures exponential vanishingof the integral kernel of e − A s for s → ∞ . The usual transgression integral expressesthe local index form Ω([0 , × ( E ⊔ B K op )) as a sum of contributions of the boundarycomponents or ends (see [ Bun , proof of Lemma 2.2.15 ]). These contributions can becalculated separately for each part.Because of the product structure we have Ω([0 , × ( E ⊔ B K op )) = 0. The con-tribution of the boundary { } × ( E ⊔ B K op ) is given by the proof of the APS-indextheorem of [ BC91 ], [
BC90b ], [
BC90a ], and it is equal to η BC ( E ⊔ B K op ) = η BC ( E ).The second equality holds true, since the Dirac operator for K op is trivial. The con-tribution of the boundary { } × ( E ⊔ B K op ) is calculated in the proof of [ Bun , Lemma2.2.15] and equal to − η (( E ⊔ B K op ) t ′ ). Therefore we have η BC ( E ) = η (( E ⊔ B K op ) t ′ )(note that we calculate modulo exact forms). We now use 2.4.10 and a relative index MOOTH K-THEORY theorem (compare (28)) in order to see that η (( E⊔ B K op ) t ′ ) − η (( E⊔ B K op ) t ) = ch dR ( index (([0 , × ( E⊔ B K op )) bt,AP S )) ∈ ch dR ( K ( B )) . Using Proposition 2.20 we get:
Corollary 5.5 . —
We have [ E ,
0] = [ K , η BC ( E )] .5.3.5. — Let p : W → B be a proper submersion with closed fibres with a smooth K -orientation represented by o . Let V be a geometric vector bundle over W , and let V denote the associated geometric family. Then we can form the geometric family E := p ! V (see Definition 3.7). Assume that the kernel of the family of Dirac operators( D ( E b )) b ∈ B has constant dimension, forming thus the kernel bundle K . Since V haszero-dimensional fibres we have ˜Ω(1 , V ) = 0. From (17) we getˆ p ! [ V , ρ ] = [ p ! V , Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )]= [ E , Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )]= [ K , η BC ( E ) + Z W/B ˆ A c ( o ) ∧ ρ + Z W/B σ ( o ) ∧ (Ω( V ) − dρ )] . ˆ K -class on S . — We construct in a natural way an element x S ∈ ˆ K ( S ) coming from thePoincar´e bundle over S × S . Let us identify S ∼ = R / Z . We consider the complexline bundle L := ( R × R / Z × C ) / Z over R / Z × R / Z , where the Z -action is givenby n ( s, t, z ) = ( s + n, t, exp( − πint ) z ). On R × R / Z × C → R × R / Z we have the Z -equivariant connection ∇ := d + 2 πisdt with curvature R ∇ = 2 πids ∧ dt . Thisconnection descends to a connection ∇ L on L . The unitary line bundle with con-nection L := ( L, h L , ∇ L ) gives a geometric family L over R / Z × R / Z . It represents v := [ L , ∈ ˆ K ( R / Z × R / Z ). Note that R ( v ) = 1 + ds ∧ dt . We now considerthe projection p : R / Z × R /Z → R / Z on the second factor. This fibre bundle has anatural smooth ˆ K -orientation ( g T v p , T h p, ˜ ∇ , S and the product structure. Moreover, T v p is trivialized by the S -action. Hence it has a preferred orientation. We takethe bounding Spin -structure on the fibres which induces the
Spin c -structure and theconnection ˜ ∇ . Definition 5.6 . —
We define x S := ˆ p ! v ∈ ˆ K ( S ) .5.4.2. — We have R ( x S ) = dt . Let t ∈ S . Then we compute t ∗ x S ∈ ˆ K ( ∗ ) ∼ = R / Z (identification again as in Lemma 5.2). Note that 0 ∗ x S is represented by the trivialline bundle over S . Since we choose the bounding spin structure, the correspondingDirac operator is invertible. Its spectrum is symmetric and its η -invariant vanishes ULRICH BUNKE & THOMAS SCHICK (compare 4.3.3). Therefore we have 0 ∗ x S = 0. It now follows by the homotopyformula (or by an explicit computation of η -invariants), that t ∗ x S = − t . (27) Let f : B → S be given. Then we define Definition 5.7 . — < f > := f ∗ x S ∈ ˆ K ( B ) . Assume now that we have two such maps f, g : B → S . As an interesting illustra-tion we characterize < f > ∪ < g > ∈ ˆ K ( B ) . It suffices to consider the universal example B = T = S × S . We consider theprojections pr i : S × S → S , i = 1 ,
2. Let x := ˆ pr ∗ x S and y := ˆ pr ∗ x S . Then wemust compute x ∪ y ∈ ˆ K ( T ). We identify T = R / Z × R / Z with coordinates s, t .First note that R ( x ∪ y ) = R ( x ) ∪ R ( y ) = ds ∧ dt . Thus the class x ∪ y − v + 1 is flat,i.e. x ∪ y − v + 1 ∈ K flat ( T ) . In fact, since K ( T ) is torsion-free, we have K flat ( T ) ∼ = H odd ( T ) / im ( ch dR ) = R / Z . In order to determine this element we must compute its holonomies along the circles S × × S . The holonomy of v along these circles is trivial. Since 0 ∗ x = 0 and0 ∗ y = 0 we see that x × y also has trivial holonomies along these circles. Thereforewe conclude Proposition 5.8 . — x ∪ y = v − f, g induce a map f × g : B → T . Corollary 5.9 . —
We have < f > ∪ < g > = ( f × g ) ∗ v − . S -valued maps and line-bundles. — Let f : B → S be a smooth map and L := ( L, ∇ L , h L ) be a hermiteanline bundle with connection over B . It gives rise to a geometric family L (see 2.1.4).We consider the smooth K -theory classes < f > and < L > := [ L , −
1. It is againinteresting to determine the class < f > ∪ < L > ∈ ˆ K ( B ) . An explicit answer is only known in special cases.First we compute the curvature: R ( < f > ∪ < L > ) = R ( < f > ) ∧ R ( < L > ) = df ∧ ( e c ( ∇ L ) − , where df := f ∗ dt and c ( ∇ L ) := − πi R ∇ L . MOOTH K-THEORY Note that the degree-one component of the odd form R ( < f > ∪ < L > )vanishes. Let now q : Σ → B be a smooth map from an oriented closed surface. Then R ( q ∗ ( < f > ∪ < L > )) = q ∗ R (( < f > ∪ < L > )) = 0. Therefore q ∗ ( < f > ∪ < L > ) ∈ ˆ K flat (Σ) ∼ = H ev (Σ , R ) / im ( ch ) ∼ = R / Z ⊕ R / Z , where the first component corresponds to H (Σ , R ) and the second to H (Σ , R ). Inorder to evaluate the first component we restrict to a point. Since the restriction of < L > to a point vanishes, the first component of q ∗ ( < f > ∪ < L > ) vanishes.Therefore it remains to determine the second component. Let us assume that q ∗ L is trivial. We choose a trivialization. Thenwe can define the transgression Chern form ˜ c ( ∇ q ∗ L , ∇ triv ) ∈ Ω (Σ) such that d ˜ c ( ∇ q ∗ L , ∇ triv ) = q ∗ c ( ∇ L ). By the homotopy formula we have q ∗ < L > = [ ∅ , − ˜ c ( ∇ q ∗ L , ∇ triv )] . In this special case we can compute q ∗ ( < f > ∪ < L > ) = q ∗ < f > ∪ q ∗ < L > = < q ∗ f > ∪ q ∗ < L > = [ ∅ , q ∗ df ∧ ˜ c ( ∇ q ∗ L , ∇ triv )] . We see that the second component is (cid:20)Z Σ q ∗ df ∧ ˜ c ( ∇ q ∗ L , ∇ triv ) (cid:21) R / Z . We do not know a good answer in the general case where q ∗ L is non-trivial. ˆ K - class on SU (2) . — Let G be a group acting on the manifold M . Definition 5.10 . —
A class x ∈ ˆ K ( M ) is called invariant, if g ∗ x = x for all x ∈ G .5.6.2. — For example, the class x S ∈ ˆ K ( S ) defined in 5.6 is not invariant underthe action L t , t ∈ S , of S on itself. Note that R ( x S ) = dt is invariant. Therefore L ∗ t x S − x S ∈ R / Z . In fact by (27) we have L ∗ t x S − x S = − t . Since dt is the only invariant form with integral one we see that the only way toproduce an invariant smooth refinement of the generator of H ( S , Z ) ∼ = Z would beto perturb x S by a class b ∈ H ( S , R / Z ). But b is of course homotopy invariant,hence L ∗ t b = b . We conclude that the generator of H ( S , Z ) (and also every non-trivial multiple) does not admit any invariant lift. ULRICH BUNKE & THOMAS SCHICK
The situation is different for simply-connected groups. Let us consider thefollowing example. The group G := SU (2) × SU (2) acts on SU (2) by ( g , g ) h := g hg − . Let vol SU (2) ∈ Ω ( SU (2)) denote the normalized volume form. Furthermorewe let i : ∗ → SU (2) denote the embedding of the identity. Proposition 5.11 . —
For k ∈ Z there exists a unique class x SU (2) ( k ) ∈ ˆ K ( SU (2)) such that R ( x SU (2) ) = k vol SU (2) and i ∗ x = 0 . This element is SU (2) × SU (2) -invariantProof . — Assume, that x, y ∈ ˆ K ( SU (2)) satisfy R ( x ) = R ( y ). Then we have x − y ∈ ˆ K flat ( SU (2)) ∼ = K flat ( S ) ∼ = R / Z . Since i ∗ x = i ∗ y = 0 we have in fact that x = y .Therefore, if the class x SU (2) ( k ) exists, then it is unique.We show the existence of an invariant class in an abstract manner. Note that k vol SU (2) represents a class ch ( Y ) for some Y ∈ K ( S ). In terms of classifyingmaps, Y for k = 1 is given by the embedding SU (2) → U (2) → U ( ∞ ) ∼ = K . Wehave the exact sequence0 → Ω ev ( SU (2)) / im ( ch dR ) a → ˆ K ( SU (2)) I → K ( SU (2)) → . Therefore we can choose any class y ∈ ˆ K ( SU (2)) such that I ( y ) = Y . Then thecontinuous group cocycle G ∋ t → c ( t ) = t ∗ y − y ∈ Ω ev ( SU (2)) / im ( ch dR ) representsan element [ c ] ∈ H c ( G, Ω ev ( SU (2)) / im ( ch dR )).We claim that this cohomology group is trivial. Note that Ω ev ( SU (2)) / im ( ch dR ) ∼ =Ω ( SU (2)) / Z ⊕ Ω ( SU (2)) / im ( d ). Since Ω ( SU (2)) / im ( d ) is a real topological vectorspace with a continuous action of the compact group G we immediately concludethat H c ( G, Ω ( SU (2)) / im ( d )) = 0 by the usual averaging argument. We consider theexact sequence of G -spaces0 → Z → Ω ( SU (2)) → Ω ( SU (2)) / Z → . Since G is simply-connected we see that taking continuous functions from G × · · · × G with values in these spaces, we obtain again exact sequences of Z -modules. It followsthat we have a long exact sequence in continuous cohomology. The relevant partreads H c ( G, Z ) → H c ( G, Ω ( SU (2))) → H c ( G, Ω ( SU (2)) / Z ) → H c ( G, Z ) . Since Z is discrete and G is connected we see that H ic ( G, Z ) = 0 for i ≥
1. Therefore, H c ( G, Ω ( SU (2))) ∼ = H c ( G, Ω ( SU (2)) / Z ) . But Ω ( SU (2)) is again a continuous representation of G on a real vector space sothat H c ( G, Ω ( SU (2))) = 0. The claim follows.We now can choose w ∈ Ω ev ( SU (2)) / im ( ch dR ) such that t ∗ w − w = t ∗ y − y for all t ∈ G . We can further assume that i ∗ w = i ∗ y by adding a constant. Then we set x SU (2) ( k ) = y − w ∈ ˆ K ( SU (2)). This element has the required properties. MOOTH K-THEORY It is an interesting problem to write down an invariant cycle which represents theclass x SU (2) . Note that x SU (2) ( k ) = kx SU (2) (1). Let Σ ⊂ SU (2) be an embedded ori-ented hypersurface. Then R ( x SU (2) (1)) | Σ = 0 so that ( x SU (2) ) | Σ ∈ ˆ K flat (Σ). Since x SU (2) (1) evaluates trivially on points we have in fact( x SU (2) (1)) | Σ ∈ ker (cid:16) ˆ K flat (Σ) → ˆ K flat ( ∗ ) (cid:17) ∼ = R / Z . This number can be determined by integration over Σ. Formally, let p : Σ → {∗} be the projection. If we choose some smooth K -orientation, then we can ask forˆ p ! ( x SU (2) (1)) | Σ ∈ ˆ K flat ( ∗ ) ∼ = R / Z . The hypersurface Σ decomposes SU (2) in twoparts SU (2) ± Σ . Let SU (2) +Σ be the part such that ∂SU (2) +Σ has the orientation givenby Σ. We choose a K -orientation o of the projection q : SU (2) +Σ → ∗ which has aproduct structure such that σ ( o ) = 0 and ˆ A c ( o ) = 1. In order to get the latterequality we choose a Spin c -structure coming from a spin structure. The smooth K -orientation of q induces a smooth K -orientation of p . Then q : SU (2) +Σ → ∗ providesa zero-bordism of Σ, and of ( x SU (2) (1)) | Σ . Therefore, we have by Proposition 5.18ˆ p ! ( x SU (2) (1)) | Σ = " ∅ , Z SU (2) +Σ R ( x SU (2) (1)) = − [ vol ( SU (2) +Σ )] R / Z , where [ λ ] R / Z denotes the class of λ ∈ R . Note that the identification ˆ K flat ( ∗ ) ∼ = R / Z is induced by a : R ∼ = Ω odd ( ∗ ) / im ( d ) → K flat ( ∗ ) given by λ [ ∅ , − λ ]. This explainsthe minus sign in the second equality above. Some of the arguments from the SU (2)-case generalize. Let G be a compactconnected and simply-connected Lie group and G/H be a homogenous space.Given Y ∈ K ( G/H ) we can find a lift y ∈ ˆ K ( G/H ). We form the cocycle G ∋ g c ( g ) := g ∗ y − y ∈ Ω( G/H ) / im ( ch dR ). Since Ω( G/H ) / im ( ch dR ) is the quotient of avector space by a lattice and G is connected and simply-connected we can use the argu-ments as in the SU (2)-case in order to conclude that H c ( G, Ω( G/H ) / im ( ch dR )) = 0.Therefore we can choose the lift y such that g ∗ y = y for all g ∈ G . In particular, R ( y ) ∈ Ω( G/H ) is now an invariant form representing ch ( Y ). Note that an invariantform is in general not determined by this condition. If we specialize to the case that
G/H is symmetric, then invariant formsexactly represent the cohomology. In this case we see that two choices of invariantlifts y , y of Y have the same curvature so that y − y ∈ ˆ K flat ( G/H ). Since the y i also have the same index, we indeed have y − y ∈ H ( G/H, R ) / im ( ch dR ). We havethus shown the following lemma. ULRICH BUNKE & THOMAS SCHICK
Lemma 5.12 . —
Assume that
G/H is a symmetric space with G connected andsimply connected. Then every Y ∈ K ( G/H ) has an invariant lift y ∈ ˆ K ( G/H ) whichis uniquely determined up to H ( G/H, R ) / im ( ch dR ) .5.7.3. — We can apply this in certain cases. First we write S n +1 ∼ = Spin (2 n +2) /Spin (2 n + 1), n ≥
1. Note that K ( S n +1 ) ∼ = Z . Since H ev ( S n +1 , R ) / im ( ch dR ) = R / Z is concentrated in degree zero we have the following result. Corollary 5.13 . —
Let n ≥ . For each k ∈ Z there is a unique x S n +1 ( k ) ∈ ˆ K ( S n +1 ) which is invariant, has index k ∈ Z ∼ = K ( S n +1 ) , and evaluates triviallyon points.5.7.4. — In the even-dimensional case we write S n ∼ = Spin (2 n + 1) /Spin (2 n ), n ≥ K ( S n ) ∼ = Z ⊕ Z and H odd ( S n , R ) / im ( ch dR ) = 0. Corollary 5.14 . —
For each k ∈ Z there is a unique x S n ( k ) ∈ ˆ K ( S n ) which isinvariant and has index k ∈ Z ∼ = ˜ K ( S n ) , and evaluates trivially on points5.7.5. — We write CP n := SU ( n +1) /S ( U (1) × U ( n )). Then H odd ( CP n , R ) / im ( ch dR ) =0. Therefore we conclude: Lemma 5.15 . —
For each Y ∈ K ( CP n ) there is a unique SU ( n +1) -invariant class y CP n ( Y ) ∈ ˆ K ( CP n ) such that I ( y CP n ( Y )) = Y .5.7.6. — Let G be a connected and simply-connected Lie group. Let T ⊂ G be amaximal torus. Then we have a G -map P : G/T × T → G , P ([ g ] , t ) := gtg − , where G acts on the left-hand side by g ([ h ] , t ) := ([ gh ] , t ), and by conjugation on the right-hand side. Let x ∈ ˆ K ∗ ( G ) be an invariant element. It is an interesting question how P ∗ x looks like.Let us consider the special case G = SU (2) and x SU (2) = x SU (2) (1) ∈ ˆ K ( SU (2)).In this case we have T = S and G/T ∼ = CP . First we compute the curvature of P ∗ x SU (2) . For this we must compute P ∗ vol SU (2) which is given by Weyl’s integrationformula. We have P ∗ vol SU (2) = vol CP ∧ (2 πt ) dt . There is a unique class z ∈ ˆ K ( S ) with curvature 4 sin (2 πt ) dt such that 0 ∗ z = 0.Furthermore, there is a unique class < L > ∈ ˆ K ( CP ) with curvature vol CP whichis in fact the class < L > considered in 5.5.1 associated to the canonical line bundle L on CP .The product < L > ∪ z has now the same curvature as P ∗ x SU (2) . We concludethat P ∗ x SU (2) − < L > ∪ z ∈ H ev ( CP × S , R ) / im ( ch dR ) . MOOTH K-THEORY Now note that H ev ( CP × S , R ) / im ( ch dR ) ∼ = (cid:0) H ( CP , R ) ⊗ H ( S , R ) ⊕ H ( CP , R ) ⊗ H ( S , R ) (cid:1) / im ( ch dR ) ∼ = R / Z ⊕ R / Z . The first component can be determined by evaluating the difference P ∗ x SU (2) − < L > ∪ z at a point. Since x SU (2) is trivial on points, this first component vanishes.The second component can be determined by evaluating P ∗ x SU (2) − < L > ∪ z at CP × { } . Note that P ∗ CP ×{ } x SU (2) = 0, since P | CP ×{ } is constant. Furthermore,0 ∗ z = 0 implies that < L > ∪ z | CP ×{ } = 0. Thus we have shown (using S ∼ = CP ): Lemma 5.16 . — P ∗ x SU (2) = x S (1) ∪ z A zero bordism of a geometric family E over B is a geometric family W over B with boundary such that E = ∂ W . The notion of a geometric family withboundary is explained in [ Bun ]. It is important to note that in our set-up a geometricfamily with boundary always has a product structure.
Proposition 5.17 . — If E admits a zero bordism W , then in ˆ K ∗ ( B ) we have theidentity [ E ,
0] = [ ∅ , Ω( W )] . Proof . — Since E admits a zero bordism we have index ( E ) = 0 so that E admits ataming E t . This taming induces a boundary taming W bt . The obstruction againstextending the boundary taming to a taming of W is index ( W bt ) ∈ K ( B ) [ Bun ,Lemma 2.2.6].Let us assume for simplicity that E is not zero-dimensional. Otherwise we mayhave to stabilize in the following assertion. Using 2.4.10 we can adjust the taming E t such that index ( W bt ) = 0. At this point we employ a version of the relative indextheorem [ Bun95 ] index ( W bt ′ ) = index ( W bt ) + index (( E × [0 , bt ) , (28)where E t and E t ′ define the boundary taming ( E × [0 , bt .If index ( W bt ) = 0, then we can extend the boundary taming W bt to a taming W t .We now apply the identity [ Bun , Thm. 2.2.13]:Ω( W ) = dη ( W t ) − η ( E t ) . Note that this equality is more precise than needed since it holds on the level of formswithout factoring by im ( d ). We see that ( E ,
0) is paired with ( ∅ , Ω( W )). This impliesthe assertion. ULRICH BUNKE & THOMAS SCHICK
Let p : W → B be a proper submersion from a manifold with boundary W which restricts to a submersion q := p | ∂W : V := ∂W → B . We assume that p has a topological K -orientation and a smooth K -orientation represented by o p which refines the topological K -orientation. We assume that the geometric data of o p has a product structure near V (see [ Bun , Section 2.1] for a detailed discussionof such product structures). Recall o p = ( g T v p , T h p, ˜ ∇ p , σ p ). By the assumption of aproduct structure we have a quadruple ( g T v q , T h q, ˜ ∇ q , σ q ) and an isomorphism of aneighbourhood of p | ∂W : ∂W → B with the bundle E × [0 , pr E → E p → B such that thegeometric data are related as follows.1. T v p |E× [0 , ∼ = pr ∗E T v q ⊕ pr ∗ [0 , T [0 ,
1) and g T v p |E× [0 , = pr ∗E g T v q + pr ∗ [0 , dr , where r ∈ [0 ,
1) is the coordinate.2. T h p |E× [0 , = pr ∗E T h q .3. ( σ p ) |E× [0 , = pr ∗E σ q .4. The Spin c -structure on T v q and the canonical Spin c -structure on T [0 ,
1) inducea
Spin c -structure on the vertical bundle T v ∼ = pr E T v E ⊕ pr ∗ [0 , T [0 ,
1) of
E × [0 , S ( T v ) = pr ∗E S c ( T v q )or pr ∗E S c ( T v q ) ⊗ C depending on the dimension of T v q . In particular, theconnection ˜ ∇ q gives rise to a connection ˜ ∇ prod . The product structure identifiesthe restricted Spin c -structure of T v p |E× [0 , with this product Spin c -structuresuch that ˜ ∇ |E× [0 , becomes ˜ ∇ prod .From this description we deduce thatˆ A c ( ˜ ∇ ) |E× [0 , = pr ∗E ˆ A c ( ˜ ∇ q ) , ˆ A c ( o p ) |E× [0 , = pr ∗E ˆ A c ( o q ) . It is now easy to see that the restriction of representatives (with product structure)preserves equivalence and gives a well-defined restriction of smooth K -orientations.We have the following version of bordism invariance of the push-forward in smooth K -theory. Proposition 5.18 . —
For y ∈ ˆ K ( W ) we set x := y | V ∈ ˆ K ( V ) . Then we have ˆ q ! ( x ) = [ ∅ , p o ! R ( y )] . MOOTH K-THEORY Proof . — Let y = [ E , ρ ]. We compute using (17), Proposition 5.17, Stokes’ theorem,Definition 3.15, and the adiabatic limit λ → q ! ( x ) = [ q λ ! E | V , Z V/B ˆ A c ( o q ) ∧ ρ + ˜Ω( λ, E | V ) + Z V/B σ ( o q ) ∧ R ( x )]= [ ∅ , Ω( p λ ! E ) + Z V/B ˆ A c ( o q ) ∧ ρ + ˜Ω( λ, E | V ) + Z V/B σ ( o q ) ∧ R ( x )] ! = [ ∅ , Z W/B (cid:16) ˆ A c ( o p ) ∧ Ω( E ) − ˆ A c ( o p ) ∧ dρ − dσ ( o p ) ∧ R ( y ) (cid:17) ]= [ ∅ , Z W/B ( ˆ A c ( o p ) − dσ ( o p )) ∧ R ( y )]= [ ∅ , p o ! R ( y )] Z /k Z -invariants. — Here we associate to a family of Z /k Z -manifolds over B a class in ˆ K flat ( B ). Definition 5.19 . —
A geometric family of Z /k Z -manifolds is a triple ( W , E , φ ) ,where W is a geometric family with boundary, E is a geometric family without bound-ary, and φ : ∂ W ∼ → k E is an isomorphism of the boundary of W with k copies of E . We define u ( W , E , φ ) := [ E , − k Ω( W )] ∈ ˆ K ( B ). Lemma 5.20 . —
We have u ( W , E , φ ) ∈ ˆ K flat ( B ) . This class is a k -torsion class.It only depends on the underlying differential-topological data.Proof . — We first compute by 5.17 ku ( W , E , φ ) = k [ E , − k Ω( W )]= [ k E , − Ω( W )]= [ ∅ , R ( u ( W , E , φ )) = 0 so that u ( W , E , φ ) ∈ ˆ K flat ( B ). Independence ofthe geometric data is now shown by a homotopy argument. We now explain the relation of this construction to the Z /k Z -index of Freed-Melrose [ FM92 ]. Lemma 5.21 . —
Let B = ∗ and dim( W ) be even. Then u ( W , E , φ ) ∈ ˆ K flat ( ∗ ) ∼ = R / Z . Let i k : Z /k Z → R / Z the embedding which sends k Z to k . Then i k ( index a ( ¯ W )) = u ( W , E , φ ) , ULRICH BUNKE & THOMAS SCHICK where i k ( index a ( ¯ W )) ∈ Z /k Z is the index of the Z /k Z -manifold ¯ W (the notation of [ FM92 ] ).Proof . — We recall the definition of index a ( ¯ W ). In our language is can be stated asfollows. Since index ( E ) = 0 we can choose a taming E t . We let k copies of E t inducethe boundary taming W bt . We have index a ( ¯ W ) = index ( W bt ) + k Z . In fact it is easy to see that a change of the taming E t leads to change of the index index ( W bt ) by a multiple of k . We can now prove the Lemma using [ Bun , Thm.2.2.18]. u ( W , E , φ ) = [ E , − k Ω( W )]= [ ∅ , − η ( E t ) − k Ω( W )]= [ ∅ , − k index ( W bt )]= a (cid:18) k index ( W bt ) (cid:19) = i k ( index a ( ¯ W )) ∈ R / Z . Spin c -bordism invariants. — Let π be a finite group. We construct a transformation φ : Ω Spin c ( BU ( n ) × Bπ ) → ˆ K flat ( ∗ ) . Let f : M → BU ( n ) × Bπ represent [ M, f ] ∈ Ω Spin c ( BU ( n ) × Bπ ). This map deter-mines a covering p : ˜ M → M and an n -dimensional complex vector bundle V → M .We choose a Riemannian metric g T M and a
Spin c -extension ˜ ∇ of the Levi-Civitaconnection ∇ T M . These structures determine a smooth K -orientation of t : M → ∗ .We further fix a metric h V and a connection ∇ V in order to define a geometric bundle V := ( V, h V , ∇ V ) and the associated geometric family V (see 2.1.4). The pull-backof g T M and ˜ ∇ via ˜ M → M fixes a smooth K -orientation of ˜ t : ˜ M → ∗ .We define the geometric families M := t ! V and ˜ M := ˜ t ! ( p ∗ V ) over ∗ . Then we set φ ([ M, f ]) := [ ˜
M ⊔ ∗ | π |M op , ∈ ˆ K flat ( ∗ ) . By a homotopy argument we see that this class is independent of the choice of geom-etry. We now argue that it only depends on the bordism class of [
M, f ].The construction is additive. Let now [
M, f ] be zero-bordant by [
W, F ]. Then wehave a zero bordism ˜ W of ˜ M over W . Note that the bundles also extend over thebordism. The local index form of ˜ W ⊔ B | π |W vanishes. We conclude by 5.17, that[ ˜ M ⊔ B | π | · M op ,
0] = 0.
MOOTH K-THEORY In this construction we can replace Eπ → Bπ by any finite covering. This construction allows the following modification. Let ρ ∈ Rep ( π ) be avirtual zero-dimensional representation of π . It defines a flat vector bundle F ρ → Bπ .To [ M, f ] we associate the geometric family M ρ := t ! ( L ), where L is the geometricfamily associated to the geometric bundle V ⊗ ( pr ◦ f ) ∗ F ρ . We define φ ρ : Ω Spin c ∗ ( BU ( n ) × Bπ ) → ˆ K flat ( ∗ )such that φ ρ [ M, f ] := [ M ρ , π is finite. This isthe construction of ρ -invariants in the smooth K -theory picture.The first construction is a special case of the second with the representation ρ = C ( π ) ⊕ ( C | π | ) op . We now discuss a parametrized version. Let B be some compact manifoldand X be some topological space. Then we can define the parametrized bordism groupΩ Spin c ∗ ( X/B ). Its cycles are pairs ( p : W → B, f : W → X ) of a proper topologically K -oriented submersion p and a continuous map f . The bordism relation is definedcorrespondingly.There is a natural transformation φ : Ω Spin c ∗ (( BU ( n ) × Bπ ) /B ) → ˆ K ∗ flat ( B ) . It associates to x = ( p : W → B, f : W → BU ( n ) × Bπ ) the class [ ˜ W ⊔ B | π | · W op , p : ˜ W → W is again the π -covering classified by pr ◦ f . We define thegeometric family W using some choice of geometric structures and the twisting bundle V , where V is classified by the first component of f . The family ˜ W is obtained from˜ W and p ∗ V using the lifted geometric structures. Again, the class φ ( x ) is flat andindependent of the choices of geometry. Using 5.17 one checks that φ passes throughthe bordism relation.Again there is the following modification. For ρ ∈ Rep ( π ) we can define φ ρ : Ω Spin c ∗ (( BU ( n ) × Bπ ) /B ) → ˆ K ∗ flat ( B ) . It associates to x = ( p : W → B, f : W → BU ( n ) × Bπ ) the class [ W ρ ] of the geomet-ric manifold W with twisting bundle V ⊗ ( pr ◦ f ) ∗ F ρ . These classes are K -theoretichigher ρ -invariants. It seems promising to use this picture to draw geometric conse-quences using these invariants. e -invariant. — A framed n -manifold M is a manifold with a trivialization T M ∼ = M × R n .More general, a bundle of framed n -manifolds over B is a fibre bundle π : E → B with a trivialization T v π ∼ = E × R n . Proposition 5.22 . —
A bundle of framed n -manifolds π : E → B has a canonicalsmooth K -orientation which only depends on the homotopy class of the framing. ULRICH BUNKE & THOMAS SCHICK
Proof . — The framing T v π ∼ = E × R n induces a vertical Riemannian metric g T v π and an isomorphism SO ( T v π ) ∼ = E × SO ( n ). Hence we get an induced verticalorientation and a Spin -structure which determines a
Spin c -structure, and thus a K -orientation of π . We choose a horizontal distribution T h π which gives rise toa connection ∇ T v π . Since our Spin c -structure comes from a Spin -structure, thisconnection extends naturally to a
Spin c -connection ˜ ∇ of trivial central curvature.The trivial connection ∇ triv on T v π induced by the framing also lifts naturally tothe trivial Spin c -connection ˜ ∇ triv . The quadruple o := ( g T v π , T h π, ˜ ∇ , ˜ˆ A c ( ˜ ∇ , ˜ ∇ triv ))defines a smooth K -orientation of π which refines the given underlying topological K -orientation.We claim that this orientation is independent of the choice of the vertical dis-tribution T h π . Indeed, if T h π is a second horizontal distribution with associated Spin c -connection ˜ ∇ ′ , then we set o ′ := ( g T v π , T h π ′ , ˜ ∇ ′ , ˆ A c ( ˜ ∇ ′ , ˜ ∇ triv )) . Since ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ triv ) − ˜ˆ A c ( ˜ ∇ , ˜ ∇ triv ) = ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ )we have o ∼ o ′ in view of the Definition 3.1.9.Let us now consider a second framing of T v π which is homotopic to the first.In induces a second trivial connection ˜ ∇ ′ triv and a metric g ′ T v π . We thereforeget a connection ˜ ∇ ′ and and a second representative of a smooth K -orientation o ′ := ( g ′ T v π , T h π, ˜ ∇ ′ , ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv )). In fact, the homotopy between the framingsprovides a connection ˜ ∇ h,triv on I × E . Since this connection is flat we see that˜ˆ A c ( ˜ ∇ ′ triv , ˜ ∇ triv ) = 0. From˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv ) = ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ ) + ˜ˆ A c ( ˜ ∇ , ˜ ∇ triv ) + ˜ˆ A c ( ˜ ∇ triv , ˜ ∇ ′ triv )we get ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ ′ triv ) − ˜ˆ A c ( ˜ ∇ , ˜ ∇ triv ) = ˜ˆ A c ( ˜ ∇ ′ , ˜ ∇ )and thus o ∼ o ′ .Since ˜ ∇ triv is flat we haveˆ A c ( o ) − dσ ( o ) = ˆ A ( ˜ ∇ ) − d ˜ˆ A ( ˜ ∇ , ˜ ∇ triv ) = 1 . Assume that the fibre dimension n satisfies n ≥
1. According to Lemma 3.16 thecurvature of ˆ π ! (1) is given by R (ˆ π ! (1)) = Z E/B ( ˆ A c ( o ) − dσ ( o )) ∧ Z E/B ∧ MOOTH K-THEORY Definition 5.23 . — If π : E → B is a bundle of framed manifolds of fibre dimension n ≥ , then we define a differential topological invariant e ( E → B ) := − ˆ π ! (1) ∈ ˆ K − nflat ( B ) . In the following we will explain in some detail that this is a higher generalizationof the Adams e -invariant. The stable homotopy groups of the sphere π n := π sn ( S )have a decreasing filtration · · · ⊆ π n ⊆ π n ⊆ π n = π n related to the MSpin-based Adams Novikov spectral sequence. The e -invariant is ahomomorphism e : π n − /π n − → R / Z . A closed framed 4 n − M represents a class [ M ] ∈ π n − underthe Pontrjagin-Thom identification of framed bordism with stable homotopy. In theindicated dimension π n − = π n − so that [ M ] is actually a boundary of a compact4 n -dimensional Spin -manifold N . As explained in [ APS75 ] (see also [
Lau99 ]) the e -invariant e [ M ] can be calculated as follows. One chooses a connection ∇ T N on T N which restricts to the trivial connection ∇ triv on T M given by the framing. Then e ([ M ]) = (cid:20)Z N ˆ A ( ∇ ) (cid:21) R / Z . We now consider q : M → ∗ as a bundle of framed manifolds over the point andidentify R / Z ∼ → ˆ K − n +1 flat ( ∗ ) by [ u ] a ( u ) = [ ∅ , − u ], u ∈ R . Lemma 5.24 . —
Under these identifications we have e ( M → ∗ ) = e ([ M ]) .Proof . — We choose a metric g T M on M which induces the representative o := ( g T M , , ˜ ∇ , ˜ˆ A c ( ˜ ∇ , ∇ triv ))of the smooth K -orientation on q . The Spin -structure of N induces a Spin c -structure. We choose a Riemannian metric g T N on N with a product struc-ture near the boundary which extends g T M and induces the
Spin - and
Spin c -connections ∇ N and ˜ ∇ N . Note that ˜ˆ A c ( ˜ ∇ N , ˜ ∇ T N ) extends ˜ˆ A c ( ˜ ∇ , ˜ ∇ triv ). Therefore o N := ( g T N , , ˜ ∇ N , ˜ˆ A c ( ˜ ∇ N , ˜ ∇ T N )) represents a smooth K -orientation of p : N → ∗ which extends the orientation o of q : M → ∗ . We can now apply the bordism formula ULRICH BUNKE & THOMAS SCHICK
Proposition 5.18 in the marked step and get e ( M → ∗ ) = − ˆ q ! (1) ! = a ( p ! ( R (1)))= "Z N/ ∗ ( ˆ A c ( o N ) − dσ ( o N )) ∧ R / Z = "Z N/ ∗ ˆ A c ( ˜ ∇ N ) − d ˜ˆ A ( ˜ ∇ N , ˜ ∇ T N ) R / Z = "Z N/ ∗ ˆ A c ( ˜ ∇ T N ) R / Z = "Z N/ ∗ ˆ A ( ∇ T N ) R / Z = e ([ M ]) . Using the method of Subsection 5.3 or the APS index theorem it is now easy toreproduce the result of [
APS75 ] e ([ M ]) = (cid:20) η ( M ) − Z M ˆ A ( ˜ ∇ , ˜ ∇ triv ) (cid:21) R / Z .
6. The Chern character and a smooth Grothendieck-Riemann-Rochtheorem6.1. Smooth rational cohomology. —
Let Z k − ( B ) be the group of smooth singular cycles on B . The picture ofˆ H ( B, Q ) as Cheeger-Simons differential charactersˆ H k ( B, Q ) ⊂ Hom ( Z k − ( B ) , R / Q )is most appropriate to define the integration map. By definition (see [ CS85 ]) ahomomorphism φ ∈ Hom ( Z k − ( B ) , R / Q ) is a differential character if and only if thereexists a form R ( φ ) ∈ Ω kd =0 ( B ) such that φ ( ∂c ) = (cid:20)Z c R ( φ ) (cid:21) R / Q (29)for all smooth k -chains c ∈ C k ( B ). It is shown in [ CS85 ] that R ( φ ) is uniquelydetermined by φ . In fact, the map R : ˆ H k ( B, Q ) → Ω kd =0 ( B ) is the curvature trans-formation in the sense of Definition 1.1.Assume that T is a closed oriented manifold of dimension n with a triangulation.Then we have a map τ : Z k − ( B ) → Z k − n ( T × B ). If σ : ∆ k − → B is a smooth MOOTH K-THEORY singular simplex, then the triangulation of T × ∆ k − gives rise to a k − n chain τ ( σ ) : = id × σ : T × ∆ → T × B . The integration( ˆ pr ) ! : ˆ H ( T × B, Q ) → ˆ H ( B, Q )is now induced by τ ∗ : Hom ( Z k − n ( T × B ) , R / Q ) → Hom ( Z k − ( B ) , R / Q ) . Alternative definitions of the integration (for proper oriented submersions) are given in[
HS05 ], [
GT00 ]. Another construction of the integration has been given in [
DL05 ],where also a projection formula (the analog of 4.5 for smooth cohomology) is proved.This picture is used in [
K¨o7 ] in particular to establish functoriality.We will also need the following bordism formula which we prove using yet an-other characterization of the push-forward. We consider a proper oriented submersion q : W → B such that dim( T v q ) = n . Let x ∈ ˆ H r ( W, Q ) and f : Σ → B be a smoothmap from a closed oriented manifold of dimension r − n −
1. We get a pull-backdiagram U F −−−−→ W y y q Σ f −−−−→ B .
The orientations of Σ and T v q induce an orientation of U . Note that f ∗ ˆ q ! ( x ) and F ∗ x are flat classes for dimension reasons. Therefore F ∗ x ∈ H r − ( U, R / Q ) and f ∗ ˆ q ! ( x ) ∈ H r − n − (Σ , R / Q ). The compatibility of the push-forward with cartesiandiagrams implies the following relation in R / Q : < f ∗ ˆ q ! ( x ) , [Σ] > = < F ∗ x, [ U ] > . If we let f : Σ → B vary, then these numbers completely characterize the push-forwardˆ p ! ( x ) ∈ ˆ H r − n ( B, Q ). We will use this fact in the argument below. Let now p : V → B be a proper oriented submersion from a manifold withboundary such that ∂V ∼ = W and p | W = q . Assume that x ∈ ˆ H ( V, Q ). Lemma 6.1 . — In ˆ H ( B, Q ) we have the equality ˆ q ! ( x | W ) = − a Z V/B R ( x ) ! . Proof . — Assume that x ∈ ˆ H r ( V, Q ). Let f : Σ → B be as above and form thecartesian diagram Z z −−−−→ V y y p Σ f −−−−→ B. ULRICH BUNKE & THOMAS SCHICK
The oriented manifold Z has the boundary ∂Z ∼ = U . Using (29) at the markedequality we calculate < f ∗ ˆ q ! ( x | W ) , [Σ] > = < F ∗ x | W , [ U ] > = < ( z ∗ x ) | U , [ U ] > ! = (cid:20)Z Z R ( z ∗ x ) (cid:21) R / Q = "Z Σ Z Z/ Σ R ( z ∗ x ) R / Q = "Z Σ f ∗ Z V/B R ( x ) R / Q = − < f ∗ a Z V/B R ( x ) ! , [Σ] > . This implies the assertion.
We start by recalling the classical smooth characteristic classes of Cheeger-Simons. A complex vector bundle V → B has Chern classes c i ∈ H i ( B, Z ), i ≥
1. Ifwe add the geometric data of a hermitean metric and a metric connection, then weget the geometric bundle V = ( V, h V , ∇ V ). In [ CS85 ] the Chern classes have beenrefined to smooth integral cohomology-valued Chern classesˆ c i ( V ) ∈ ˆ H i ( B, Z )(see 1.2.1 for an introduction to smooth ordinary cohomology). In particular, theclass ˆ c ( V ) ∈ ˆ H ( B, Z ) classifies isomorphism classes of hermitean line bundles withconnection.The embedding Z ֒ → Q induces a natural map ˆ H ( B, Z ) → ˆ H ( B, Q ), and we letˆ c Q ( V ) ∈ ˆ H ( B, Q ) denote the image of ˆ c ( V ) ∈ ˆ H ( B, Z ) under this map. The smooth Chern character ˆ ch which we will construct is a natural trans-formation ˆ ch : ˆ K ( B ) → ˆ H ( B, Q )of smooth cohomology theories. In particular, this means that the following diagramscommute (compare Definition 1.3)Ω( B ) / im ( d ) a / / ˆ K ( B ) I / / ˆ ch (cid:15) (cid:15) K ( B ) ch (cid:15) (cid:15) Ω( B ) / im ( d ) a / / ˆ H ( B, Q ) I / / H ( B, Q ) , ˆ K ( B ) R / / ˆ ch (cid:15) (cid:15) Ω d =0 ( B )ˆ H ( B, Q ) R / / Ω d =0 ( B ) . (30) MOOTH K-THEORY In addition we require that the even and odd Chern characters are related bysuspension, which in the smooth case amounts to the commutativity of the followingdiagram ˆ K ( S × B ) ( ˆ pr ) ! (cid:15) (cid:15) ˆ ch / / ˆ H ev ( S × B, Q ) ( ˆ pr ) ! (cid:15) (cid:15) ˆ K ( B ) ˆ ch / / ˆ H odd ( B, Q ) . (31)The smooth K -orientation of pr : S × B → B is as in 4.3.2. Theorem 6.2 . —
There exists a unique natural transformation ˆ ch : ˆ K ( B ) → ˆ H ( B, Q ) such that (30) and (31) commute. Note that naturality means that ˆ ch ◦ f ∗ = f ∗ ◦ ˆ ch for every smooth map f : B ′ → B .The proof of this theorem occupies the remainder of the present subsection. Proposition 6.3 . —
If the smooth Chern character ˆ ch exists, then it is unique.Proof . — Assume that ˆ ch and ˆ ch ′ are two smooth Chern characters. Consider thedifference ∆ := ˆ ch − ˆ ch ′ . It follows from the diagrams above that ∆ factors throughan odd natural transformation¯∆ : K ( B ) → H ( B, R / Q ) . Indeed, the left diagram of (30) gives a factorization K ( B ) → ( im : Ω( B ) / im ( d ) → ˆ H ( B, Q )) , and the right square in (30) refines it to ¯∆. We now use the following topological fact. Let P be a space of the homotopytype of a countable CW -complex. It represents a contravariant set-valued functor W P ( W ) := [ W, P ] on the category of compact manifolds. We further considersome abelian group V . Lemma 6.4 . —
A natural transformation of functors N : P ( B ) → H j ( B, V ) on thecategory of compact manifolds is necessarily induced by a class N ∈ H j ( P, V ) .Proof . — There exists a countable directed diagram M of compact manifolds suchthat hocolim M ∼ = P in the homotopy category. Hence we have a short exact sequence0 → lim H ( M , V ) → H ( P, V ) → lim H ( M , V ) → . If x ∈ P ( P ) is the tautological class, then the pull-back of N ( x ) to the system M gives an element in lim H ( M , V ). A preimage in H ( P, V ) induces the naturaltransformation. ULRICH BUNKE & THOMAS SCHICK
In our application, P = Z × BU , and the relevant cohomology H odd ( Z × BU, R / Q )is trivial. Therefore ¯∆ : K ( B ) → H odd ( B, R / Q ) vanishes Next we observe that ( ˆ pr ) ! : ˆ K ( S × B ) → ˆ K ( B ) is surjective. In fact, wehave ( ˆ pr ) ! ( pr ∗ x S ∪ pr ∗ ( x )) = x (32)by the projection formula 4.5 and ˆ p ! ( x S ) = 1 for p : S → ∗ , where x S ∈ ˆ K ( S ) wasdefined in 5.6. Hence (31) implies that ¯∆ : K ( B ) → H ev ( B, R / Q ) vanishes, too. In view of Proposition 6.3 it remains to show the existence of the smoothChern character. We first construct the even partˆ ch : ˆ K ( B ) → ˆ H ev ( B, Q )using the splitting principle. We will define ˆ ch as a natural transformation of functorssuch that the following conditions hold.1. ˆ ch [ L ,
0] = e ˆ c Q ( L ) ∈ ˆ H ev ( B, Q ), where L is the geometric family given by ahermitean line bundle with connection L , and ˆ c Q ( L ) ∈ ˆ H ( B, Q ) is derivedfrom the Cheeger-Simons Chern class which classifies the isomorphism class of L (6.2.1).2. R ◦ ˆ ch = R
3. ˆ ch ◦ a = a Once this is done, the resulting ˆ ch automatically satisfies (30). For this it sufficesto show that ch ◦ I = I ◦ ˆ ch . We consider the following diagramˆ K ( B ) R ) ) ˆ ch / / I (cid:15) (cid:15) ˆ H ( B, Q ) I (cid:15) (cid:15) R / / Ω d =0 ( B ) (cid:15) (cid:15) K ( B ) ch / / H ( B, Q ) i / / H ( B, R )The outer square and the right square commute. It follows from 2. that the uppertriange commutes. Since i is injective we conclude that the left square commutes,too. In the construction of the Chern character ˆ ch we will use the splittingprinciple. If x ∈ ˆ K ( B ), then there exists a Z / Z -graded hermitean vector bundlewith connection V = ( V, h V , ∇ V ) such that x = [ V , ρ ] for some ρ ∈ Ω odd ( B ) / im ( d ),where V is the zero-dimensional geometric family with underlying Dirac bundle V .We will call V the splitting bundle for x . Let F ( V ± ) → B be the bundle of fullflags on V ± and p : F ( V ) := F ( V + ) × B F ( V − ) → B . Then we have a decomposition p ∗ V ± ∼ = ⊕ L ∈ I ± L for some ordered finite sets I ± of line bundles over F ( V ). For L ∈ I ± let L denote the bundle with the induced metric and connection, and let L be the corresponding zero-dimensional geometric family. Then we have p ∗ x = MOOTH K-THEORY P L ∈ I + [ L , − P L ∈ I − [ L ,
0] + a ( σ ) for some σ ∈ Ω odd ( F ( V )) / im ( d ). The propertiesabove thus uniquely determine p ∗ ˆ ch ( x ). Lemma 6.5 . —
The following pull-back operations are injective: p ∗ : H ∗ ( B, Q ) → H ∗ ( F ( V ) , Q ) , p ∗ : H ∗ ( B, R ) → H ∗ ( F ( V ) , R )3. p ∗ : H ∗ ( B, R / Q ) → H ∗ ( F ( V ) , R / Q )4. p ∗ : ˆ H ∗ ( B, Q ) → ˆ H ∗ ( F ( V ) , Q )5. p ∗ : Ω( B ) → Ω( F ( V )) .Proof . — The assertion is a classical consequence of the Leray-Hirsch theorem inthe cases 1., 2., and 3. In case 5., it follows from the fact that p is surjective anda submersion. It remains to discuss the case 4. Let x ∈ ˆ H ∗ ( B, Q ). Assume that p ∗ x = 0. Then in particular p ∗ R ( x ) = R ( p ∗ x ) = 0 so that from 5. also R ( x ) = 0.Thus x ∈ H ( B, R / Q ). We now apply 3. and see that p ∗ x = 0 implies x = 0.In view of Proposition 6.3 we see that a natural transformation ˆ ch : ˆ K ( B ) → ˆ H ev ( B, Q ) is uniquely determined by the conditions 1., 2., and 3. formulated in 6.2.6. Proposition 6.6 . —
There exists a natural transformation ˆ ch : ˆ K ( B ) → ˆ H ev ( B, Q ) which satisfies the conditions 1. to 3. formulated in 6.2.6. We give the proof of this Proposition in the next couple of subsections. Let x :=[ E , ρ ] ∈ ˆ K ( B ), and V → B be a splitting bundle for x with bundle of flags p : F ( V ) → B . We choose a geometry V := ( V, h V , ∇ V ) and let V denote the associated geometricfamily (5) . In order to avoid stabilizations we can and will always assume that E hasa non-zero dimensional component. Then we have p ∗ I ( x ) = X ǫ ∈{± } ,L ∈ I ǫ ǫI ([ L , . We define F := F B,ǫ ∈{± } ,L ∈ I ǫ L ǫ . Then we can find a taming ( p ∗ E ⊔ F ( V ) F op ) t , and p ∗ x = X ǫ ∈{± } ,L ∈ I ǫ ǫ ([ L , − a ( p ∗ ρ − η (( p ∗ E ⊔ F ( V ) F op ) t )) . (5) It was suggested by the referee that one should use the Chern character ˆ ch ( V ) ∈ ˆ H ev ( B, Q )constructed in [ CS85 ]. The Ansatz would beˆ ch ( x ) := ˆ ch ( V ) + η (( E ⊔ B V op ) t ) . In order to show that this is independent of the choice of V one would need to show an equation likeˆ ch ( V ) − ˆ ch ( V ′ ) = a ( η (( V op ⊔ V ′ ) t )) . Since after all we know that the Chern character exists this equation is true, but we do not know a simple direct proof. Therefore we opted for the variant to give a complete and independent proof. ULRICH BUNKE & THOMAS SCHICK
We now set p ∗ ˆ ch ( x ) = ˆ ch ( p ∗ x ) := X ǫ ∈{± } ,L ∈ I ǫ ǫ exp(ˆ c Q ( L )) + a ( η (( p ∗ E ⊔ F ( V ) F op ) t )) − a ( p ∗ ρ ) . This construction a priori depends on the choices of the representative of x , thesplitting bundle V → B , and the taming ( E ⊔ F ( V ) F op ) t . In this paragraph we show that this construction is independent of thechoices.
Proposition 6.7 . —
Assume that there exists a class z ∈ ˆ H ev ( B, Q ) such that p ∗ z = X ǫ ∈{± } ,L ∈ I ǫ ǫ exp(ˆ c Q ( L )) + a ( η (( p ∗ E ⊔ F ( V ) F op ) t )) − a ( p ∗ ρ ) for one set of choices. Then z is determined by x ∈ ˆ K ( B ) .Proof . — If ( E ′ , ρ ′ ) is another representative of x , then we have index ( E ) = index ( E ′ ). Therefore we can take the same splitting bundle for E ′ . The followingLemma (together with Lemma 6.5) shows that z does not depend on the choice ofthe representative of x . Lemma 6.8 . —
We have a ( η (( p ∗ E ⊔ F ( V ) F op ) t ) − p ∗ ρ ) = a ( η (( p ∗ E ′ ⊔ F ( V ) F op ) t ) − p ∗ ρ ′ ) Proof . — In fact, by Lemma 2.21 there is a taming ( E ′ ∪ E op ) t such that ρ ′ − ρ = η (( E ′ ∪ E op ) t ). Therefore the assertion is equivalent to a (cid:2) η (cid:0) ( p ∗ E ⊔ F ( V ) F op ) t (cid:1) − η (cid:0) ( p ∗ E ′ ⊔ F ( V ) F op ) t (cid:1) + p ∗ η (cid:0) ( E ′ ⊔ F ( V ) E op ) t (cid:1)(cid:3) = 0 . But this is true since this sum of η -forms represents a rational cohomology class ofthe form ch dR ( ξ ). This follows from 2.4.10 and the fact p ∗ E ⊔ F ( V ) F op ⊔ F ( V ) p ∗ E ′ op ⊔ F ( V ) F ⊔ F ( V ) p ∗ E ′ ⊔ F ( V ) p ∗ E op admits another taming with vanishing η -form (as in the proof of Lemma 2.11). Next we discuss what happens if we vary the splitting bundle. Thus let V ′ → B be another Z / Z -graded bundle which represents index ( E ). Let p ′ : F ( V ′ ) → B be the associated splitting bundle. Lemma 6.9 . —
Assume that we have classes c, c ′ ∈ ˆ H ( B, Q ) such that p ∗ c = X ǫ ∈{± } ,L ∈ I ǫ ǫ exp(ˆ c Q ( L )) + a (cid:0) η (cid:0) ( p ∗ E ⊔ F ( V ) F op ) t (cid:1) − p ∗ ρ (cid:1) and p ′∗ c ′ = X ǫ ∈{± } ,L ∈ I ′ ǫ ǫ exp(ˆ c Q ( L ′ )) + a (cid:0) η (cid:0) ( p ′∗ E ⊔ F ( V ′ ) F ′ op ) t (cid:1) − p ′∗ ρ (cid:1) . Then we have c = c ′ . MOOTH K-THEORY Proof . — Note that the right-hand sides depend on the geometric bundles V , V ′ since they depend on the induced connections on the line bundle summands. We firstdiscuss a special case, namely that V ′ is obtained from V by stabilization, i.e. V ′ = V ⊕ B × ( C m ⊕ ( C m ) op ). In this case there is a natural embedding i : F ( V ) ֒ → F ( V ′ )which is induced by extension of the flags in V by the standard flag in C m . We canfactor p = p ′ ◦ i . Furthermore, there exists subsets S ǫ ⊂ I ′ ǫ of line bundles (the last m line bundles in the natural order) and a natural bijection I ′ ǫ ∼ = I ǫ ⊔ S ǫ . If L ∈ S ǫ ,then i ∗ L is trivial with the trivial connection. We thus have p ∗ ( c ′ − c ) = a [ i ∗ η (( p ′∗ E ∪ F ′ op ) t ) − η (( p ∗ E ∪ F op ) t )]It is again easy to see that this difference of η -forms represents a rational cohomologyclass in the image of ch dR . Therefore, p ∗ ( c ′ − c ) = 0 and hence c = c ′ by Lemma 6.5.Since the bundle V represents the index of E , two choices are always stably isomor-phic as hermitean bundles. Using the special case above we can reduce to the casewhere V and V ′ only differ by the connection.We argue as follows. We have p ∗ R ( c ′ − c ) = R ( p ∗ ( c ′ − c )) = 0 by an explicitcomputation. Therefore c ′ − c ∈ H odd ( B, R / Q ). Since any two connections on V canbe connected by a family we conclude that p ∗ ( c ′ − c ) = 0 by a homotopy argument.The assertion now follows.This finishes the proof of Proposition 6.7. In order to finish the construction of the Chern character in the even caseit remains to verify the existence clause in Proposition 6.7. Let x := [ E , ρ ] ∈ ˆ K ( B ) besuch that E has a non-zero dimensional component. Let V → B be a splitting bundleand p : F ( V ) → B be as above. Lemma 6.10 . —
We have z := X ǫ ∈{± } ,L ∈ I ǫ ǫ exp(ˆ c Q ( L )) + a [ η (( p ∗ E ∪ F op ) t ) − p ∗ ρ ] ∈ im ( p ∗ ) . Proof . — We use a Mayer-Vietoris sequence argument. Let us first recall the Mayer-Vietoris sequence for smooth rational cohomology. Let B = U ∪ V be an open coveringof B . Then we have the exact sequence · · · → H ( U ∩ V, R / Q ) → ˆ H ( B, Q ) → ˆ H ( U, Q ) ⊕ ˆ H ( V, Q ) → ˆ H ( U ∩ V, Q ) → H ( B, Q ) → . . . which continues to the left and right by the Mayer-Vietoris sequences of H ( . . . , R / Q )and H ( . . . , Q ).We choose a finite covering of B by contractible subsets. Let U be one of these.Note that index ( E ) | U ∈ Z . Thus x | U = [ U × W, θ ] for some form θ and Z / Z -gradedvector space W . Then we have by 1. and 3. that c U : = ˆ ch ( x | U ) = dim( W ) − a ( θ ).This can be seen using the splitting bundle F ( B × C n ). Moreover, p ∗ c U = p ∗ [dim( W ) − a ( θ )] = z | p − U by Proposition 6.7. ULRICH BUNKE & THOMAS SCHICK
Assume now that we have already constructed c V ∈ ˆ H ( V, Q ) such that p ∗ c V = z | p − V , where V is a union V of these subsets. Let U be the next one in the list.We show that we can extend c V to c V ∪ U . We have ( c U ) | U ∩ V = ( c V ) | U ∩ V by theinjectivity of the pull-back p ∗ : ˆ H ( U ∩ V, Q ) → ˆ H ( p − ( U ∩ V ) , Q ), Lemma 6.5. TheMayer-Vietoris sequence implies that we can extend c V by c U to U ∪ V . We now construct the odd part of the Chern character. In fact, by (31)and (32) we are forced to defineˆ ch : ˆ K ( B ) → ˆ H odd ( B, Q )by ˆ ch ( x ) := ( ˆ pr ) ! ( ˆ ch ( x S ∪ x )) . Lemma 6.11 . —
The diagrams (30) and (31) commute.Proof . — The even case of (30) has been checked already. The diagram (31) com-mutes by construction. The odd case of (30) follows from the Projection formula 4.5and the even case.This finishes the proof of Theorem 6.2
Note that ˆ H ( B, Q ) is a Q -vector space, and that the sequence (1) is anexact sequence of Q -vector spaces. The Chern character extends to a rational versionˆ ch Q : ˆ K Q ( B ) → ˆ H ( B, Q ) , where ˆ K Q ( B ) := ˆ K ( B ) ⊗ Z Q . Proposition 6.12 . — ˆ ch Q : ˆ K Q ( B ) → ˆ H ( B, Q ) is an isomorphism.Proof . — By (30) we have the commutative diagram K Q ( B ) ch Q (cid:15) (cid:15) ch dR / / Ω( B ) / im ( d ) a / / ˆ K Q ( B ) ˆ ch Q (cid:15) (cid:15) I / / K Q ( B ) ch Q (cid:15) (cid:15) / / H ( B, Q ) / / Ω( B ) / im ( d ) / / ˆ H ( B, Q ) I / / H ( B, Q ) / / , whose horizontal sequences are exact. Since ch Q : K Q ( B ) → H ( B, Q ) is an isomor-phism we conclude that ˆ ch Q is an isomorphism by the Five Lemma. MOOTH K-THEORY We can extend ˆ K Q to a smooth cohomology theory if we define the structuremaps as follows:1. R : ˆ K Q ( B ) → Ω d =0 ( B ) is the rational extension of R : ˆ K ( B ) → Ω d =0 ( B ).2. I : ˆ K Q ( B ) I ⊗ id Q → K ( B ) Q ch Q → H ( B, Q ),3. a : Ω( B ) / im ( d ) a → ˆ K ( B ) ···⊗ → ˆ K Q ( B ).The commutative diagrams (30) now imply: Corollary 6.13 . —
The rational Chern character induces an isomorphism of smoothcohomology theories refining the isomorphism ch Q : K Q → H Q (in the sense of Defi-nition 1.3).6.3.3. — Proposition 6.14 . —
The smooth Chern character ˆ ch : ˆ K ( B ) → ˆ H ( B, Q ) is a ring homomorphism.Proof . — Since the target of ˆ ch is a Q -vector space it suffices to show thatˆ ch Q : ˆ K Q ( B ) → ˆ H ( B, Q ) is a ring homomorphism. Using that ˆ ch Q is an isomorphismof smooth extensions of rational cohomology we can use the rational Chern characterin order to transport the product on ˆ K Q ( B ) to a second product ∪ K on ˆ H ( B, Q ). Itremains to show that ∪ and ∪ K coincide. Hence the following Lemma finishes theproof of Proposition 6.14. Lemma 6.15 . —
There is a unique product on smooth rational cohomology.Proof . — Assume that we have two products ∪ k , k = 0 ,
1. We consider the bilineartransformation B : ˆ H ( B, Q ) × ˆ H ( B, Q ) → ˆ H ( B, Q ) given by( x, y ) B ( x, y ) := x ∪ y − x ∪ y . We first consider the curvature. Since a product is compatible with the curvature(1.2, 2.) we get R ( B ( x, y )) = R ( x ∪ y ) − R ( x ∪ y ) = R ( x ) ∧ R ( y ) − R ( x ) ∧ R ( y ) = 0 . Therefore, by (1) the bilinear form factors over an odd transformation B : ˆ H ( B, Q ) × ˆ H ( B, Q ) → H ( B, R / Q ) . Furthermore, for ω ∈ Ω( B ) / im ( d ) we have by 1.2, 2. B ( a ( ω ) , y ) = a ( ω ) ∪ y − a ( ω ) ∪ y = a ( ω ∧ R ( y )) − a ( ω ∧ R ( y )) = 0 . Similarly, B ( x, a ( ω )) = 0. Again by (1) B has a factorization over a natural bilineartransformation ¯ B : H ( B, Q ) × H ( B, Q ) → H ( B, R / Q ) . ULRICH BUNKE & THOMAS SCHICK
We consider the restriction ¯ B p,q of ¯ B to H p ( B, Q ) × H q ( B, Q ).The functor from finite CW -complexes to sets W → H p ( W, Q ) × H q ( W, Q )is represented by a product of Eilenberg MacLane spaces P p,q := H Q p × H Q q . The spaces H Q p , and hence P has the homotopy type of countable CW -complexes.Therefore we can apply Lemma 6.4 and conclude that ¯ B p,q is induced by a cohomologyclass b ∈ H ( P p,q , R / Q ). We finish the proof of Lemma 6.15 by showing that b = 0. Tothis end we analyse the candidates for b and show that they vanish either for degreereasons, or using the fact that ¯ B p,q is bilinear.Consider a homomorphism of Q -vector spaces w : R / Q → Q . It induces a transfor-mation w ∗ : H ( B, R / Q ) → H ( B, Q ). In particular we can consider w ∗ b ∈ H ( P p,q , Q ).1. First of all if p, q are both even, then w ∗ b ∈ H odd ( P p,q, , Q ) vanishes since P p,q does not have odd-degree rational cohomology at all.2. Assume now that p, q are both odd. The odd rational cohomology of P p,q isadditively generated by the classes 1 × x q and x p ×
1, where x p ∈ H p ( H Q p , Q )and x q ∈ H q ( H Q q , Q ). It follows that w ∗ b = c · x p × d · × x q for some rational constants c, d . Consider odd classes u p ∈ H p ( B, Q ) and v q ∈ H q ( B, Q ). The form of b implies that w ∗ ◦ ¯ B p,q ( u p , v q ) = c · u p × d · × v q . This can only be bilinear if all c and d vanish. Hence b = 0.3. Finally we consider the case that p is even and q is odd (or vice versa, q is evenand p is odd). In this case b is an even class. The even cohomology of P p,q isadditively generated by the classes x np × n ≥
0. Therefore w ∗ b = P n ≥ c n x np × c n , n ≥
0. Let u p ∈ H p ( B, Q ) and v q ∈ H q ( B, Q ).Then we have w ∗ ◦ ¯ B p,q ( u p , v q ) = X n ≥ c n u np . This is only bilinear if c n = 0 for all n ≥
0, hence w ∗ b = 0.Since we can choose w ∗ : R / Q → Q arbitrary we conclude that b = 0.This also finishes the proof of the Proposition 6.14. MOOTH K-THEORY Let p : W → B be a proper submersion with a smooth K -orientation o . TheRiemann Roch theorem asserts the commutativity of a diagramˆ K ( W ) ˆ ch −−−−→ ˆ H ( W, Q ) y p ! y ˆ p A ! ˆ K ( B ) ˆ ch −−−−→ ˆ H ( B, Q ) . Here ˆ p A ! is the composition of the cup product with a smooth rational cohomologyclass ˆˆ A c ( o ) and the push-forward in smooth rational cohomology. The RiemannRoch theorem refines the characteristic class version of the ordinary index theoremfor families.We will first give the details of the definition of the push-forward ˆ p A ! . In order toshow the Riemann Roch theorem we then show that the difference∆ := ˆ ch ◦ ˆ p ! − ˆ p A ! ◦ ˆ ch vanishes.This is proved in several steps. First we use the compatibilites of the push-forwardwith the transformations a, I, R in order to show that ∆ factors over a map¯∆ : K ( W ) → H ( B, R / Q ) . In the next step we show that ∆ is natural with respect to the pull-back of fibrebundles, and that it does neither depend on the smooth nor on the topological K -orientations of p .We then show that ∆ vanishes in the special case that B = ∗ . The argument isbased on the bordism invariance Proposition 5.18 and some calculation of rational Spin c -bordism groups.Finally we use the functoriality of the push-forward Proposition 3.23 in order toreduce the case of a general B to the special case of a point. We consider a proper submersion p : W → B with closed fibres with asmooth K -orientation represented by o = ( g T v p , T h p, ˜ ∇ , σ ). In the following we definea refinement ˆˆ A ( o ) ∈ ˆ H ev ( W, Q ) of the form ˆ A c ( o ) ∈ Ω ev ( W ). The geometric dataof o determines a connection ∇ T v p (see 2.2.4, 3.1.3) and hence a geometric bundle T v p := ( T v p, g T v p , ∇ T v p ). According to [ CS85 ] we can define Pontrjagin classesˆ p i ( T v p ) ∈ ˆ H i ( W, Z ) , i ≥ . The
Spin c -structure gives rise to a hermitean line bundle L → W with connection ∇ L (see 3.1.6). A choice of a local spin structure amounts to a choice of a localsquare root L of L (this bundle was considered already in 3.1.3) such that S c ( T v p ) ∼ = S ( T v p ) ⊗ L as hermitean bundles with connections. We set L := ( L , h L , ∇ L ). Inparticular, we have 12 πi R ˜ ∇ L = 2 c ( ˜ ∇ ) . ULRICH BUNKE & THOMAS SCHICK
Again using [
CS85 ] we get a classˆ c ( L ) ∈ ˆ H ( W, Z )with curvature R (ˆ c ( L )) = 2 c ( ˜ ∇ ). Inserting the classes ˆ p i ( T v p ) into that ˆ A -series ˆ A ( p , p , . . . ) ∈ Q [[ p , p . . . ]]we can define ˆˆ A ( T v p ) := ˆ A (ˆ p ( T v p ) , ˆ p ( T v p ) , . . . ) ∈ ˆ H ev ( W, Q ) . (33)Let ˆ c Q ( L ) ∈ ˆ H ( W, Q ) denote the image of ˆ c ( L ) under the natural mapˆ H ( W, Z ) → ˆ H ( W, Q ). Definition 6.16 . —
We define ˆˆ A c ( o ) := ˆˆ A ( T v p ) ∧ e ˆ c Q ( L ) ∈ ˆ H ev ( W, Q ) . Note that R ( ˆˆ A c ( o )) = ˆ A c ( o ). Lemma 6.17 . —
The class (6) ˆˆ A c ( o ) − a ( σ ( o )) ∈ ˆ H ev ( W, Q ) only depends on the smooth K -orientation represented by o .Proof . — This is a consequence of the homotopy formula Lemma 2.22. Given tworepresentatives o , o of a smooth K -orientation we can choose a representative ˜ o of asmooth K -orientation on id R × p : R × W → R × B which restricts to o k on { k } × B , k = 0 ,
1. The construction of the class ˆˆ A c ( o ) is compatible with pull-back. Thereforeby the definition of the transgression form 3.4 we haveˆˆ A c ( o ) − ˆˆ A c ( o ) = i ∗ ˆˆ A c (˜ o ) − i ∗ ˆˆ A c (˜ o ) = a "Z [0 , × W/W R ( ˆˆ A c (˜ o )) = a h ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) i . By the definition of equivalence of representatives of smooth K -orientations we have σ ( o ) − σ ( o ) = ˜ˆ A c ( ˜ ∇ , ˜ ∇ ) . Therefore ˆˆ A c ( o ) − a ( σ ( o )) = ˆˆ A c ( o ) − a ( σ ( o )) . (6) This class is denoted by A ( p ) in the abstract and 1.1.9. MOOTH K-THEORY We use the class ˆˆ A c ( o ) ∈ ˆ H ev ( W, Q ) in order to define the push-forwardˆ p A ! := ˆ p ! ([ ˆˆ A c ( o ) − a ( σ ( o ))] ∪ . . . ) : ˆ H ( W, Q ) → ˆ H ( B, Q ) , (34)where ˆ p ! : ˆ H ( W, Q ) → ˆ H ( B, Q ) is the push-forward in smooth rational cohomology(see 6.1.1) fixed by the underlying ordinary orientation of p . By Lemma 6.17 alsoˆ p A ! only depends to the smooth K -orientation of p and not on the choice of therepresentative.If f : B ′ → B is a smooth map then we consider the pull-back diagram W ′ p ′ (cid:15) (cid:15) F / / W p (cid:15) (cid:15) B ′ f / / B .
The smooth K -orientation o of p induces (see 3.2.4) a smooth K -orientation o ′ of p ′ .We have ˆˆ A ( o ′ ) = F ∗ ˆˆ A ( o ) and ˆ p ′ A ! ◦ F ∗ = f ∗ ◦ ˆ p A ! . As in 3.3.3 we consider the composition of proper smoothly K -orientedsubmersions W q p / / B r / / A .
The composition q := r ◦ p has an induced smooth K -orientation (Definition 3.21and Lemma 3.22). In this situation we have push-forwards ˆ p A ! , ˆ r A ! and ˆ q A ! in smoothrational cohomology given by (34). Lemma 6.18 . —
We have the equality ˆ r A ! ◦ ˆ p A ! = ˆ q A ! of maps ˆ H ( W, Q ) → ˆ H ( B, Q ) .Proof . — We choose representatives of smooth K -orientations o p of p and o r of r ,and we let o λq := o p ◦ λ o r be the composition. We consider the class (see Definition3.21)ˆˆ A c ( o λq ) − a ( σ ( o λq ))= ˆˆ A c ( o λq ) − a (cid:16) σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ λq ) − dσ ( o p ) ∧ p ∗ σ ( o r ) (cid:17) . By Lemma 6.17 and Lemma 3.22 this class is independent of λ . If we let λ →
0, thenthe connection ∇ T v q tends to the direct sum connection ∇ T v p ⊕ p ∗ ∇ T v r . Furthermore, ULRICH BUNKE & THOMAS SCHICK the transgression ˜ˆ A c ( ˜ ∇ adia , ˜ ∇ λq ) tends to zero. Therefore lim λ → [ ˆˆ A c ( o λq ) − a ( σ ( o λq ))]= ˆˆ A c ( o p ) ∪ p ∗ ˆˆ A c ( o r ) − a (cid:16) σ ( o p ) ∧ p ∗ ˆ A c ( o r ) + ˆ A c ( o p ) ∧ p ∗ σ ( o r ) − dσ ( o p ) ∧ p ∗ σ ( o r ) (cid:17) = ( ˆˆ A c ( o p ) − a ( σ ( o p ))) ∪ p ∗ ( ˆˆ A c ( o r ) − a ( σ ( o r ))) . For x ∈ ˆ H ( W, Q ) we get using the projection formula and the functorialty ˆ q ! = ˆ r ! ◦ ˆ p ! for the push-forward in smooth rational cohomologyˆ r A ! ◦ ˆ p A ! ( x ) = ˆ r ! (cid:16)h ˆˆ A c ( o r ) − a ( σ ( o r )) i ∪ ˆ p ! (cid:16)h ˆˆ A c ( o p ) − a ( σ ( o p )) i ∪ x (cid:17)(cid:17) = ˆ q ! (cid:16) p ∗ h ˆˆ A c ( o r ) − a ( σ ( o r )) i ∪ h ˆˆ A c ( o p ) − a ( σ ( o p )) i ∪ x (cid:17) = ˆ q ! (cid:16) ( ˆˆ A c ( o aq ) − a ( σ ( o aq ))) ∪ x (cid:17) = ˆ q A ! ( x ) . Recall Definition 3.18 that the smooth K -orientation determines a push-down ˆ p ! : ˆ K ( W ) → ˆ K ( B ) . We can now formulate the index theorem.
Theorem 6.19 . —
The following square commutes ˆ K ( W ) ˆ ch −−−−→ ˆ H ( W, Q ) y ˆ p ! y ˆ p A ! ˆ K ( B ) ˆ ch −−−−→ ˆ H ( B, Q ) . Proof . — We consider the difference∆ := ˆ ch ◦ ˆ p ! − ˆ p A ! ◦ ˆ ch . It suffices to show that ∆ = 0.
Let x ∈ ˆ K ( W ). Lemma 6.20 . —
We have R (∆( x )) = 0 .Proof . — This Lemma is essentially equivalent to the local index theorem. We haveby Definition 3.15 and Lemma 3.16 R ( ˆ ch ◦ ˆ p ! ( x )) = R (ˆ p ! ( x )) = p ! ( R ( x )) = Z W/B (cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ R ( x ) . MOOTH K-THEORY On the other hand, since R (cid:16) ˆˆ A c ( o ) − a ( σ ( o )) (cid:17) = ˆ A c ( o ) − dσ ( o ) we get R (cid:16) ˆ p A ! ◦ ˆ ch ( x ) (cid:17) = Z W/B (cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ R ( ˆ ch ( x )) = Z W/B (cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ R ( x ) . Therefore R (∆( x )) = 0. Lemma 6.21 . —
We have I (∆( x )) = 0 Proof . — This is the usual index theorem. Indeed, I ( ˆ ch ◦ ˆ p ! ( x )) = ch ◦ I (ˆ p ! ( x )) = Z W/B ˆ A c ( T v p ) ∪ ch ( I ( x ))and I (cid:16) ˆ p A ! ◦ ˆ ch ( x ) (cid:17) = Z W/B ˆ A c ( T v p ) ∪ I ( ˆ ch ( x )) = Z W/B ˆ A c ( T v p ) ∪ ch ( I ( x )) . The equality of the right-hand sides proves the Lemma. Alternatively one couldobserve that the Lemma is a consequence of Lemma 6.20.
Let ω ∈ Ω( W ) / im ( d ). Lemma 6.22 . —
We have ∆( a ( ω )) = 0 .Proof . — We have by Proposition 3.19ˆ ch ◦ ˆ p ! ( a ( ω )) = ˆ ch ◦ a ( p ! ( ω )) = a Z W/B (cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ ω ! . On the other hand, by (30) and h ˆˆ A c ( o ) − a ( σ ( o )) i ∪ a ( ω ) = a (cid:16) R (cid:16) ˆˆ A ( o ) − a ( σ ( o )) (cid:17) ∧ ω (cid:17) = a (cid:16)(cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ ω (cid:17) , ˆ p A ! ◦ ˆ ch ( a ( ω )) = ˆ p A ! ( a ( ω )) = a Z W/B (cid:16) ˆ A c ( o ) − dσ ( o ) (cid:17) ∧ ω ! . Let o , o represents two smooth refinements of the same topological K -orientation of p . Assume that ∆ k is defined with the choice o k , k = 0 , Lemma 6.23 . —
We have ∆ = ∆ . ULRICH BUNKE & THOMAS SCHICK
Proof . — We can assume that o k = ( g T v p , T h p, ˜ ∇ , σ k ) for σ k ∈ Ω odd ( W ) / im ( d ).Then we have for x ∈ ˆ K ( W )∆ ( x ) − ∆ ( x ) = − a Z W/B ( σ − σ ) ∧ R ( x ) ! + Z W/B a ( σ − σ ) ∪ ˆ ch ( x )= − a Z W/B ( σ − σ ) ∧ R ( x ) ! + Z W/B a h ( σ − σ ) ∧ R ◦ ˆ ch ( x ) i = 0since R ◦ ˆ ch ( x ) = R ( x ) and a ◦ R W/B = R W/B ◦ a . It follows from Lemma 6.20 and (1) that ∆ factorizes through a transfor-mation ∆ : ˆ K ( W ) → H ( B, R / Q ) . By Lemma 6.22 and 2.20 the map ∆ factors over a map¯∆ : K ( W ) → H ( B, R / Q ) . This map only depends on the topological K -orientation of p . It is our goal to showthat ¯∆ = 0. Next we want to show that the transformation ¯∆ is natural. For themoment we write ∆ p := ¯∆. Let f : B ′ → B be a smooth map and form the cartesiandiagram W ′ p ′ (cid:15) (cid:15) F / / W p (cid:15) (cid:15) B ′ f / / B .
The map p ′ is a proper submersion with closed fibres which has an induced topological K -orientation. Lemma 6.24 . —
We have the equality of maps K ( W ) → H ( B ′ , R / Q )∆ p ′ ◦ F ∗ = f ∗ ◦ ∆ p . Proof . — This follows from the naturality of ˆ ch , ˆ p ! , and ˆ p A ! with respect to the base B . Lemma 6.25 . — If pr : S × B → B is the trivial bundle with the topological K -orientation given by the bounding spin structure, then ∆ pr : K ( S × B ) → H odd ( B, R / Q ) vanishes. MOOTH K-THEORY Proof . — The odd Chern character is defined such that for x ∈ K ( S × B ) we haveˆ ch (( ˆ pr ) ! x ) = ( ˆ pr ) ! ˆ ch ( x ) (see (31)). With the choice of the smooth K -orientationof pr given in 4.3.2 we have ˆˆ A ( o ) − a ( σ ( o )) = 1 so that ˆ p A ! = ˆ p ! . This implies theLemma. The group H ( W, Z ) acts simply transitive on the set of Spin c -structures of T v p . Let Q → W be a unitary line bundle classified by c ( Q ) ∈ H ( W, Z ). We choosea hermitean connection ∇ Q and form the geometric line bundle Q := ( Q, h Q , ∇ Q ).Let o := ( T v p, T h p, ˜ ∇ , ρ ) represent a smooth K -orientation refining the given topo-logical K -orientation of p . Note that ˜ ∇ is completely determined by the Cliffordconnection on the Spinor bundle S c ( T v p ). The spinor bundle of the shift of the topo-logical K -orientation by c ( Q ) is given by S c ( T v p ) ′ = S c ( T v p ) ⊗ Q . We constructa corresponding smooth K -orientation o ′ = ( T v p, T h p, ˜ ∇ ⊗ ∇ Q , ρ ). We let ˆ p ! and ˆ p ′ ! denote the corresponding push-forwards in smooth K -theory. Let Q be the geometricfamily over W with zero-dimensional fibre given by the bundle Q (see 2.1.4). Thepush-forwards ˆ p ! and ˆ p ′ ! are now related as follows: Lemma 6.26 . — ˆ p ′ ! ( x ) = ˆ p ! ([ Q , ∪ x ) , ∀ x ∈ ˆ K ( W ) . Proof . — Let x = [ E , ρ ]. By an inspection of the constructions leading to Definition3.7 we see that p ′ λ ! E = p λ ! ( Q × W E ) . Furthermore we have c ( ˜ ∇ ⊗ ∇ Q ) = c ( ˜ ∇ ) + c ( ∇ Q ) so thatˆ A c ( o ′ ) = ˆ A c ( o ) ∧ e c ( ∇ Q ) . On the other hand, since Ω( Q ) = e c ( ∇ Q ) we have[ Q , ∪ [ E , ρ ] = [ Q × W E , e c ( ∇ Q ) ∧ ρ ]Using the explicit formula (17) we getˆ p ′ ! ([ E , ρ ]) − ˆ p ! ([ Q , ∪ [ E , ρ ]) = [ ∅ , ˜Ω ′ ( λ, E ) − ˜Ω( λ, E )]for all small λ >
0. Since both transgression forms vanish in the limit λ = 0 we getthe desired result.In the notation of 6.4.2 we have L ′ = L ⊗ Q . Thereforeˆ c Q ( L ′ ) = ˆ c Q ( L ) + 2ˆ c Q ( Q )and hence we can express ˆ p ′ ,A ! according to (34) asˆ p ′ A ! ( x ) = ˆ p ! h(cid:16) ˆˆ A c ( o ) ∪ e ˆ c Q ( Q ) − a ( σ ( o )) (cid:17) ∪ x i . ULRICH BUNKE & THOMAS SCHICK
As before, let p : W → B be a proper oriented submersion which admitstopological K -orientations. Lemma 6.27 . — If ∆ p = 0 for some topological K -orientation of p , then it vanishesfor every topological K -orientation of p .Proof . — We fix the K -orientation of p such that ∆ p = 0 and let p ′ denote the samemap with the topological K -orientation shifted by c ( Q ) ∈ H ( W, Z ). We continue touse the notation of 6.4.14. We choose a representative o of a smooth K -orientation of p refining the topological K -orientation. For simplicity we take σ ( o ) = 0. Furthermore,we take o ′ as above. Using ˆ ch ([ Q , e ˆ c Q ( Q ) and the multiplicativity of the Cherncharacter we getˆ p ′ A ! ◦ ˆ ch ( x ) − ˆ ch ◦ ˆ p ′ ! ( x ) = ˆ p ! h ˆˆ A c ( o ) ∪ e ˆ c Q ( Q ) ∪ ˆ ch ( x ) i − ˆ ch ◦ ˆ p ! ([ Q , ∪ x )= ˆ p ! h ˆˆ A c ( o ) ∪ ˆ ch ([ Q , ∪ ˆ ch ( x ) i − ˆ p A ! ◦ ˆ ch ([ Q , ∪ x )= ˆ p A ! ◦ ˆ ch ([ Q , ∪ x ) − ˆ p A ! ◦ ˆ ch ([ Q , ∪ x )= 0 . We now consider the special case that B = ∗ and W is an odd-dimensional Spin c -manifold. Since H ( ∗ , R / Q ) ∼ = R / Q we get a homomorphism∆ p : K ( W ) → R / Q . Proposition 6.28 . — If B ∼ = ∗ , then ∆ p = 0 Proof . — First note that ∆ p is trivial on K ( W ) for degree reasons. It thereforesuffices to study ∆ p : K ( W ) → R / Q . Let x ∈ K ( W ) be classified by ξ : W → Z × BU . It gives rise to an element [ ξ ] ∈ Ω Spin c dim( W ) ( Z × BU ) of the Spin c -bordismgroup of Z × BU . Lemma 6.29 . — If [ ξ ] = 0 , then ∆ p = 0 .Proof . — Assume that [ ξ ] = 0. In this case there exists a compact Spin c -manifold V with boundary ∂V ∼ = W (as Spin c -manifolds), and a map ν : V → Z × BU such that ν | ∂V = ξ .We can choose a Z / Z -graded vector bundle E → V which represents the classof ν in K ( V ). We refine E to a geometric bundle E := ( E, h E , ∇ E ) and form theassociated geometric family E with zero-dimensional fibre.We choose a representative ˜ o of a smooth K -orientation of the map q : V → ∗ which refines the topological K -orientation given by the Spin c -structure and whichhas a product structure near the boundary. For simplicity we assume that σ (˜ o ) = 0.The restriction of ˜ o to the boundary ∂V defines a smooth K -orientation of p . MOOTH K-THEORY We let ˆ y := [ E , ∈ ˆ K ( V ), and we define ˆ x := ˆ y | ∂V such that I (ˆ x ) = x . ByProposition 5.18 we haveˆ ch ◦ ˆ p ! (ˆ x ) = ˆ ch ◦ ˆ p ! (ˆ y | W ) = ˆ ch ([ ∅ , q ! ( R (ˆ y ))]) = − a (cid:18)Z V ˆ A c (˜ o ) ∧ R (ˆ y ) (cid:19) . On the other hand, the bordism formula for the push-forward in smooth rationalcohomology, Lemma 6.1, givesˆ p A ! ◦ ˆ ch (ˆ x ) = ˆ p ! (cid:16) ˆˆ A c ( o ) ∪ ˆ ch (ˆ x ) (cid:17) = ˆ p ! (cid:16) ˆˆ A c (˜ o ) | W ∪ ˆ ch (ˆ y ) | W (cid:17) = − a (cid:18)Z V ˆ A c (˜ o ) ∧ R (ˆ y ) (cid:19) . These two formulas imply that ∆ p = 0. We now finish the proof of Proposition 6.28. We claim that there exists c ∈ N such that c [ ξ ] = 0. In view of Lemma 6.29 we then have0 = ∆ cp = c ∆ p , and this implies the Proposition since the target R / Q of ∆ p is a Q -vector space.Note that the graded ring Ω Spin c ∗ ⊗ Q is concentrated in even degrees. Using thatΩ SO ∗ ⊗ Q is concentrated in even degrees, one can see this as follows. In [ Sto68 ,p. 352] it is shown that the homomorphism
Spin c → U (1) × SO induces an injectionΩ Spin c ∗ → Ω SO ∗ ( BU (1)). Since H ∗ ( BU (1) , Z ) ∼ = Z [ z ] with deg( z ) = 2 lives in evendegrees, we see using the Atiyah-Hirzebruch spectral sequence that Ω SO ( BU (1)) ⊗ Q lives in even degrees, too. This implies that Ω Spin c ∗ ⊗ Q is concentrated in even degrees.Since H ∗ ( Z × BU, Z ) is also concentrated in even degrees it follows again from theAtiyah-Hirzebruch spectral sequence that Ω Spin c ∗ ( Z × BU ) ⊗ Q is concentrated in evendegrees.Since [ ξ ] is of odd degree we conclude the claim that c [ ξ ] = 0 for an appropriate c ∈ N .This finishes the proof of Proposition 6.28. We now consider the general case. Let p : W → B be a proper submersionwith closed fibres with a topological K -orientation. Proposition 6.30 . —
We have ∆ p = 0 . We give the proof in the next couple of subsections.
For a closed oriented manifold Z let PD : H ∗ ( Z, Q ) ∼ → H ∗ ( Z, Q ) denote thePoincar´e duality isomorphism. Lemma 6.31 . —
The group H ∗ ( B, Q ) is generated by classes of the form f ∗ (cid:16) PD ( ˆ A c ( T Z )) (cid:17) ,where Z is a closed Spin c -manifold and f : Z → B . ULRICH BUNKE & THOMAS SCHICK
Proof . — We consider the sequence of transformations of homology theoriesΩ
Spin c ∗ ( B ) α → K ∗ ( B ) β → H ∗ ( B, Q ) . The transformation α is the K -orientation of the Spin c -cobordism theory, and β is thehomological Chern character. We consider all groups as Z / Z -graded. The homolog-ical Chern character is a rational isomorphism. Furthermore one knows by [ BD82 ],[
BHS ] that Ω
Spin c ∗ ( B ) α → K ∗ ( B ) is surjective. It follows that the composition β ◦ α : Ω Spin c ( B ) ⊗ Q → H ∗ ( B, Q )is surjective. An explicit description of β ◦ α is given as follows. Let x ∈ Ω Spin c ( B )be represented by a map f : Z → B from a closed Spin c -manifold Z to B . Let PD : H ∗ ( Z, Q ) ∼ → H ∗ ( Z, Q ) denote the Poincar´e duality isomorphism. Then we have β ◦ α ( x ) = f ∗ (cid:16) PD ( ˆ A c ( T Z )) (cid:17) . For the proof of Proposition 6.30 we first consider the case that p haseven-dimensional fibres, and that x ∈ K ( W ). By Lemma 6.31, in order to show that∆ p ( x ) = 0, it suffices to show that all evaluations ∆ p ( x ) (cid:16) f ∗ ( PD ( ˆ A c ( T Z ))) (cid:17) vanish.In the following, if x denotes a K -theory class, then ˆ x denotes a smooth K -theoryclass such that I (ˆ x ) = x .We choose a representative o q of a smooth K -orientation which refines the topo-logical K -orientation of the map q : Z → ∗ induced by the Spin c -structure on T Z .Furthermore, we consider the diagram with a cartesian square V s $ $ r (cid:15) (cid:15) F / / W p (cid:15) (cid:15) Z q (cid:15) (cid:15) f / / B ∗ . MOOTH K-THEORY In the present case ∆ p ( x ) ∈ H odd ( B, R / Q ), and we can assume that Z is odd-dimensional. We calculate∆ p ( x ) (cid:16) f ∗ ( PD ( ˆ A c ( T Z ))) (cid:17) = f ∗ ∆ p ( x ) (cid:16) PD ( ˆ A c ( T Z )) (cid:17) Lemma . = ∆ r ( F ∗ x ) (cid:16) PD ( ˆ A c ( T Z )) (cid:17) = ( ˆ A c ( ∇ T Z ) ∪ ∆ r ( F ∗ x ))[ Z ]= Z Z ˆ A c ( o ) ∧ ∆ r ( F ∗ x )= ˆ q ! (cid:16) ˆˆ A c ( o q ) ∪ ∆ r ( F ∗ x ) (cid:17) = ˆ q A ! (∆ r ( F ∗ ˆ x ))= ˆ q A ! h ˆ ch ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ r A ! ◦ ˆ ch ( F ∗ ˆ x ) i = ˆ q A ! ◦ ˆ ch ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ ch ( F ∗ ˆ x ) P roposition . = ˆ ch ◦ ˆ q ! ◦ ˆ r ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ ch ( F ∗ ˆ x )= ˆ ch ◦ ˆ s ! ( F ∗ ˆ x ) − ˆ s A ! ◦ ˆ ch ( F ∗ ˆ x )= ∆ s ( F ∗ x ) P roposition . = 0 . We thus have shown that0 = ∆ p : K ( W ) → H odd ( B, R / Q )if p has even-dimensional fibres. If p has odd-dimensional fibres and x ∈ K ( W ), then we can choose y ∈ K ( S × W ) such that ( ˆ pr ) ! ( y ) = x . Since p ◦ pr has even-dimensional fibreswe get using the Lemmas 6.18 and 3.23∆ p ( x ) = ˆ ch ◦ ˆ p ! ◦ ( ˆ pr ) ! (ˆ y ) − ˆ p A ! ◦ ˆ ch ◦ ( ˆ pr ) ! (ˆ y ) Lemma . = ˆ ch ◦ ( \ p ◦ pr ) ! (ˆ y ) − ˆ p A ! ◦ ( ˆ pr ) A ! ◦ ˆ ch (ˆ y )= ˆ ch ◦ ( \ p ◦ pr ) ! (ˆ y ) − ( \ p ◦ pr ) A ! ◦ ˆ ch (ˆ y )= ∆ p ◦ pr ( y )= 0 . Therefore 0 = ∆ p : K ( W ) → H odd ( B, R / Q )if p has odd-dimensional fibres. ULRICH BUNKE & THOMAS SCHICK
Let us now consider the case that p has even-dimensional fibres, and that x ∈ K ( W ). In this case we consider the diagram S × W Pr −−−−→ W y t := id S × p y p S × B pr −−−−→ B .
We choose a class y ∈ K ( S × W ) such that ( Pr ) ! ( y ) = x . We further choose asmooth refinement ˆ y ∈ ˆ K ( S × W ) of y and set ˆ x := ( ˆ Pr ) ! (ˆ y ). Then we calculateusing the Lemmas 6.18 and 3.23∆ p ( x ) = ˆ ch ◦ ˆ p ! (ˆ x ) − ˆ p A ! ◦ ˆ ch (ˆ x )= ˆ ch ◦ ˆ p ! ◦ ( ˆ Pr ) ! (ˆ y ) − ˆ p A ! ◦ ˆ ch ◦ ( ˆ Pr ) ! (ˆ y ) Lemma . = ˆ ch ◦ ˆ p ! ◦ ( ˆ Pr ) ! (ˆ y ) − ˆ p A ! ◦ ( ˆ Pr ) A ! ◦ ˆ ch ◦ (ˆ y )= ˆ ch ◦ ( \ p ◦ Pr ) ! (ˆ y ) − ( \ p ◦ Pr ) A ! ◦ ˆ ch (ˆ y )= ˆ ch ◦ ( \ pr ◦ t ) ! (ˆ y ) − ( \ pr ◦ t ) A ! ◦ ˆ ch (ˆ y )= ˆ ch ◦ ˆ pr ◦ ˆ t ! (ˆ y ) − ˆ pr A ◦ ˆ t A ! ◦ ˆ ch (ˆ y ) Lemma . = ( ˆ pr ) A ! h ˆ ch ◦ ˆ t ! (ˆ y ) − ˆ t A ! ◦ ˆ ch (ˆ y ) i = ( ˆ pr ) A ! ◦ ∆ t ( y ) = 0 . Therefore 0 = ∆ p : K ( W ) → H ev ( B, R / Q )if p has even-dimensional fibres. In the final case p has odd-dimensional fibres and x ∈ K ( W ). In this casewe consider the sequence of projections S × S × W pr → S × W pr → W .
We choose a class y ∈ K ( S × S × W ) such that ( pr ◦ pr ) ! ( y ) = x . We furtherchoose a smooth refinement ˆ y ∈ ˆ K ( S × S × W ) of y and set ˆ x := ( \ pr ◦ pr ) ! (ˆ y ).Then we calculate using the already known cases and the Lemmas 6.18 and 3.23,∆ p ( x ) = ˆ ch ◦ ˆ p ! (ˆ x ) − ˆ p A ! ◦ ˆ ch (ˆ x )= ˆ ch ◦ ˆ p ! ◦ ( ˆ pr ) ! ◦ ( ˆ pr ) ! (ˆ y ) − ˆ p A ! ◦ ˆ ch ◦ ( ˆ pr ) ! ◦ ( ˆ pr ) ! (ˆ y )= ˆ ch ◦ ( \ p ◦ pr ) ! ◦ ( ˆ pr ) ! (ˆ y ) − ˆ p A ! ◦ ˆ ch ◦ ( \ pr ◦ pr ) ! (ˆ y )= ( \ p ◦ pr ) A ! ◦ ˆ ch ◦ ( ˆ pr ) ! (ˆ y ) − ˆ p A ! ◦ ( \ pr ◦ pr ) A ! ◦ ˆ ch (ˆ y )= ( \ p ◦ pr ) A ! ◦ ∆ pr (ˆ y ) Lemma . = 0 . This finishes the proof of Theorem 6.19. MOOTH K-THEORY
7. Conclusion
We have now constructed a geometric model for smooth K-theory, built out ofgeometric families of Dirac-type operators. We equipped it with a compatible multi-plicative structure, and we have given an explicit construction of a push-down map forfibre bundles with all the expected properties. For the verification of these propertieswe heavily used local index theory.We presented a collection of natural examples of smooth K-theory classes andshowed in particular that several known secondary analytic-geometric invariants canbe understood in this framework very naturally. This involved also the considerationof bordisms in this framework.Finally, we constructed a smooth lift of the Chern character and proved a smoothversion of the Grothendieck-Riemann-Roch theorem. This also involved certain con-siderations from homotopy theory which are special to K-theory.Important open questions concern the construction of equivariant versions of thistheory, or even better versions which work for orbifolds or similar singular spaces.In a different direction, we have addressed the construction of geometric modelsof smooth bordism theories along similar lines in [
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Ulrich Bunke , NWF I - Mathematik, Johannes-Kepler-Universit¨at Regensburg, 93040 Regensburg,GERMANY • E-mail : [email protected]