An index theorem in differential K-theory
aa r X i v : . [ m a t h . DG ] M a r An Index Theorem in Differential K -Theory D ANIEL
S. F
REED J OHN L OTT
Let π : X Ñ B be a proper submersion with a Riemannian structure. Given adifferential K -theory class on X , we define its analytic and topological indices asdifferential K -theory classes on B . We prove that the two indices are the same. To our teacher Isadore Singer on the occasion of his 85th birthday
Let π : X Ñ B be a proper submersion of relative dimension n . The Atiyah-Singerfamilies index theorem [7] equates the analytic and topological index maps, definedon the topological K -theory of the relative tangent bundle. Suppose that the relativetangent bundle has a spin c -structure. This orients the map π in K -theory, and the indexmaps may be expressed as pushforwards K p X ; Z q Ñ K (cid:1) n p B ; Z q . The topologicalindex map π top (cid:6) , which preceded the index theorem, is due to Atiyah and Hirzebruch[2]. The analytic index map π an (cid:6) is defined in terms of Dirac-type operators as follows.Let E Ñ X be a complex vector bundle representing a class r E s P K p X ; Z q . Choosea Riemannian structure on the relative tangent bundle, a spin c -lift of the resultingLevi-Civita connection, and a connection on E . This geometric data determines afamily of fiberwise Dirac-type operators, parametrized by B . The analytic index π an (cid:6) r E s P K (cid:1) n p B ; Z q is the homotopy class of that family of Fredholm operators; it isindependent of the geometric data. The families index theorem asserts π an (cid:6) (cid:16) π top (cid:6) . Inthe special case when B is a point, one recovers the original integer-valued Atiyah-Singer index theorem [6].The work of Atiyah and Singer has led to many other index theorems. In this paperwe prove a geometric extension of the Atiyah-Singer theorem, in which K -theory isreplaced by differential K -theory. Roughly speaking, differential K -theory combinestopological K -theory with differential forms. We define analytic and topological push-forwards in differential K -theory. The analytic pushforward ind an is constructed using D. S. Freed and J. Lott the Bismut superconnection and local index theory techniques. The topological push-forward ind top is constructed as a refinement of the Atiyah-Hirzebruch pushforward intopological K -theory. Our main result is the following theorem. Theorem
Let π : X Ñ B be a proper submersion of relative dimension n equippedwithaRiemannian structure and adifferential spin c -structure. Then ind an (cid:16) ind top ashomomorphisms q K p X q Ñ q K (cid:1) n p B q on differential K -theory.This theorem provides a topological formula (with differential forms) for geometricinvariants of Dirac-type operators. We illustrate this for the determinant line bundle(Section 8) and the reduced eta-invariant (Section 9).In the remainder of the introduction we describe the theorem and its proof in moredetail. We also give some historical background. K -theory Let u be a formal variable of degree 2 and put R (cid:16) R r u , u (cid:1) s . For any manifold X ,its differential K -theory q K (cid:13)p X q fits into the commutative square(1–1) q K (cid:13)p X q Ω (cid:13)p X ; R q K K (cid:13)p X ; Z q H (cid:13)p X ; R q . The bottom map is the Chern character ch : K (cid:13)p X ; Z q Ñ H p X ; R q(cid:13) ; the formalvariable u encodes Bott periodicity. Also, Ω p X ; R q(cid:13) K denotes the closed differentialforms whose cohomology class lies in the image of the Chern character. The rightvertical map is defined by the de Rham theorem. One can define differential K -theoryby positing that (1–1) be a homotopy pullback square [28], which is the precise sensein which differential K -theory combines topological K -theory with differential forms.We use a geometric model for differential K -theory, defined by generators and relations.A generator E of q K p X q is a quadruple p E , h E , ∇ E , φ q , where E Ñ X is a complexvector bundle, h E is a Hermitian metric, ∇ E is a compatible connection, and φ P Ω p X ; R q(cid:1) { Image p d q . The relations come from short exact sequences of Hermitianvector bundles. There is a similar description of q K (cid:1) p X q , in which p E , h E , ∇ E , φ q isadditionally equipped with a unitary automorphism U E P Aut p E q . One can then useperiodicity to define q K r p X q for any integer r . ifferential index theorem Let π : X Ñ B be a proper submersion. A Riemannian structure on π consists ofan inner product on the vertical tangent bundle and a horizontal distribution on X .A differential spin c -structure on π is a topological spin c -structure together with aunitary connection on the characteristic line bundle associated to the spin c -structure.This geometric data determines a local index form Todd p X { B q P Ω p X ; R q , the spin c -version of the ˆ A -form.Suppose first that the fibers of π have even dimension n . Then there is a diagram(1–2) 0 K (cid:1) p X ; R { Z q ind an ind top j q K p X q ? ? ω Ω p X ; R q K ³ X { B Todd p X { B q^(cid:1) K (cid:1) n (cid:1) p B ; R { Z q j q K (cid:1) n p B q ω Ω p B ; R q(cid:1) nK K -theory. The left vertical arrows are the topologicalindex ind top , defined by a construction in generalized cohomology theory, and theanalytic index ind an , defined in [33]. The main theorem of [33] is the equality of thesearrows. Our analytic and topological indices are defined to fill in the “?”’s in the middlevertical arrows subject to the condition that the resulting two diagrams (with analyticand topological indices) commute. Note that the right vertical arrow depends on thegeometric structures; hence the same is true for the middle vertical arrows.The analytic index is based on Quillen’s notion of a superconnection [42] as generalizedto the infinite-dimensional setting by Bismut [10]. To define it in our finite-dimensionalmodel of differential K -theory we use the Bismut-Cheeger eta form [11], which me-diates between the Chern character of the Bismut superconnection and the Cherncharacter of the finite-dimensional index bundle. The resulting Definition 3–11 is thena simple extension of the R { Z analytic index in [33].As in topological K -theory, to define the topological index we factor π as the com-position of a fiberwise embedding X Ñ S N (cid:2) B and the projection S N (cid:2) B Ñ B ,broadly basing our construction on [28, 32, 40]. However, our differential K -theorypushforward for an embedding, given in Definition 4–13, is new and of independentinterest. For our proof of the main theorem we want the image to be defined in termsof currents instead of differential forms, so the embedding pushforward lands in the“currential” K -theory of S N (cid:2) B . The definition uses the Bismut-Zhang current [13],which essentially mediates between the Chern character of a certain superconnectionand a cohomologous current supported on the image of the embedding. A K ¨unneth D. S. Freed and J. Lott decomposition of the currential K -theory of S N (cid:2) B is used to give an explicit formulafor the projection pushforward. The topological index is the composition of the em-bedding and projection pushforwards, with a modification to account for a discrepancyin the horizontal distributions. The topological index does not involve any spectralanalysis, but does involve differential forms, so may be more appropriately termed the“differential topological index”.For proper submersions with odd fiber dimension, we introduce suspension and desus-pension maps between even and odd differential K -theory groups. We use them todefine topological and analytic pushforward maps for the odd case in terms of thosefor the even case.The preceding constructions apply when B is compact. To define the index maps fornoncompact B , we take a limit over an exhaustion of B by compact submanifolds.This depends on a result of independent interest which we prove in the appendix: q K (cid:13)p B q is isomorphic to the inverse limit of the differential K -theory groups of compactsubmanifolds. The commutativity of the right-hand square of (1–2) for both the analytic and topo-logical pushforwards, combined with the exactness of the bottom row, implies that forany E P q K p X q we have ind an p E q(cid:1) ind top p E q (cid:16) j p T q for a unique T P K (cid:1) n (cid:1) p B ; R { Z q .We now use the basic method of proof in [33] to show that T vanishes. Namely, itsuffices to demonstrate the vanishing of the pairing of T with any element of the K -homology group K (cid:1) n (cid:1) p B ; Z q . Such pairings are given by reduced eta-invariants,assuming the family of Dirac-type operators has vector bundle kernel. After somerewriting we are reduced to proving an identity involving a reduced eta-invariant of S N (cid:2) B , a reduced eta-invariant of X , and the eta form on B . The relation between thereduced eta-invariant of X and the eta form on B is an adiabatic limit result of Dai [17].The new input is a theorem of Bismut-Zhang which relates the reduced eta-invariantof X to the reduced eta-invariant of S N (cid:2) B [13]. To handle the case when the rank ofthe kernel is not locally constant we follow a perturbation argument from [33], whichuses a lemma of Mischenko-Fomenko [39]. Karoubi’s description of K -theory with coefficients [30] combines vector bundles,connections, and differential forms into a topological framework. His model of ifferential index theorem K (cid:1) p X ; C { Z q is essentially the same as the kernel of the map ω in (1–2). (Her-mitian metrics can be added to his model to get R { Z -coefficients.) Inspired by thiswork, Gillet and Soul´e [23] defined a group ˆ K p X q in the holomorphic setting which isa counterpart of differential K -theory in the smooth setting. Faltings [18] and Gillet-R ¨ossler-Soul´e [24] proved an arithmetic Riemann-Roch theorem about these groups.Using Karoubi’s description of K -theory with coefficients, the second author proved anindex theorem in R { Z -valued K -theory [33]. Based on the Gillet-Soul´e work, he alsoconsidered what could now be called differential flat K -theory p K R p X q and differentialL-theory p L ǫ p X q [34].Differential K -theory has an antecedent in the differential character groups of Cheeger-Simons [16], which are isomorphic to integral differential cohomology groups. Inde-pendent of the developments in the last paragraph, and with physical motivation, thefirst author sketched a notion of differential K -theory q K p X q [19, 22]. In retrospect, dif-ferential K-theory can also be seen as a case of Karoubi’s multiplicative K-theory [31]for a particular choice of subcomplexes. Generalized differential cohomology wasdeveloped by Hopkins and Singer [28]. In particular, they defined differential orienta-tions and pushforwards in their setting, and constructed a pushforward in differential K -theory for such a map [28, Example 4.85] (but with a more elaborate notion of“differential spin c -structure”). Klonoff [32] constructed an isomorphism between thedifferential K -theory group q K p X q defined by Hopkins-Singer and the one given bythe finite-dimensional model used in this paper. (We remark that the argument in [32]relies on a universal connection which is not proved to exist; Ortiz [40, § 3.3 ] givesa modification of the argument which bypasses this difficulty.) A topological indexmap for proper submersions is also developed in [32, 40] and it fits into a commutativediagram (1–2). It has many of the same ingredients as the topological index definedhere and most likely agrees with it, but we have not checked the details. One notabledifference is our use of currents, a key element in our proof of the main theorem. In[32], Klonoff proves a version of Proposition 8–2 and Corollary 9–35.There are many other recent works on differential K -theory, among which we onlymention two. Bunke and Schick [15] use a different definition of q K (cid:13)p X q in whichthe generators are fiber bundles over X with a Riemannian structure and a differentialspin c -structure. They prove a rational Riemann-Roch-type theorem with value inrational differential cohomology. This theorem is the result of applying the differentialChern character to our index theorem; see Subsection 8.3. In a different direction,Simons and Sullivan [43] prove that the differential form φ in the definition of q K p X q can be removed, provided that one modifies the relations accordingly. In this way,differential K -theory really becomes a K -theory of vector bundles with connection. D. S. Freed and J. Lott
However, in our approach to the analytic and topological indices it is natural to includethe form φ .The differential index theorem, or rather its consequence for determinant line bundles,is used in Type I string theory to prove the anomaly cancellation known as the “Green-Schwarz mechanism” [19], at least on the level of isomorphism classes. Indeed, thisapplication was one motivation to consider a differential K-theory index theorem, for thefirst author. The differential K -theory formula for the determinant line bundle reducesin special low dimensional cases to a formula in a simpler differential cohomologytheory. There is a two-dimensional example relevant to worldsheet string theory [21,§ 5] and an example in four-dimensional gauge theory [20, § 2]. We begin in Section 2 with some basic material about characteristic forms, differential K -theory and reduced eta-invariants. We also establish our notation. In Section 3 wedefine the analytic index for differential K -theory, in the case of vector bundle kernel. InSection 4 we construct the pushforward for differential K -theory under an embeddingwhich is provided with a Riemannian structure and a differential spin c -structure onits normal bundle. It lands in the currential K -theory of the image manifold. InSection 5 we construct the topological index. In Section 6 we prove our main theoremin the case of vector bundle kernel. The general case is covered in Section 7. Therelationship of our index theorem to determinant line bundles, differential Riemann-Roch theorems, and indices in Deligne cohomology is the subject of Section 8. InSection 9 we describe how to extend the results of the preceding sections, which werefor even differential K -theory and even relative dimension, to the odd case by means ofsuspensions and desuspensions. In the appendix we prove that the differential K -theoryof a noncompact manifold may be computed as a limit of the differential K -theory ofcompact submanifolds.More detailed explanations appear at the beginnings of the individual sections.The first author thanks Michael Hopkins and Isadore Singer for early explorations,as well as Kiyonori Gomi for more recent discussions. We also thank the referee forconstructive suggestions which improved the paper. ifferential index theorem In this section we review some standard material and clarify notation.In Subsection 2.1 we describe the Chern character and the relative Chern-Simons form.One slightly nonstandard point is that we include a formal variable u of degree 2 sothat the Chern character preserves integer cohomological degree as opposed to only amod 2 degree.In Subsection 2.2 we define differential K -theory in even degrees using a model whichinvolves vector bundles, connections and differential forms. We will also need a slightextension of differential K -theory in which differential forms are replaced by de Rhamcurrents. This “currential” K-theory is introduced in Subsection 2.3.In Subsection 2.4 we show that on a compact odd-dimensional spin c -manifold, theAtiyah-Patodi-Singer reduced η -invariant gives an invariant of currential K -theory.Finally, in Subsection 2.5 we recall Quillen’s definition of superconnections and theirassociated Chern character forms. Define the Z -graded real algebra(2–1) R (cid:13) (cid:16) R r u , u (cid:1) s , deg u (cid:16) . It is isomorphic to K (cid:13)p pt; R q .Let X be a smooth manifold. Let Ω p X ; R q(cid:13) denote the Z -graded algebra of differentialforms with coefficients in R ; we use the total grading. Let H p X ; R q(cid:13) denote thecorresponding cohomology groups. Let R u : Ω p X ; R q(cid:13) b C Ñ Ω p X ; R q(cid:13) b C be themap which multiplies u by 2 π i .We take K p X ; Z q to be the homotopy-invariant K -theory of X , i.e. K p X ; Z q (cid:16)r X , Z (cid:2) BGL p8 , C qs . Note that one can carry out all of the usual K -theory constructionswithout any further assumption on the manifold X , such as compactness or finitetopological type. For example, given a complex vector bundle over X , there is alwaysanother complex vector bundle on X so that the direct sum is a trivial bundle [38,Problem 5-E].We can describe K p X ; Z q as an abelian group generated by complex vector bundles E over X equipped with Hermitian metrics h E . The relations are that E (cid:16) E (cid:0) E D. S. Freed and J. Lott whenever there is a short exact sequence of Hermitian vector bundles, meaning thatthere is a short exact sequence(2–2) 0 ÝÑ E i ÝÑ E j ÝÑ E ÝÑ i and j (cid:6) are isometries. Note that in such a case, we get an orthogonal splitting E (cid:16) E ` E .Let ∇ E be a compatible connection on E . The corresponding Chern character form is(2–3) ω p ∇ E q (cid:16) R u tr (cid:1) e (cid:1) u (cid:1) p ∇ E q (cid:9) P Ω p X ; R q . It is a closed form whose de Rham cohomology class ch p E q P H p X ; R q is independentof ∇ E . The map ch : K p X ; Z q ÝÑ H p X ; R q becomes an isomorphism after tensoringthe left-hand side with R . We also put(2–4) c p ∇ E q (cid:16) (cid:1) π i u (cid:1) tr (cid:0)p ∇ E q (cid:8) P Ω p X ; R q . We can represent K p X ; Z q using Z { Z -graded vector bundles. A generator of K p X ; Z q is then a Z { Z -graded complex vector bundle E (cid:16) E (cid:0) ` E (cid:1) on X , equippedwith a Hermitian metric h E (cid:16) h E (cid:0) ` h E (cid:1) . Choosing unitary connections ∇ E (cid:8) andletting str denote the supertrace, we put(2–5) ω p ∇ E q (cid:16) R u str (cid:1) e (cid:1) u (cid:1) p ∇ E q (cid:9) P Ω p X ; R q . More generally, if r is even then by Bott periodicity, we can represent a generator of K r p X ; Z q by a complex vector bundle E on X , equipped with a Hermitian metric h E .Again, we choose a compatible connection ∇ E . In order to define the Chern characterform, we put(2–6) ω p ∇ E q (cid:16) u r { R u tr (cid:1) e (cid:1) u (cid:1) p ∇ E q (cid:9) P Ω p X ; R q r , and similarly for Z { Z -graded generators of K r p X ; Z q . Remark 2–7
Note the factor of u r { . It would perhaps be natural to insert the formalvariable u r { in front of E but we will refrain from doing so. In any given case, itshould be clear from the context what the degree is.If ∇ E and ∇ E are two connections on a vector bundle E then there is an explicitrelative Chern-Simons form CS p ∇ E , ∇ E q P Ω p X ; R q(cid:1) { Image p d q . It satisfies(2–8) dCS p ∇ E , ∇ E q (cid:16) ω p ∇ E q (cid:1) ω p ∇ E q . ifferential index theorem More generally, if(2–9) 0 ÝÑ E ÝÑ E ÝÑ E ÝÑ t ∇ E i u i (cid:16) then there is anexplicit relative Chern-Simons form CS p ∇ E , ∇ E , ∇ E q P Ω p X ; R q(cid:1) { Image p d q . Itsatisfies(2–10) dCS p ∇ E , ∇ E , ∇ E q (cid:16) ω p ∇ E q (cid:1) ω p ∇ E q (cid:1) ω p ∇ E q . To construct CS p ∇ E , ∇ E , ∇ E q , put W (cid:16) r , s (cid:2) X and let p : W Ñ X be theprojection map. Put F (cid:16) p (cid:6) E . Let ∇ F be a unitary connection on F which equals p (cid:6) ∇ E near t u (cid:2) X and which equals p (cid:6)p ∇ E ` ∇ E q near t u (cid:2) X . Then(2–11) CS p ∇ E , ∇ E , ∇ E q (cid:16) » ω p ∇ F q P Ω p X ; R q(cid:1) { Image p d q . If W is a real vector bundle on X with connection ∇ W then we put(2–12) ˆ A p ∇ W q (cid:16) R u d det (cid:2) u (cid:1) Ω W { u (cid:1) Ω W { (cid:10) P Ω p X ; R q , where Ω W is the curvature of ∇ W .Suppose that W is an oriented R n -vector bundle on X with a Euclidean metric h W and a compatible connection ∇ W . Let B Ñ X denote the principal SO p n q -bundleon X to which W is associated. We say that W has a spin c -structure if the principal SO p n q -bundle B Ñ X lifts to a principal Spin c p n q -bundle F Ñ X . Let S W Ñ X bethe complex spinor bundle on X that is associated to F . It is Z { Z -graded if n iseven and ungraded if n is odd. Let L W Ñ X denote the characteristic line bundle on X that is associated to F Ñ X by the homomorphism Spin c p n q Ñ U p q . (Recall thatSpin c p n q (cid:16) Spin p n q (cid:2) Z { Z U p q ; the indicated homomorphism is trivial on the Spin p n q factor and is the square on the U p q factor.) Choose a unitary connection ∇ L W on L W . Then ∇ W and ∇ L W combine to give a connection on F and hence an associatedconnection p ∇ W on S W . We write(2–13) Todd p p ∇ W q (cid:16) ˆ A p ∇ W q ^ e c (cid:2) ∇ LW (cid:10) P Ω p X ; R q The motivation for our notation comes from the case when W is the underlying realvector bundle of a complex vector bundle W . If W has a unitary structure then W inherits a spin c -structure. If ∇ W is a unitary connection on W then S W (cid:21) Λ , (cid:6)p W q inherits a connection p ∇ W and Todd p p ∇ W q equals the Todd form of ∇ W [26, Chapter1.7]. D. S. Freed and J. Lott
Definition 2–14
The differential K-theory group q K p X q is the abelian group comingfrom the following generators and relations. The generators are quadruples E (cid:16)p E , h E , ∇ E , φ q where • E isa complex vector bundle on X . • h E is aHermitian metric on E . • ∇ E is an h E -compatible connection on E . • φ P Ω p X ; R q(cid:1) { Image p d q .The relations are E (cid:16) E (cid:0) E whenever there is a short exact sequence (2–2) ofHermitian vector bundles and φ (cid:16) φ (cid:0) φ (cid:1) CS p ∇ E , ∇ E , ∇ E q .Hereafter, when we speak of a generator of q K p X q , we will mean a quadruple E (cid:16)p E , h E , ∇ E , φ q as above.There is a homomorphism ω : q K p X q ÝÑ Ω p X ; R q , given on generators by ω p E q (cid:16) ω p ∇ E q (cid:0) d φ .There is an evident extension of the definition of q K to manifolds-with-boundary.We can also represent q K p X q using Z { Z -graded vector bundles. A generator of q K p X q is then a quadruple consisting of a Z { Z -graded complex vector bundle E on X , a Hermitian metric h E on E , a compatible connection ∇ E on E and an element φ P Ω p X , R q(cid:1) { Image p d q .One can define q K (cid:13)p X q by a general construction [28]; it is a 2-periodic generalizeddifferential cohomology theory. Remark 2–15
The abelian group defined in Definition 2–14 is isomorphic to thatdefined in [28]; see [32, 40] for a proof.We use the following model in arbitrary even degrees. For any even r , a generator of q K r p X q is a quadruple E (cid:16) p E , h E , ∇ E , φ q where φ P Ω p X , R q r (cid:1) { Image p d q has totaldegree r (cid:1)
1. For such a quadruple, we put(2–16) ω p E q (cid:16) u r { R u tr (cid:1) e (cid:1) u (cid:1) p ∇ E q (cid:9) (cid:0) d φ P Ω p X ; R q r , and similarly for Z { Z -graded generators of q K r p X q . Remark 2–17
As in Remark 2–7, it would perhaps be natural to insert the formalvariable u r { in front of p E , h E , ∇ E q but we will refrain from doing so. ifferential index theorem Let Ω p X ; R q(cid:13) K denote the union of affine subspaces of closed forms whose de Rhamcohomology class lies in the image of ch : K (cid:13)p X ; Z q ÝÑ H p X ; R q(cid:13) . There are exactsequences(2–18) 0 ÝÑ K (cid:13)(cid:1) p X ; R { Z q i ÝÝÑ q K (cid:13)p X q ω ÝÝÑ Ω p X ; R q(cid:13) K ÝÑ ÝÑ Ω p X ; R q(cid:13)(cid:1) Ω p X ; R q(cid:13)(cid:1) K j
ÝÝÑ q K (cid:13)p X q c ÝÝÑ K (cid:13)p X ; Z q ÝÑ . Also, q K (cid:13)p X q is an algebra, with the product on q K p X q given by (cid:0) E , h E , ∇ E , φ (cid:8) (cid:4) (cid:0) E , h E , ∇ E , φ (cid:8) (cid:16) (2–20) (cid:0) E b E , h E b h E , ∇ E b I (cid:0) I b ∇ E , φ ^ ω p ∇ E q (cid:0) ω p ∇ E q ^ φ (cid:0) φ ^ d φ (cid:8) . Then with respect to the exact sequences (2–18) and (2–19),(2–21) i p x q ˇ b (cid:16) i (cid:0) xc p ˇ b q(cid:8) , j p α q ˇ b (cid:16) j (cid:0) α ^ ω p ˇ b q(cid:8) . We now describe how a differential K-theory class changes under a deformation of itsHermitian metric, its unitary connection and its differential form.
Lemma 2–22
For i P t , u ,let A i : X Ñ r , s (cid:2) X betheembedding A i p x q (cid:16) p i , x q .Given E E , h E , ∇ E , φ
1q P q K pr , s (cid:2) X q , put E i (cid:16) A (cid:6) i E K p X q . Then E (cid:16) E (cid:0) j p³ ω p E . Proof
We can write E as the pullback of a vector bundle on X , under the projectionmap r , s (cid:2) X Ñ X . Thereby, E and E get identified with a single vector bundle E .After performing an automorphism of E , we can also assume that h E is the pullbackof a Hermitian metric h E on E . Since(2–23) ω p E
1q (cid:16) ω p ∇ E
1q (cid:0) d φ ω p ∇ E
1q (cid:0) dt ^ B t φ d X φ , we have(2–24) » ω p E
1q (cid:16) CS p ∇ E , ∇ E q (cid:0) φ (cid:1) φ (cid:1) d X » φ , from which the lemma follows. Remark 2–25
There is an evident extension of Lemma 2–22 to the case when E E , h E , ∇ E , φ
1q P q K r pr , s (cid:2) X q for r even. D. S. Freed and J. Lott
Let δ Ω p p X q denote the p -currents on X , meaning δ Ω p p X q (cid:16) (cid:1) Ω dim p X q(cid:1) pc p X ; o q(cid:9)(cid:6) ,where o is the flat orientation R -bundle on X . We think of an element of δ Ω p p X q as a p -form on X whose components, in a local coordinate system, are distributional.Consider the cocomplex δ Ω p X ; R q(cid:13) (cid:16) δ Ω p X q b R equipped with the differential d ofdegree 1. In the definition of q K p X q , suppose that we take φ P δ Ω p X q(cid:1) { Image p d q .Let δ q K (cid:13)p X q denote the ensuing “currential” K-theory groups. With an obvious meaningfor δ Ω p X ; R q nK , there are exact sequences(2–26) 0 ÝÑ K (cid:13)(cid:1) p X ; R { Z q i ÝÝÑ q K (cid:13)p X q ω ÝÝÑ δ Ω p X ; R q(cid:13) K ÝÑ ÝÑ δ Ω p X ; R q(cid:13)(cid:1) δ Ω p X ; R q(cid:13)(cid:1) K j
ÝÝÑ q K (cid:13)p X q c ÝÝÑ K (cid:13)p X ; Z q ÝÑ . However, δ q K (cid:13)p X q is not an algebra, since we can’t multiply currents. Suppose that X is a closed odd-dimensional spin c manifold. Let L X denote thecharacteristic line bundle of the spin c structure. We assume that X is equipped witha Riemannian metric g TX and a unitary connection ∇ L X on L X . Let S X denote thespinor bundle on X . Given a generator E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) for δ q K p X q , let D X , E bethe Dirac-type operator acting on smooth sections of S X b E . Let η p D X , E q denote itsreduced eta-invariant, i.e.(2–28) η p D X , E q (cid:16) η p D X , E q (cid:0) dim p Ker p D X , E qq p mod Z q . Definition 2–29
Givenagenerator E for δ q K p X q ,define η p X , E q P u (cid:1) dim p X q(cid:0) (cid:4) p R { Z q by(2–30) η p X , E q (cid:16) u (cid:1) dim p X q(cid:0) η p D X , E q (cid:0) » X Todd (cid:1) p ∇ TX (cid:9) ^ φ p mod u (cid:1) dim p X q(cid:0) (cid:4) Z q . Note that ³ X Todd (cid:1) p ∇ TX (cid:9) ^ φ is a real multiple of u (cid:1) dim p X q(cid:0) for dimensional reasons.Note also that η p X , E q generally depends on the geometric structure of X .We now prove some basic properties of η p X , E q . ifferential index theorem Proposition 2–31 (1) Let W be an even-dimensional compact spin c -manifold-with-boundary. Supposethat W isequipped withaRiemannianmetric g TW anda unitary connection ∇ L W , which are products near B W . Let F be a generatorfor q K p W q which is aproduct near B W and let E be itspullback to B W . Then(2–32) η pB W , E q (cid:16) » W Todd (cid:1) p ∇ TW (cid:9) ^ ω p F q p mod u (cid:1) dim pB W q(cid:0) (cid:4) Z q . (2) Theassignment E ÝÑ η p X , E q factorsthroughahomomorphism η : δ q K p X q Ñ u (cid:1) dim p X q(cid:0) (cid:4) p R { Z q .(3) If a P K (cid:1) p X ; R { Z q ,then(2–33) η p X , i p a qq (cid:16) u (cid:1) dim p X q(cid:0) xr X s , a y , where r X s P K (cid:1) p X ; Z q is the (periodicity-shifted) fundamental class in K-homology and xr X s , a y P R { Z is the result of the pairing between K (cid:1) p X ; Z q and K (cid:1) p X ; R { Z q . Proof
For part (1), write F (cid:16) p F , h F , ∇ R , Φ q . By the Atiyah-Patodi-Singer indextheorem [3],(2–34) u dim p W q » W Todd (cid:1) p ∇ TW (cid:9) ^ ω p ∇ F q (cid:1) η p D X , E q P Z . (Note that ³ W Todd (cid:1) p ∇ TW (cid:9) ^ ω p ∇ E q is a real multiple of u (cid:1) dim p W q for dimensionalreasons.) As » W Todd (cid:1) p ∇ TW (cid:9) ^ ω p F q (cid:16) » W Todd (cid:1) p ∇ TW (cid:9) ^ p ω p ∇ F q (cid:0) d Φ q (2–35) (cid:16) » W Todd (cid:1) p ∇ TW (cid:9) ^ ω p ∇ F q(cid:0)»B W Todd (cid:1) p ∇ T B W (cid:9) ^ Φ , part (1) follows.To prove part (2), suppose first that we have a relation E (cid:16) E (cid:0) E for q K p X q .Put W (cid:16) r , s (cid:2) X , with a product metric. If p : W Ñ X is the projection map,put F (cid:16) p (cid:6) E and h F (cid:16) p (cid:6) h E . Let ∇ F be a unitary connection on F whichequals p (cid:6) ∇ E near t u (cid:2) X and which equals p (cid:6)p ∇ E ` ∇ E q near t u (cid:2) X . Choose Φ P Ω p W ; R q(cid:1) { Im p d q which equals p (cid:6) φ near t u(cid:2) X and which equals p (cid:6)p φ (cid:0) φ q D. S. Freed and J. Lott near t u (cid:2) X . Using part (1), η p X , E q (cid:1) η p X , E q (cid:1) η p X , E q (cid:16) » W Todd (cid:1) p ∇ TW (cid:9) ^ p ω p ∇ F q (cid:0) d Φ q (2–36) (cid:16) » X » Todd (cid:1) p ∇ TX (cid:9) ^ p ω p ∇ F q (cid:0) d Φ q(cid:16) » X Todd (cid:1) p ∇ TX (cid:9) ^ (cid:0) CS (cid:0) ∇ E , ∇ E , ∇ E (cid:8) (cid:0) φ (cid:1) φ (cid:1) φ (cid:8)(cid:16) u (cid:1) dim p X q(cid:0) (cid:4) p R { Z q . This shows that η extends to a map q K p X q Ñ u (cid:1) dim p X q(cid:0) (cid:4) p R { Z q .The argument easily extends if we use currents instead of forms, thereby proving part(2) of the proposition.Part (3) follows from [33, Proposition 3]. Remark 2–37
To prove part (2) of Proposition 2–31, we could have used the varia-tional formula for η [4], which is more elementary than the Atiyah-Patodi-Singer indextheorem.More generally, if p E , h E , ∇ E , φ q is a generator for δ q K r p X q then we define η p X , E q P u r (cid:1) dim p X q(cid:1) (cid:4) p R { Z q by(2–38) η p X , E q (cid:16) u r (cid:1) dim p X q(cid:1) η p D X , E q (cid:0) » X Todd (cid:1) p ∇ TX (cid:9) ^ φ p mod u r (cid:1) dim p X q(cid:1) (cid:4) Z q . Define the auxiliary ring(2–39) R R r u { , u (cid:1) { s , where u { is a formal variable of degree 1 and u (cid:1) { its inverse. Then R € R .If E is a Z Z -graded vector bundle on X then the Ω p X ; R -module Ω p X ; E b R ofdifferential forms with values in E b R is p Z (cid:2) Z (cid:2) Z Z q -graded: by form degree,degree in R , and degree in E . We use a quotient p Z (cid:2) Z Z q -grading: the integer degreeis the sum of the form degree and the degree in R , while the mod 2 degree is thedegree in E plus the mod two form degree. Definition 2–40
A superconnection A on E is a graded Ω p X ; R -derivation of Ω p X ; E b R ofdegree p , q . ifferential index theorem Note that we can uniquely write(2–41) A (cid:16) u { ω (cid:0) ∇ (cid:0) u (cid:1) { ω (cid:0) u (cid:1) ω (cid:0) (cid:4) (cid:4) (cid:4) , where ∇ is an ordinary connection on E (which preserves degree) and ω j is anEnd p E q -valued j -form on X which is an even endomorphism if j is odd and an oddendomorphism if j is even. The powers of u are related to the standard scaling of asuperconnection. The Chern character of A is defined by(2–42) ω p A q (cid:16) R u Str e u (cid:1) A P Ω p M ; R q . Notice that the curvature A has degree p , q so u (cid:1) A of degree p , q can beexponentiated. Also, there are no fractional powers of u in the result since the supertraceof an odd endomorphism of E vanishes. In this section we define the analytic pushforward of a differential K-theory class undera proper submersion. This is an extension of the analytic pushforward in R { Z -valuedK-theory that was defined in [33, Section 4]. The geometric assumptions are that wehave a proper submersion π : X Ñ B of relative dimension n , with n even, which isequipped with a Riemannian structure on the fibers and a differential spin c -structure(in a sense that will be made precise below).Given a differential K-theory class E (cid:16) p E , h E , ∇ E , φ q on X , there is an ensuing family D V of vertical Dirac-type operators. In this section we assume that Ker p D V q forms avector bundle on B . (This assumption will be lifted in Section 7). In Definition 3–11we define the analytic pushforward ˇ π (cid:6)p E q P q K (cid:1) n p B q of E , using the Bismut-Cheegereta form.For later purposes, we will want to extend the definition of the analytic pushforwardto certain currential K-theory classes. To do so, we have to make a compatibilityassumption between the singularities of the current φ and the fibration π . This isphrased in terms of the wave front set of the current φ , which is a subset of T (cid:6) X thatmicrolocally measures the singularity locus of φ . For a fiber bundle π : X Ñ B wedefine an analog WF q K p X q of q K p X q using currents φ whose wave front set has zerointersection with the conormal bundle of the fibers. Roughly speaking, this meansthat the singularity locus of φ meets the fibers of π transversely, so we can integrate φ fiberwise to get a smooth form on B . We then define the analytic pushforward on WF q K p X q . D. S. Freed and J. Lott
Let π : X Ñ B be a proper submersion of relative dimension n , with n even. Recallthat this is the same as saying that π : X Ñ B is a smooth fiber bundle with compactfibers of even dimension n . Let T V X (cid:16) Ker p d π q denote the relative tangent bundle on X .We define a Riemannian structure on π to be a pair consisting of a vertical metric g T V X and a horizontal distribution T H X on X . This terminology is justified by the existenceof a certain connection on T V X which restricts to the Levi-Civita connection on eachfiber of π [10, Definition 1.6]. We recall the definition. Let g TB be a Riemannianmetric on B . Using g TB and the Riemannian structure on π , we obtain a Riemannianmetric g TX on X . Let ∇ TX be its Levi-Civita connection. Let P : TX Ñ T V X beorthogonal projection. Definition 3–1
Theconnection ∇ T V X on T V X is ∇ T V X (cid:16) P (cid:5) ∇ TX (cid:5) P . Itisindependentofthe choice of g TB .Suppose the map π is spin c -oriented in the sense that T V X has a spin c -structure, withcharacteristic hermitian line bundle L V X Ñ X . A differential spin c -structure on π isin addition a unitary connection on L V X . Let S V X denote the associated spinor bundleon X . The connections on T V X and L V X induce a connection p ∇ T V X on S V X .Define π (cid:6) : Ω p X ; R q(cid:13) Ñ Ω p B ; R q(cid:13)(cid:1) n by(3–2) π (cid:6)p φ q (cid:16) » X { B Todd (cid:1) p ∇ T V X (cid:9) ^ φ. Note that our π (cid:6) differs from the de Rham pushforward by the factor of Todd (cid:1) p ∇ T V X (cid:9) .It will simplify later formulas if we use our slightly unconventional definition.We recall that there is a notion of the wave front set of a current on X ; it is the unionof the wave front sets of its local distributional coefficients [29, Chapters 8.1 and 8.2].The wave front set is a subset of T (cid:6) X . Let N (cid:6) V X (cid:16) π (cid:6) T (cid:6) B € T (cid:6) X be the conormalbundle of the fibers. Let WF Ω p X ; R q(cid:13) denote the subspace of δ Ω p X ; R q(cid:13) consistingof elements whose wave front set intersects N (cid:6) V X only at the zero section of N (cid:6) V X . By[29, Theorem 8.2.12], equation (3–2) defines a map(3–3) π (cid:6) : WF Ω p X ; R q(cid:13) Ñ Ω p B ; R q(cid:13)(cid:1) n . Let WF q K p X q be the abelian group whose generators are quadruples E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) with φ P WF Ω p X ; R q(cid:1) { Image p d q , and with relations as before. Then there are exact ifferential index theorem sequences(3–4) 0 ÝÑ K (cid:13)(cid:1) p X ; R { Z q i ÝÝÑ WF q K (cid:13)p X q ω ÝÝÑ WF Ω p X ; R q(cid:13) K ÝÑ ÝÑ WF Ω p X ; R q(cid:13)(cid:1) WF Ω p X ; R q(cid:13)(cid:1) K j
ÝÝÑ WF q K (cid:13)p X q c ÝÝÑ K (cid:13)p X ; Z q ÝÑ . Here we use the fact that if α P WF Ω p X ; R q(cid:13) and α P Image p d : δ Ω p X ; R q(cid:13)(cid:1) Ñ δ Ω p X ; R q(cid:13)q then α P Image p d : WF Ω p X ; R q(cid:13)(cid:1) Ñ WF Ω p X ; R q(cid:13)q .Given a Riemannian structure on π and a generator E for q K p X q , we want to definea pushforward of E that lives in q K (cid:1) n p B q . Write E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) . Let H denotethe (possibly infinite dimensional) vector bundle on B whose fiber H b at b P B is thespace of smooth sections of p S V X b E q(cid:7)(cid:7) X b . The bundle H is Z { Z -graded. For s ¡ Bismut superconnection A s is(3–6) A s (cid:16) su { D V (cid:0) ∇ H (cid:1) s (cid:1) u (cid:1) { c p T q . Here D V is the Dirac-type operator acting on H b , ∇ H is a certain unitary connectionon H constructed from p ∇ T V X , ∇ E and the mean curvature of the fibers, and c p T q isthe Clifford multiplication by the curvature 2-form T of the fiber bundle. For moreinformation, see [9, Proposition 10.15]. We use powers of s in (3–6) in order tosimplify calculations, as compared to the powers of s { used by some other authors,but there is no essential difference.Now assume that Ker p D V q forms a smooth vector bundle on B , necessarily Z { Z -graded. There are an induced L -metric h Ker p D V q and a compatible projected connection ∇ Ker p D V q . Note that r Ker p D V qs lies in K (cid:1) n p B q . Then(3–7) lim s Ñ u (cid:1) n { R u STr (cid:1) e (cid:1) u (cid:1) A s (cid:9) (cid:16) π (cid:6)p ω p ∇ E qq , while(3–8) lim s Ñ8 u (cid:1) n { R u STr (cid:1) e (cid:1) u (cid:1) A s (cid:9) (cid:16) ω p ∇ Ker p D V qq ;see [9, Chapter 10]. Note that the preceding two equations lie in forms of totaldegree (cid:1) n .The Bismut-Cheeger eta-form [11] is(3–9) ˜ η (cid:16) u (cid:1) n { R u » 8 STr (cid:2) u (cid:1) dA s ds e (cid:1) u (cid:1) A s (cid:10) ds P Ω p B ; R q(cid:1) n (cid:1) { Image p d q . It satisfies(3–10) d ˜ η (cid:16) π (cid:6)p ω p ∇ E qq (cid:1) ω p ∇ Ker p D V qq . D. S. Freed and J. Lott
Definition 3–11
Given a generator E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) for q K p X q , and assumingKer p D V q isavector bundle, wedefine the analytic index ind an p E q P q K (cid:1) n p B q by(3–12) ind an p E q (cid:16) (cid:1) Ker p D V q , h Ker p D V q , ∇ Ker p D V q , π (cid:6)p φ q (cid:0) r η (cid:9) . It follows from Theorem 6–2 below that the assignment E Ñ ind an p E q factors througha map from q K p X q to q K (cid:1) n p B q .Given a generator E of WF q K p X q , we define ind an p E q P q K (cid:1) n p B q by the same formula(3–12). Lemma 3–13 If E is a generator for WF q K p X q then ω p ind an p E qq (cid:16) π (cid:6)p ω p E qq in Ω p B ; R q(cid:1) n . Proof
From (3–10),(3–14) ω p ind an p E qq (cid:16) ω p ∇ Ker p D V qq(cid:0) d p π (cid:6)p φ q(cid:0) r η q (cid:16) π (cid:6)p ω p ∇ E q(cid:0) d φ q (cid:16) π (cid:6)p ω p E qq , which proves the lemma. In this section we define a pushforward on differential K -theory under a proper em-bedding ι : X Ñ Y of manifolds. The definition uses the data of a generator E (cid:16) p E , h E , ∇ E , φ q of q K p X q and a Riemannian structure on the normal bundle ν of the embedding.To motivate our definition, let us recall how to push forward ordinary K -theory under ι [1]. Suppose that the normal bundle p : ν Ñ X has even dimension r and is endowedwith a spin c -structure. Let S ν Ñ X denote the corresponding Z { Z -graded spinorbundle on X . Clifford multiplication by an element in ν gives an isomorphism, on thecomplement of the zero-section in ν , between p (cid:6) S ν (cid:0) and p (cid:6) S ν (cid:1) . The K -theory Thomclass U K is the corresponding relative class in K r p D p ν q , S p ν q ; Z q , where D p ν q denotesthe closed disk bundle of ν and S p ν q (cid:16) B D p ν q is the sphere bundle.Given a vector bundle E on X , the Thom homomorphism K p X ; Z q Ñ K r p D p ν q , S p ν q ; Z q sends r E s to p (cid:6)r E s (cid:4) U K . Transplanting this to a closed tubular neighborhood T of X in Y , we obtain a relative K -theory class in K p T , B T ; Z q . Then excision in K -theorydefines an element ι (cid:6)r E s P K r p Y ; Z q , which is the K -theory pushforward of r E s .Applying the Chern character, one finds that ch p ι (cid:6)r E sq is the extension to Y of the ifferential index theorem cohomology class p (cid:6) ch pr E sqY U H Todd p ν q P H (cid:13)p D p ν q , S p ν q ; Q q , where U H P H r p D p ν q , S p ν q ; Z q is the Thom class in cohomology.In order to push forward classes in differential K -theory, we will need to carry alongdifferential form information in the K -theory pushforward. There are differential formdescriptions of the Thom class in cohomology, but they are not very convenient forour purposes. Instead we pass to currents and simply write the Thom homomorphismin real cohomology as the map which sends a differential form ω on X to the current ω ^ δ X on Y . Following this line of reasoning, the pushforward under ι of a differential K -theory class on X is a currential K -theory class on Y . An important ingredient inits definition is a certain current γ defined by Bismut-Zhang [13]. Let ι : X ãÑ Y be a proper embedding of manifolds. Let r be the codimension of X in Y . We assume that r is even. Let δ X P δ Ω p Y q r denote the current of integration on X .Let ν (cid:16) ι (cid:6) TY { TX be the normal bundle to X . We define a Riemannian structureon ι to be a metric g ν on ν and a compatible connection ∇ ν on ν . Suppose themap ι carries a differential spin c -structure , in the sense that ν has a spin c -structurewith characteristic hermitian line bundle L ν Ñ X and that the line bundle is endowedwith a unitary connection ∇ L ν . Let S ν Ñ X be the spinor bundle of ν . Then S ν inherits a connection p ∇ ν . Let c p ξ q denote Clifford multiplication by ξ P ν on S ν . Let p : ν Ñ X be the vector bundle projection. Then there is a self-adjoint oddendomorphism c P End p p (cid:6) S ν q which acts on p p (cid:6) S ν q ξ (cid:21) S ν as Clifford multiplicationby ξ P ν .There is a pushforward map ι (cid:6) : Ω p X ; R q(cid:13) Ñ δ Ω p Y ; R q(cid:13)(cid:0) r given by(4–1) ι (cid:6)p φ q (cid:16) φ Todd (cid:1) p ∇ ν (cid:9) ^ δ X . Note that our ι (cid:6) differs from the de Rham pushforward by the factor of Todd (cid:1) p ∇ ν (cid:9) .It will simplify later formulas if we use our slightly unconventional definition.Given a Riemannian structure on ι , we want to define a map ˇ ι (cid:6) : q K p X q ÝÑ δ q K r p Y q .To do so, we use a construction of Bismut and Zhang [13]. Let F be a Z { Z -graded vector bundle on Y equipped with a Hermitian metric h F . We assume that weare given an odd self-adjoint endomorphism V of F which is invertible on Y (cid:1) X ,and that Ker p V q has locally constant rank along X . Then Ker p V q restricts to a D. S. Freed and J. Lott Z { Z -graded vector bundle on X . It inherits a Hermitian metric h Ker p V q from F .Let P Ker p V q denote orthogonal projection from F (cid:7)(cid:7) X to Ker p V q . If F has an h F -compatible connection ∇ F then Ker p V q inherits an h Ker p V q -compatible connectiongiven by ∇ Ker p V q (cid:16) P Ker p V q ∇ F P Ker p V q . Given a connection ∇ F on F , a point x P X and a vector ξ P ν x , lift ξ to an element ˆ ξ P T x Y and put(4–2) B ξ V (cid:16) P Ker p V q (cid:1) ∇ F ˆ ξ V (cid:9) P Ker p V q . Then B ξ V is an odd self-adjoint endomorphism of Ker p V q which is independent ofthe choices of ∇ F and ˆ ξ . There is a well-defined odd self-adjoint endomorphism B V P End p p (cid:6) Ker p V qq so that B V acts on p p (cid:6) Ker p V qq ξ (cid:21) Ker p V q by B ξ V . Lemma 4–3 [13,Remark1.1]Givena Z { Z -gradedvectorbundle E on X ,equippedwith a Hermitian metric h E and a compatible connection ∇ E , there are F , h F , ∇ F , V on Y so that(4–4) (cid:1) S ν b E , h S ν b h E , p ∇ ν b Id (cid:0) Id b ∇ F , c b Id (cid:9) (cid:21) (cid:1) Ker p V q , h Ker p V q , ∇ Ker p V q , B V (cid:9) . Proof
Let D p ν q denote the closed unit disk bundle of ν . Put S p ν q (cid:16) B D p ν q .Then there is a diffeomorphism σ : T Ñ D p ν q between D p ν q and a closed tubularneighborhood T of X in Y . The Z { Z -graded vector bundle W (cid:16) σ (cid:6) p (cid:6)p S ν b E q on T is equipped with an isomorphism J : W (cid:0)(cid:7)(cid:7)B T Ñ W (cid:1)(cid:7)(cid:7)B T on B T given by σ (cid:6) c .By the excision isomorphism in K-theory, K p Y , Y (cid:1) T q (cid:21) K p T , B T q . This meansthat after stabilization, W can be extended to a Z { Z -graded vector bundle F on Y which is equipped with an isomorphism between F (cid:0)(cid:7)(cid:7) Y (cid:1) T and F (cid:1)(cid:7)(cid:7) Y (cid:1) T . More explicitly,let R be a vector bundle on T so that W (cid:1) ` R is isomorphic to R N (cid:2) T , for some N .Then W (cid:1) ` R extends to a trivial R N -vector bundle F (cid:1) on Y . Let F (cid:0) be the resultof gluing the vector bundle W (cid:0) ` R (on T ) with R N (cid:2) Y (cid:1) T (on Y (cid:1) T ), using theclutching isomorphism(4–5) p W (cid:0) ` R q(cid:7)(cid:7)B T J ` Id ÝÑ p W (cid:1) ` R q(cid:7)(cid:7)B T ÝÑ R N (cid:2) B T along B T .Let h R be a Hermitian inner product on R and let ∇ R be a compatible connection.Choose h F (cid:8) and ∇ F (cid:8) to agree with h W (cid:8)` R and ∇ W (cid:8)` R on T . Let V P End p F (cid:0) , F (cid:1)q be the result of gluing σ (cid:6) c (cid:7)(cid:7) W (cid:0) ` Id R (on T ) with the identity map R N Ñ R N (on Y (cid:1) T ). Put V (cid:16) V ` V (cid:6) P End p F q . Then p F , h F , ∇ F , V q satisfies the claims of thelemma. ifferential index theorem Hereafter we assume that p F , h F , ∇ F , V q satisfies Lemma 4–3. Note that r F s lies in K r p Y q . For s ¡
0, define a superconnection C s on F by(4–6) C s (cid:16) su { V (cid:0) ∇ F . Then(4–7) lim s Ñ u r { R u str (cid:1) e (cid:1) u (cid:1) C s (cid:9) (cid:16) ω p ∇ F q . Also, from [13, Theorem 1.2],(4–8) lim s Ñ8 u r { R u str (cid:1) e (cid:1) u (cid:1) C s (cid:9) (cid:16) ω p ∇ E q Todd (cid:1) p ∇ ν (cid:9) ^ δ X as currents. Definition 4–9 [13,Definition 1.3] Define γ P δ Ω p Y ; R q r (cid:1) { Image p d q by(4–10) γ (cid:16) u r { R u » 8 str (cid:2) u (cid:1) dC s ds e (cid:1) u (cid:1) C s (cid:10) ds (cid:16) u r { R u » 8 str (cid:1) u (cid:1) { Ve (cid:1) u (cid:1) C s (cid:9) ds . The integral on the right-hand side of (4–10) is well-defined, as a current on Y , by [13,Theorem 1.2]. By [13, Remark 1.5], γ is a locally integrable differential form on Y whose wave front set is contained in ν (cid:6) . Proposition 4–11 [13,Theorem 1.4] Wehave(4–12) d γ (cid:16) ω p ∇ F q (cid:1) ω p ∇ E q Todd (cid:1) p ∇ ν (cid:9) ^ δ X . Definition 4–13
Given a generator E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) for q K p X q , define ˇ ι (cid:6)p E q P δ q K r p Y q to bethe element represented bythe quadruple(4–14) ˇ ι (cid:6)p E q (cid:16) (cid:0) F , h F , ∇ F , ι (cid:6)p φ q (cid:1) γ (cid:8) . Lemma 4–15 ω p ˇ ι (cid:6)p E qq (cid:16) ι (cid:6)p ω p E qq . Proof
We have(4–16) ω p ˇ ι (cid:6)p E qq (cid:16) ω p ∇ F q (cid:0) d p ι (cid:6)p φ q (cid:1) γ q (cid:16) ω p ∇ E q (cid:0) d φ Todd (cid:1) p ∇ ν (cid:9) ^ δ X (cid:16) ι (cid:6)p ω p E qq , which proves the lemma. D. S. Freed and J. Lott
Proposition 4–17 (1) The pushforward ˇ ι (cid:6)p E q P δ q K r p Y q is independent of thechoices of F , h F , ∇ F and V ,subject to (4–4).(2) The assignment E Ñ ˇ ι (cid:6)p E q factors through amap ˇ ι (cid:6) : q K p X q Ñ δ q K r p Y q . Proof
Let G be a complex vector bundle on Y with Hermitian metric h G and com-patible connection ∇ G . If we put F F (cid:8) ` G , h F h F (cid:8)` G , ∇ F ∇ F (cid:8)` G and V V ` (cid:2) I G I G (cid:10) then it is easy to check that γ does not change. Clearly (cid:1) F , h F , ∇ F , ι (cid:6)p φ q (cid:1) γ (cid:9) equals (cid:0) F , h F , ∇ F , ι (cid:6)p φ q (cid:1) γ (cid:8) in δ q K r p Y q .To prove part (1), suppose that for i P t , u , (cid:0) F i , h F i , ∇ F i , V i (cid:8) are two different choicesof data as in the statement of the proposition. Since F i , (cid:0) (cid:1) F i , (cid:1) represent the sameclass in K r p Y q for i P t , u , the preceding paragraph implies that we can stabilizeto put ourselves into the situation that F i , (cid:0) (cid:16) F (cid:0) and F i , (cid:1) (cid:16) F (cid:1) for some fixed Z { Z -graded vector bundle F (cid:8) on Y .For t P r , s , let (cid:0) h F p t q , ∇ F p t q , V p t q(cid:8) be a smooth 1-parameter family of data inter-polating between (cid:0) h F , ∇ F , V (cid:8) and (cid:0) h F , ∇ F , V (cid:8) . Consider the product embedding ι : r , s (cid:2) X Ñ r , s (cid:2) Y . Let E be the pullback of E to r , s (cid:2) X . Let F be thepullback of F to r , s (cid:2) Y and let (cid:1) h F , ∇ F , V be the ensuing data on F comingfrom the 1-parameter family. Construct γ δ Ω pr , s (cid:2) Y ; R q r (cid:1) { Image p d q from(4–10). Put(4–18) ι E
1q (cid:16) p F , h F , ∇ F , ι φ
1q (cid:1) γ . By Remark 2–25 (or more precisely its extension to currential K-theory),(4–19) (cid:0) F , h F , ∇ F , ι (cid:6)p φ q (cid:1) γ (cid:8) (cid:1) (cid:0) F , h F , ∇ F , ι (cid:6)p φ q (cid:1) γ (cid:8) (cid:16) j (cid:2)» ω p ι E in δ q K r p Y q . However, using Lemma 4–15, ω p ι E ι ω p E is the pullback of ι (cid:6)p ω p E qq from Y to r , s (cid:2) Y . In particular, ³ ω p ι E
0. This proves part (1)of the proposition.To prove part (2), suppose that we have a relation E (cid:16) E (cid:0) E in q K p X q coming froma short exact sequence (2–2). Let p : r , s (cid:2) X Ñ X be the projection map. Put E p (cid:6) E . Let ∇ E be a unitary connection on E which is p (cid:6) ∇ E near t u (cid:2) X and whichis p (cid:6)p ∇ E ` ∇ E q near t u (cid:2) X . Choose φ Ω pr , s (cid:2) X ; R q(cid:1) { Image p d q whichequals p (cid:6) φ near t u (cid:2) X and which equals p (cid:6)p φ (cid:0) φ q near t u (cid:2) X . Consider theproduct embedding ι : r , s (cid:2) X Ñ r , s (cid:2) Y and construct ι E
1q P δ q K pr , s (cid:2) Y q ifferential index theorem as in Definition 4–13. For i P t , u , let A i : Y Ñ r , s (cid:2) Y be the embedding A i p y q (cid:16) p i , y q . From Lemmas 2–22 and 4–15,(4–20) A (cid:6) ι E
1q (cid:1) A (cid:6) ι E
1q (cid:16) j (cid:2)» ι ω p E j (cid:2) ι (cid:6) » ω p E , since the relation E (cid:16) E (cid:0) E and Lemma 2–22 imply that ³ ω p E vanishes. Hence ι (cid:6)p E q (cid:16) ι (cid:6)p E q (cid:0) ι (cid:6)p E q . This proves part (2) of the proposition. In this section we define the topological index in differential K-theory. We first considertwo fiber bundles X Ñ B and X Ñ B , each equipped with a Riemannian structureand a differential spin c structure in the sense of the previous section. We now assumethat we have a fiberwise isometric embedding ι : X Ñ X . The preceding sectionconstructed a pushforward ˇ ι (cid:6) : q K p X q Ñ δ q K r p X q . To define the topological index wewill eventually want to compose ˇ ι (cid:6) with the pushforward under the fibration X Ñ B .However, there is a new issue because the horizontal distributions on the two fiberbundles X and X need not be compatible. Hence we define a correction form r C and,in Definition 5–7, a modified embedding pushforward ˇ ι mod (cid:6) : q K p X q Ñ WF q K r p X q .To define the topological index, we specialize to the case when X is S N (cid:2) B forsome even N , equipped with a Riemannian structure coming from a fixed Riemannianmetric on S N and the product horizontal distribution. In this case we show that thepushforward of WF q K r p S N (cid:2) B q from S N (cid:2) B to B , as defined in Definition 3–11,can be written as an explicit map ˇ π prod (cid:6) : WF q K r p S N (cid:2) B q Ñ q K r (cid:1) N p B q in terms of aK ¨unneth-type formula for WF q K r p S N (cid:2) B q . This shows that ˇ π prod (cid:6) can be computedwithout any spectral analysis and, in particular, can be defined without the assumptionabout vector bundle kernel. Relabeling X as X , we then define the topological indexind top : q K p X q Ñ q K (cid:1) n p B q by ind top (cid:16) ˇ π prod (cid:6) (cid:5) ˇ ι mod (cid:6) . Let π : X Ñ B and π : X Ñ B be fiber bundles over B , with compact fibers X , b and X , b of even dimension n and n , respectively. Let ι : X Ñ X be afiberwise embedding of even codimension r , i.e., ι is an embedding, π (cid:5) ι (cid:16) π , and n (cid:16) n (cid:0) r . Let ν be the normal bundle of X in X . There is a short exact sequence(5–1) 0 ÝÑ T V X ÝÑ ι (cid:6) T V X ÝÑ ν ÝÑ , D. S. Freed and J. Lott of vector bundles on X . Suppose that π and π have Riemannian structures. Definition 5–2
The map ι is compatible with the Riemannian structures on π and π if foreach b P B , ι b : X , b Ñ X , b isan isometric embedding.The intersection of T H X with TX defines a horizontal distribution p T H X q(cid:7)(cid:7) X on X .We do not assume that it coincides with T H X . It follows that the orthogonal projectionof ι (cid:6) ∇ T V X to T V X is not necessarily equal to ∇ T V X .In the rest of this section we assume that ι is compatible with the Riemannian structureson π and π . Then ν inherits a Riemannian structure from (5–1), which is split byidentifying ν as the orthogonal complement to T V X in ι (cid:6) T V X . Namely, the metric g ν is the quotient inner product from g T V X and the connection ∇ ν is compressed from ι (cid:6) ∇ T V X .We also assume a certain compatibility of the differential spin c -structures on π , π and ι . To describe this compatibility, recall the discussion of spin c -structures fromSubsection 2.1. Over X we have principal bundles F , F ν and ι (cid:6) F , with structuregroups Spin c p n q , Spin c p r q and Spin c p n q , respectively. They project to the orientedorthonormal frame bundles B , B ν and ι (cid:6) B . The embedding ι gives a reduction B (cid:2) B ν ãÑ ι (cid:6) B of ι (cid:6) B which is compatible with the inclusion SO n (cid:2) SO r ãÑ SO n .Then we postulate that we have a lift(5–3) F (cid:2) F ν Ñ ι (cid:6) F of B (cid:2) B ν ãÑ ι (cid:6) B which is compatible with the homomorphism Spin c p n q (cid:2) Spin c p r q Ñ Spin c p n q . (The kernel of this homomorphism is a U p q -factor embeddedanti-diagonally.) Finally, we suppose that the three spin c -connections are compatible inthe sense that the U p q -connection on ι (cid:6) Ker p F Ñ B q pulls back under (5–3) to thetensor product of the U p q -connections on Ker p F Ñ B q and Ker p F ν Ñ B q . Said interms of the characteristic line bundles, there is an isomorphism L T V X b L ν Ñ ι (cid:6) L T V X which is compatible with the metrics and connections.We now prove a lemma which shows that the elements of the image of ˇ ι (cid:6) have goodwave front support. Lemma 5–4 ν (cid:6) € T (cid:6) X (cid:7)(cid:7) X intersects N (cid:6) V X (cid:16) π (cid:6) T (cid:6) B € T (cid:6) X only in the zerosection. Proof
Suppose that x P X and ξ P ν (cid:6) x X p N (cid:6) V X q x . Then ξ annihilates T x X and p ι (cid:6) T V X q x . Since T x X X p ι (cid:6) T V X q x (cid:16) T Vx X , it follows easily that T x X (cid:0)p ι (cid:6) T V X q x (cid:16) p ι (cid:6) TX q x . Thus ξ vanishes. ifferential index theorem Hence for a generator E of q K p X q , the element ˇ ι (cid:6)p E q P δ q K r p X q is the image of aunique element in WF q K r p X q , which we will also call ˇ ι (cid:6)p E q .We now define a certain correction term to take into account the possible non-compatibility between the horizontal distributions on X and X . That is, using (5–1),we construct an explicit form r C P Ω p X ; R q(cid:1) { Image p d q so that(5–5) d r C (cid:16) ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) (cid:1) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) . Namely, put W (cid:16) r , s (cid:2) X and let p : W Ñ X be the projection map. Put F (cid:16) p (cid:6) ι (cid:6) T V X . Consider a spin c -connection p ∇ F on F which is ι (cid:6) p ∇ T V X near t u (cid:2) X and which is p ∇ T V X ` p ∇ ν near t u (cid:2) X . Then r C (cid:16) ³ Todd (cid:1) p ∇ F (cid:9) P Ω p X ; R q(cid:1) { Image p d q . Lemma 5–6
Suppose that p T H X q(cid:7)(cid:7) X (cid:16) T H X . Then(1) The orthogonal projection of ι (cid:6) ∇ T V X to T V X equals ∇ T V X ,and(2) r C (cid:16) Proof
Suppose that p T H X q(cid:7)(cid:7) X (cid:16) T H X . Choose a Riemannian metric g TB on B and construct g TX , ∇ TX , ∇ T V X , g TX , ∇ TX and ∇ T V X as in Definition 3–1. Let P : ι (cid:6) TX Ñ TX be orthogonal projection. By naturality, ∇ TX (cid:16) P (cid:5) ι (cid:6) ∇ TX (cid:5) P and ∇ T V X (cid:16) P (cid:5) ι (cid:6) ∇ T V X (cid:5) P . This proves part (1) of the lemma.As ι (cid:6) p ∇ T V X (cid:16) p ∇ T V X ` p ∇ ν , it follows that r C (cid:16)
0. This proves part (2) of thelemma.
Definition 5–7
Definethe modified pushforward ˇ ι mod (cid:6) p E q P WF q K r p X q by(5–8) ˇ ι mod (cid:6) p E q (cid:16) ˇ ι (cid:6)p E q (cid:1) j (cid:4)(cid:5) r C ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ^ ω p E q ^ δ X (cid:12)(cid:13) . Lemma 5–9 (5–10) ω p ˇ ι mod (cid:6) p E qq (cid:16) Todd (cid:1) p ∇ T V X (cid:9) ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) ^ ω p E q ^ δ X . D. S. Freed and J. Lott
Proof
We have ω p ˇ ι mod (cid:6) p E qq (cid:16) ω p ˇ ι (cid:6)p E qq (cid:1) d r C ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ^ ω p E q ^ δ X (5–11) (cid:16) ω p E q Todd (cid:1) p ∇ ν (cid:9) ^ δ X (cid:1) ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) (cid:1) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ^ ω p E q ^ δ X (cid:16) Todd (cid:1) p ∇ T V X (cid:9) ι (cid:6) Todd (cid:1) p ∇ T V X (cid:9) ^ ω p E q ^ δ X . This proves the lemma.In the next lemma we consider the submersion pushforward in the case of a productbundle, under the assumption that the differential K-theory class on the total space hasan almost-product form.
Lemma 5–12
Let Z be a compact Riemannian spin c -manifold of even dimension n with a unitary connection ∇ L Z on the characteristic line bundle L Z . Let π Z : Z Ñ ptbe the map to a point. Let B be any manifold. Let π prod : Z (cid:2) B Ñ B be projectionon the second factor. Let T Hprod p Z (cid:2) B q be the product horizontal distribution on thefiber bundle Z (cid:2) B Ñ B . Let p : Z (cid:2) B Ñ Z be projection on the first factor.Suppose that E Z (cid:16) p E Z , h E Z , ∇ E Z , φ Z q and E B (cid:16) p E B , h E B , ∇ E B , φ B q are generatorsfor q K n p Z q and q K r (cid:1) n p B q , respectively, for some even integer r . Let π Z (cid:6)p E Z q P K p pt q denote the K-theory pushforward of r E Z s P K n p Z q under the map π Z : Z Ñ pt. (Wecan identify π Z (cid:6)p E Z q with ³ Z Todd p p ∇ TZ q ^ ω p ∇ E Z q (cid:16) Index p D Z , E Z q P Z .) Given φ P WF Ω p Z (cid:2) B q r (cid:1) { Image p d q ,put E (cid:16) p p (cid:6) E Z q (cid:4) pp π prod q(cid:6) E B q (cid:0) j p φ q . Then(5–13) ˇ π prod (cid:6) E (cid:16) π Z (cid:6)p E Z q (cid:4) E B (cid:0) j p π prod (cid:6) p φ qq in q K r (cid:1) n p B q . Proof
Using (2–20), we can write E (cid:16) (cid:1) p (cid:6) E Z b p π prod q(cid:6) E B , p (cid:6) h E Z b p π prod q(cid:6) h E B , p (cid:6) ∇ E Z b I (cid:0) I b p π prod q(cid:6) ∇ E B , (5–14) p (cid:6) φ Z ^ p π prod q(cid:6) ω p ∇ E B q (cid:0) p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B (cid:0) p (cid:6) φ Z ^ p π prod q(cid:6) d φ B (cid:0) φ (cid:9) . ifferential index theorem Also, in this product situation, we have ∇ T V p Z (cid:2) B q (cid:16) p (cid:6) ∇ TZ , Ker p D V q (cid:16) Ker p D Z , E Z qb E B , h Ker p D V q (cid:16) h Ker p D Z , EZ q b h E B and ∇ Ker p D V q (cid:16) I Ker p D Z , EZ q b ∇ E B . Regarding the etaform, as r D V , ∇ H s (cid:16) p ∇ H q (cid:16) T (cid:16)
0, we have(5–15)˜ η (cid:16) u r (cid:1) n R u » 8 STr (cid:2) u (cid:1) dA s ds e (cid:1) u (cid:1) A s (cid:10) ds (cid:16) u r (cid:1) n R u » 8 STr (cid:1) u (cid:1) { D V e (cid:1) s p D V q (cid:9) ds (cid:16) Ω p X ; R q r (cid:1) n (cid:1) { Image p d q , for parity reasons. Thenˇ π prod (cid:6) E (cid:16) (cid:2) Ker p D Z , E Z q b E B , h Ker p D Z , EZ qb h E B , I Ker p D Z , EZ qb ∇ E B , (5–16) π prod (cid:6) (cid:1) p (cid:6) φ Z ^ p π prod q(cid:6) ω p ∇ E B q (cid:0) p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B (cid:0) p (cid:6) φ Z ^ p π prod q(cid:6) d φ B (cid:0) φ (cid:9)(cid:9)(cid:16) (cid:2) Ker p D Z , E Z q b E B , h Ker p D Z , EZ qb h E B , I Ker p D Z , EZ qb ∇ E B ,π Z (cid:6)p ω p ∇ E Z qq (cid:4) φ B (cid:0) π prod (cid:6) p φ q(cid:9)(cid:16) π Z (cid:6)p E Z q (cid:4) E B (cid:0) j p π prod (cid:6) p φ qq . This proves the lemma.The next lemma is a technical result, which will be used later, about the functorialityof reduced eta invariants with respect to product structures.
Lemma 5–17
UnderthehypothesesofLemma5–12,supposeinadditionthat B isanodd-dimensional closed spin c -manifold, equipped with a Riemannian metric g TB andaunitary connection ∇ L B . Then η p B , ˇ π prod (cid:6) E q (cid:16) η p Z (cid:2) B , E q in u r (cid:1) n (cid:1) dim p B q(cid:1) (cid:4) p R { Z q . Proof
Using (5–16), we have η p B , ˇ π prod (cid:6) E q (cid:16) u r (cid:1) n (cid:1) dim p B q(cid:1) (cid:4) Index p D Z , E Z q (cid:4) η p D B , E B q(cid:0) (5–18) » B Todd p p ∇ TB q ^ (cid:1)p π Z (cid:6)p ω p ∇ E Z qq (cid:4) φ B (cid:0) π prod (cid:6) p φ q(cid:9) . By separation of variables, it is easy to show that(5–19) Index p D Z , E Z q (cid:4) η p D B , E B q (cid:16) η p D Z (cid:2) B , p (cid:6) E Z b π (cid:6) E B q . D. S. Freed and J. Lott
Next, » B Todd p p ∇ TB q ^ p π Z (cid:6)p ω p ∇ E Z qq (cid:4) φ B (cid:16) (cid:2)» Z Todd p p ∇ TZ q ^ ω p ∇ E Z q(cid:10) (cid:4) » B Todd p p ∇ TB q ^ φ B (5–20) (cid:16) » Z (cid:2) B Todd p p ∇ T p Z (cid:2) B qq ^ p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B . Also,(5–21) » B Todd p p ∇ TB q ^ π prod (cid:6) p φ q (cid:16) » Z (cid:2) B Todd p p ∇ T p Z (cid:2) B qq ^ φ. Hence η p B , ˇ π prod (cid:6) E q (cid:16) u r (cid:1) n (cid:1) dim p B q(cid:1) (cid:4) η p D Z (cid:2) B , p (cid:6) E Z b π (cid:6) E B q(cid:0) (5–22) » Z (cid:2) B Todd p p ∇ T p Z (cid:2) B qq ^ (cid:1) p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B (cid:0) φ (cid:9) . On the other hand, from (5–14), η p Z (cid:2) B , E q (cid:16) u r (cid:1) n (cid:1) dim p B q(cid:1) (cid:4) η p D Z (cid:2) B , p (cid:6) E Z b π (cid:6) E B q (cid:0) » Z (cid:2) B Todd p p ∇ T p Z (cid:2) B qq^ (5–23) (cid:1) p (cid:6) φ Z ^ p π prod q(cid:6) ω p ∇ E B q (cid:0) p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B (cid:0) p (cid:6) φ Z ^ p π prod q(cid:6) d φ B (cid:0) φ (cid:9)(cid:16) u r (cid:1) n (cid:1) dim p B q(cid:1) (cid:4) η p D Z (cid:2) B , p (cid:6) E Z b π (cid:6) E B q(cid:0)» Z (cid:2) B Todd p p ∇ T p Z (cid:2) B qq ^ (cid:1) p (cid:6) ω p ∇ E Z q ^ p π prod q(cid:6) φ B (cid:0) φ (cid:9) . This proves the lemma.We now work towards the construction of the topological index, beginning with a resultabout embedding in spheres.
Lemma 5–24
Suppose that π : X Ñ B is a fiber bundle with X compact and even-dimensionalfibersofdimension n . Supposethat π hasaRiemannianstructure. Given N even,let π prod : S N (cid:2) B Ñ B be the product bundle. Then for large N ,there are anembedding ι : X Ñ S N (cid:2) B and a Riemannian metric on S N (independent of b P B )so that ι is compatible with the Riemannian structures on π and π prod . (In applyingDefinition 5–2,wetake X (cid:16) X and X (cid:16) S N (cid:2) B .) ifferential index theorem Proof
Let g TB be any Riemannian metric on B . Using the Riemannian structure on π , there is a corresponding Riemannian metric g TX on X . Let e : X Ñ S N be anyisometric embedding of X into an even-dimensional sphere with some Riemannianmetric. Put ι p x q (cid:16) p e p x q , π p x qq P S N (cid:2) B .Next, we establish a K ¨unneth-type formula for the differential K-theory of S N (cid:2) B .We endow S N with an arbitrary Riemannian metric and an arbitrary unitary connection ∇ L SN on its characteristic line bundle L S N . Lemma 5–25
Given E P WF q K r p S N (cid:2) B q with r and N even, consider the fibering π prod : S N (cid:2) B Ñ B . Let p : S N (cid:2) B Ñ S N be projection onto the first factor.Thentherearegenerators t E S N i u i (cid:16) for q K N p S N q ,generators t E Bi u i (cid:16) for q K r (cid:1) N p B q ,and φ P WF Ω p S N (cid:2) B q r (cid:1) { Image p d q so that(5–26) E (cid:16) ¸ i (cid:16) p p (cid:6) E S N i q (cid:4) pp π prod q(cid:6) E Bi q (cid:0) j p φ q . Proof
By the K ¨unneth formula in K-theory, we can write(5–27) c p E q (cid:16) ¸ i (cid:16) p (cid:6) e S N i (cid:4) p π prod q(cid:6) e Bi for additive generators t e S N i u i (cid:16) of K N p S N q and classes t e Bi u i (cid:16) in K r (cid:1) N p B q . Lift the e i ’s to differential K-theory classes E i . Then the exact sequence (3–5) implies theexistence of φ . Remark 5–28
It is possible to replace the compact manifold S N in Lemma 5–25with the noncompact affine space A N , provided that we use currential K -theory withcompact supports. In that case the summation in (5–26) would only have a singleterm, and we could remove the assumption that X is compact in Lemma 5–24. Wechose to avoid introducing compact supports, at the expense of having a slightly morecomplicated lemma.Using Lemma 5–12, we obtain an explicit formula for the pushforward, under theproduct submersion S N (cid:2) B Ñ B , of a differential K-theory class of the type consideredin Lemma 5–25. We now show that the result is independent of the particular K ¨unneth-type representation chosen. D. S. Freed and J. Lott
Lemma 5–29
Givenagenerator E for WF q K r p S N (cid:2) B q with r and N even,write E asin(5–26). Applythemap ˇ π prod (cid:6) inLemma5–12to E inthecase Z (cid:16) S N ,togetanelementof q K r (cid:1) N p B q . Thentheresultfactorsthroughamap ˇ π prod (cid:6) : WF q K r p S N (cid:2) B q Ñ q K r (cid:1) N p B q ,which isindependent of theparticular decomposition (5–26)chosen. Proof
We refer to the notation in the proof of Lemma 5–25. Let t , x u be anadditive basis of K p S N q , where 1 is the trivial bundle of rank 1 and x has rank 0.Choose e S N (cid:16) u N { e S N (cid:16) u N { x , where u denotes the Bott element in K-theory.Without loss of generality, we can assume that x is chosen so that π S N (cid:6) p u N { x q (cid:16) P Z .Given a differential K-theory class E as in (5–26), Lemma 5–12 implies that(5–30) ˇ π prod (cid:6) E (cid:16) E B (cid:0) j p π prod (cid:6) p φ qq . Now a different decomposition, as in (5–26), of the same differential K-theory class E , can only arise by the changes(5–31) E S N i ÝÑ E S N i (cid:0) j p α S N i q , E Bi ÝÑ E Bi (cid:0) j p α Bi q ,φ ÝÑ φ (cid:1) ¸ i (cid:16) (cid:16) p (cid:6) α S N i ^ p π prod q(cid:6) ω p E Bi q (cid:0) p (cid:6) ω p E S N i q ^ p π prod q(cid:6) α Bi (cid:24) for some α S N i P Ω p S N ; R q N (cid:1) { Image p d q and α Bi P Ω p B ; R q r (cid:1) N (cid:1) { Image p d q Theensuing change in the right-hand side of (5–30) is(5–32) j p α B q(cid:1) ¸ i (cid:16) j (cid:1) π prod (cid:6) (cid:1) p (cid:6) α S N i ^ p π prod q(cid:6) ω p E Bi q (cid:0) p (cid:6) ω p E S N i q ^ p π prod q(cid:6) α Bi (cid:9)(cid:9) . As π S N (cid:6) (cid:1) α S N i (cid:9) (cid:16) π S N (cid:6) (cid:1) ω p E S N q(cid:9) (cid:16) π S N (cid:6) (cid:1) ω p E S N q(cid:9) (cid:16)
1, the expression in (5–32)vanishes. The lemma follows.The point of Lemma 5–29 is that it gives us a well-defined map ˇ π prod (cid:6) : WF q K r p S N (cid:2) B q Ñq K r (cid:1) N p B q which agrees with the pushforward defined in Section 3 when applied toelements of WF q K r p S N (cid:2) B q that are written in the form (5–26), and which can becomputed explicitly, but does not need any spectral analysis. In particular, ˇ π prod (cid:6) isdefined without any condition about vector bundle kernel. (Note that if E is a generalHermitian vector bundle on S N (cid:2) B and ∇ E is a general compatible connection on E then there is no reason that Ker p D V q should form a vector bundle on B .)We now define the topological index for compact base spaces B ; the extension toproper submersions with noncompact B is described at the end of Section § 7. ifferential index theorem Definition 5–33
Let π : X Ñ B be a fiber bundle with X compact. Put n (cid:16) dim p X q (cid:1) dim p B q , which we assume to be even. Suppose that π has a Riemannianstructure. Construct N and ι from Lemma 5–24. Given a generator E for q K p X q ,construct ˇ ι mod (cid:6) p E q P WF q K N (cid:1) n p S N (cid:2) B q from Definition 5–7. Write ˇ ι mod (cid:6) p E q as inequation(5–26). UsingLemma5–29,definethetopological index ind top p E q P q K (cid:1) n p B q by(5–34) ind top p E q (cid:16) ˇ π prod (cid:6) p ˇ ι mod (cid:6) p E qq . Lemma 5–35 (5–36) ω p ind top p E qq (cid:16) π (cid:6)p ω p E qq . Proof
From Lemmas 3–13 and 5–9, ω p ind top p E qq (cid:16) ω p ˇ π prod (cid:6) p ˇ ι mod (cid:6) p E qqq (5–37) (cid:16) π prod (cid:6) p ω p ˇ ι mod (cid:6) p E qqq(cid:16) π prod (cid:6) (cid:4)(cid:5) Todd (cid:1) p ∇ T V X (cid:9) ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B q(cid:9) ^ ω p E q ^ δ X (cid:12)(cid:13)(cid:16) π (cid:6)p ω p E qq . This proves the lemma.
Proposition 5–38
The following diagram commutes :(5–39) 0 Ω p X ; R q(cid:1) Ω p X ; R q(cid:1) K π (cid:6) j q K p X q ind top c K p X ; Z q ind top Ω p B ; R q(cid:1) n (cid:1) Ω p B ; R q(cid:1) n (cid:1) K j q K (cid:1) n p B q c K (cid:1) n p B ; Z q . Proof
The right-hand square commutes from our construction of ind top : q K p X q Ñq K (cid:1) n p B q ; see the discussion at the beginning of Section 4 of the K-theory pushforwardunder an embedding. To see that the left-hand square commutes, suppose that φ P D. S. Freed and J. Lott Ω p X ; R q(cid:1) Ω p X ; R q(cid:1) K . Thenind top p j p φ qq (cid:16) ˇ π prod (cid:6) p ˇ ι mod (cid:6) p j p φ qqq (5–40) (cid:16) ˇ π prod (cid:6) (cid:4)(cid:5) j (cid:4)(cid:5) φ Todd (cid:1) p ∇ ν (cid:9) ^ δ X (cid:1) r C ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ^ d φ ^ δ X (cid:12)(cid:13)(cid:12)(cid:13)(cid:16) j (cid:4)(cid:5)» X (cid:4)(cid:5) ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1) r C Todd (cid:1) p ∇ ν (cid:9) ^ d φ (cid:12)(cid:13)(cid:12)(cid:13)(cid:16) j (cid:4)(cid:5)» X ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B (cid:9) (cid:1) d r C Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:12)(cid:13)(cid:16) j (cid:2)» X Todd (cid:0) T V X (cid:8) ^ φ (cid:10) (cid:16) j p π (cid:6)p φ qq . This proves the lemma.From what has been said so far, the map ind top : q K p X q Ñ q K (cid:1) n p B q depends onthe Riemannian structure on π and, possibly, on the embedding ι . We prove inCorollary 7–33 that it is in fact independent of ι . In this section we prove our index theorem for families of Dirac operators, under theassumption of vector bundle kernel and compact base space.In terms of the diagram(6–1) 0 K (cid:1) p X ; R { Z q j q K p X q ind an ind top ω Ω p X ; R q K π (cid:6) K (cid:1) n (cid:1) p B ; R { Z q j q K (cid:1) n p B q ω Ω p B ; R q(cid:1) nK , we know that if E P q K p X q then ω p ind an p E q (cid:1) ind top p E qq (cid:16)
0. Hence ind an p E q (cid:1) ind top p E q is the image under j of a unique element in K (cid:1) n (cid:1) p B ; R { Z q . We nowapply the method of proof of [33, Section 4] to prove that the difference vanishes, ifferential index theorem by computing its pairings with elements of K (cid:1) n (cid:1) p B q . From Lemma 2–31(2), suchpairings are given by reduced eta-invariants. As in [33, Section 4], the pairing withan element of K (cid:1) n (cid:1) p B q becomes a computation of reduced η -invariants on X aftertaking adiabatic limits. A new ingredient is the use of the main theorem of [13] inorder to relate the reduced eta-invariants of a manifold and an embedded submanifold. Theorem 6–2
Let π : X Ñ B beafiberbundlewithcompactfibersofevendimension.Suppose that π is equipped with a Riemannian structure and a differential spin c structure. Assume that X is compact and that Ker p D V q Ñ B is a vector bundle. Thenfor all E P q K p X q wehave ind an p E q (cid:16) ind top p E q . Proof
The short exact sequence (2–26), along with Lemmas 3–13 and 5–35, impliesthat ind an p E q (cid:1) ind top p E q lifts uniquely to an element T of K (cid:1) n (cid:1) p B ; R { Z q . We wantto show that this element vanishes. To do so, we use the method of proof of [33, Section4]. From the universal coefficient theorem and the divisibility of R { Z , it suffices toshow that for all α P K (cid:1) n (cid:1) p B ; Z q , the pairing x α, T y vanishes in R { Z . From [27], K (cid:1) n (cid:1) p B ; Z q is generated by elements of the form α (cid:16) f (cid:6)r M s where M is a closedodd-dimensional spin c -manifold, r M s P K (cid:1) n (cid:1) p M ; Z q is the fundamental class of M (shifted from K dim p M qp M ; Z q to K (cid:1) n (cid:1) p M ; Z q using Bott periodicity) and f : M Ñ B is a smooth map. (The argument in [33, Section 4] used instead the Baum-Douglasdescription of K-homology [8], which essentially involves an additional vector bundleon M .) As x α, T y (cid:16) xr M s , f (cid:6) T y , we can effectively pull everything back to M and so reduce to considering the case when B is an arbitrary closed odd-dimensionalspin c -manifold.Now suppose that E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) . Recall the construction of ˇ ι (cid:6)p E q (cid:16) (cid:0) F , h F , ∇ F , ι (cid:6)p φ q (cid:0) γ (cid:8) from Definition 4–13.In the rest of this proof, all equalities will be taken modulo the integers, so will be writtenas congruences. We equip B with a Riemannian metric g TB , and the characteristic linebundle L B with a unitary connection ∇ L B . We equip the fiber bundle S N (cid:2) B Ñ B with the product horizontal connection T Hprod p S N (cid:2) B q . Then S N (cid:2) B has the productRiemannian metric, from which the submanifold X acquires a Riemannian metric.By Proposition 2–31 and Lemma 5–17,(6–3) u (cid:1) dim p X q(cid:0) xr B s , T y (cid:17) C (cid:1) C D. S. Freed and J. Lott in u (cid:1) dim p X q(cid:0) (cid:4) p R { Z q , where C (cid:17) η p B , ind an p E qq (6–4) (cid:17) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:0) (cid:9) (cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:1) (cid:9) (cid:0) » B Todd (cid:1) p ∇ TB (cid:9) ^ p π (cid:6)p φ q (cid:0) r η q(cid:17) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:0) (cid:9) (cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:1) (cid:9) (cid:0) » B Todd (cid:1) p ∇ TB (cid:9) ^ r η (cid:0)» X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) ^ Todd (cid:1) p ∇ T V X (cid:9) ^ φ and, using Lemma 5–17, C (cid:17) η p B , ind top p E qq (cid:17) η p S N (cid:2) B , ˇ ι mod (cid:6) p E qq (6–5) (cid:17) u (cid:1) dim p X q(cid:0) η (cid:1) D S N (cid:2) B , F (cid:0)(cid:9) (cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D S N (cid:2) B , F (cid:1)(cid:9) (cid:0) » S N (cid:2) B Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) ^(cid:4)(cid:5) ι (cid:6) φ (cid:1) γ (cid:1) r C ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B q(cid:9) ^ Todd (cid:1) p ∇ ν (cid:9) ^ ω p E q ^ δ X (cid:12)(cid:13)(cid:17) u (cid:1) dim p X q(cid:0) η (cid:1) D S N (cid:2) B , F (cid:0)(cid:9) (cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D S N (cid:2) B , F (cid:1)(cid:9) (cid:0)» X ι (cid:6) Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1) » S N (cid:2) B Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) ^ γ (cid:1)» X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ p ω p ∇ E q (cid:0) d φ q . From [13, Theorem 2.2],(6–6) η (cid:1) D S N (cid:2) B , F (cid:0)(cid:9) (cid:1) η (cid:1) D S N (cid:2) B , F (cid:1)(cid:9) (cid:17) η (cid:0) D X , E (cid:8) (cid:0) u dim p X q(cid:0) » S N (cid:2) B Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) ^ γ. ifferential index theorem Thus C (cid:17) u (cid:1) dim p X q(cid:0) η (cid:0) D X , E (cid:8) (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q (cid:0) » X ι (cid:6) Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1) (6–7) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ d φ. Now » X ι (cid:6) Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ d φ (cid:17) (6–8) » X ι (cid:6) Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ d r C ^ φ (cid:17)» X ι (cid:6) Todd (cid:1) p ∇ T p S N (cid:2) B q(cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ φ (cid:1)» X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ (cid:1) ι (cid:6) Todd (cid:1) p ∇ T V p S N (cid:2) B q(cid:9) (cid:1) Todd (cid:1) p ∇ T V X (cid:9) ^ Todd (cid:1) p ∇ ν (cid:9)(cid:9) ^ φ (cid:17)» X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) ^ Todd (cid:1) p ∇ T V X (cid:9) ^ φ. Then C (cid:1) C (cid:17) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:0) (cid:9) (cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:1) (cid:9) (cid:0) » B Todd (cid:1) p ∇ TB (cid:9) ^ r η (cid:1) (6–9) u (cid:1) dim p X q(cid:0) η (cid:0) D X , E (cid:8) (cid:0) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q . The next lemma, stated in terms of bordisms, shows that C (cid:1) C is unchanged bycertain perturbations. D. S. Freed and J. Lott
Lemma 6–10
Suppose that B (cid:16) B B for some even-dimensional compact spin c -manifold B . Supposethatthestructures, g TB , ∇ L B , π : X Ñ B , T H X , ι : X Ñ S N (cid:2) B , E Ñ X and ∇ E extendtostructures g TB , ∇ L B , π : X B , T H X , ι : X S N (cid:2) B , E X and ∇ E over B , which are product-like near B (cid:16) B B . Suppose thatKer p D V q1 forms a Z { Z -graded vector bundle on B . Then C (cid:1) C (cid:17) Proof
From Lemma 2–31,(6–11) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:0) (cid:9)(cid:1) u (cid:1) dim p X q(cid:0) η (cid:1) D B , Ker p D V q(cid:1)(cid:9) (cid:17) » B Todd p p ∇ TB ω (cid:1) ∇ Ker p D V q1(cid:9) and(6–12) u (cid:1) dim p X q(cid:0) η (cid:0) D X , E (cid:8) (cid:17) » X Todd p p ∇ TX
1q ^ ω (cid:1) ∇ E . Also, » B Todd (cid:1) p ∇ TB (cid:9) ^ r η (cid:17) » B Todd (cid:1) p ∇ TB d r η (6–13) (cid:17) » X π Todd (cid:1) p ∇ TB Todd (cid:1) p ∇ T V X ω p ∇ E B Todd (cid:1) p ∇ TB ω (cid:1) ∇ Ker p D V q1 (cid:9) and » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q (cid:17) (6–14) » X π Todd (cid:1) p ∇ TB Todd (cid:1) p ∇ ν d r C ω p ∇ E
1q (cid:17)» X π Todd (cid:1) p ∇ TB Todd (cid:1) p ∇ ν ι Todd (cid:1) p ∇ T V p S N (cid:2) B Todd (cid:1) p ∇ T V X Todd (cid:1) p ∇ ν ω p ∇ E
1q (cid:17)» X Todd (cid:1) p ∇ TX ω p ∇ E
1q (cid:1) » X π Todd (cid:1) p ∇ TB Todd (cid:1) p ∇ T V X ω p ∇ E . The lemma follows from combining equations (6–11)-(6–14).Continuing with the proof of Theorem 6–2, we apply Lemma 6–10 with B , s(cid:2) B ,so B B B (cid:1) B . If p : r , s (cid:2) B Ñ B is the projection map then we take all of ifferential index theorem the structures on B to be pullbacks under p of the corresponding structures on B ,except for the horizontal distribution T H X . Note that the property of having vectorbundle kernel is independent of the choice of horizontal distribution. We choose T H X to equal p (cid:6) T H X near t u (cid:2) B , and to equal p (cid:6)p T Hprod p S N (cid:2) B qq(cid:7)(cid:7) X near t u (cid:2) B . ThenLemma 6–10 implies the computation of C (cid:1) C for B equals that for B . Thuswithout loss of generality, we can assume that T H X (cid:16) p T Hprod p S N (cid:2) B qq(cid:7)(cid:7) X . In this case, r C vanishes from Lemma 5–6.Next, we apply Lemma 6–10 with B , s (cid:2) B and with all of the structureson B pulling back from B , except for the Riemannian metrics. Given ǫ ¡
0, let ρ : r , s Ñ R (cid:0) be a smooth function which is ǫ near t u and which is 1 near t u .Multiply the fiberwise metrics for the Riemannian structures π : r , s(cid:2) X Ñ r , s(cid:2) B and π prod : r , s (cid:2) S N (cid:2) B Ñ r , s (cid:2) B by a factor ρ p t q , for t P r , s . By doingso, we do not alter the property of having vector bundle kernel. Then Lemma 6–10implies the computation of C (cid:1) C for B equals that for B . That is, C (cid:1) C isunchanged after scaling the metrics by ǫ .Hence it suffices to compute C (cid:1) C in the limit when ǫ Ñ
0. In this case, it is known[17, Theorem 0.1], [33, Section 4] that(6–15)lim ǫ Ñ η (cid:0) D X , E (cid:8) (cid:17) η (cid:1) D B , Ker p D V q(cid:0)(cid:9) (cid:1) η (cid:1) D B , Ker p D V q(cid:1)(cid:9) (cid:0) u dim p X q(cid:0) » B Todd (cid:1) p ∇ TB (cid:9) ^ r η in R { Z . Thus C (cid:1) C (cid:17)
0. The theorem follows.
Corollary 6–16 (1) Theassignment E Ñ ind an p E q factorsthroughamap q K p X q Ñq K (cid:1) n p B q .(2) The map ind top : q K p X q Ñ q K (cid:1) n p B q is independent of the choice of embedding ι . Proof
Part (1) follows from Theorem 6–2 and the fact that the assignment E Ñ ind top p E q factors through a map q K p X q Ñ q K (cid:1) n p B q . Part (2) follows from Theorem6–2 and the fact that ind an is independent of the choice of embedding ι . In this section we complete the proof of the differential K -theory index theorem.In general, the kernels of a family of Dirac operators need not form a vector bundle.In such a case, the basic idea is to perform a finite-rank perturbation of the operators, D. S. Freed and J. Lott in order to effectively reduce to the case of vector bundle kernel. One way to do this,used in [7] is to enlarge the domain of p D V q(cid:0) by the sections of a trivial bundle over B , in order to make a finite rank change so that p D V q(cid:0) becomes surjective; this impliesvector bundle kernel. We instead follow the method of [33, Section 5], which uses alemma of Mischenko-Fomenko (Lemma 7–11) to find a finite rank subbundle of theinfinite rank bundle H which captures the index. Adding on this finite rank subbundle,with the opposite grading, allows one to alter the operator to make it invertible.An additional technical issue arises in trying to construct the eta form. We want tomake the D V -term in the integrand invertible for large s , but we want to keep thesmall- s asymptotics of the unperturbed Bismut superconnection. As in [33, Section5], we use the trick of “time-varying η -forms”, which originated in [37].In Subsection 7.1 we recall some facts about “time-varying η -invariants” and “time-varying η -forms”. In Subsection 7.2 we review the Mischenko-Fomenko result andand construct the analytic pushforward in the general case (Definition 7–25). Afterthese preliminaries, in Subsection 7.3 we prove the general index theorem along thelines of the argument in the previous section. Finally, in Subsection 7.4 we use the limittheorem in the appendix to extend the theorem to proper fiber bundles with arbitrarybase. We first review some material from [33] about eta invariants and eta forms, which isan adaptation of [11] to the time-varying case.Let B be a closed odd-dimensional manifold. Let D be a smooth 1-parameter familyof first-order self-adjoint elliptic pseudodifferential operators D p s q on B , such that • There are a δ ¡ D on X such that for s P p , δ q , we have D p s q (cid:16) sD . • There are a ∆ ¡ D on X such that for s P p ∆ , , we have D p s q (cid:16) sD .For z P C with Re p z q ¡¡
0, put(7–1) η p D qp z q (cid:16) ? π » 8 s z Tr (cid:2) dD p s q ds e (cid:1) D p s q (cid:10) ds . Lemma 7–2 [33,Lemma2] η p D qp z q extendstoameromorphicfunctionon C whichisholomorphic near z (cid:16) ifferential index theorem Define the eta-invariant of D by(7–3) η p D q (cid:16) η p D qp q and define the reduced eta-invariant of D by(7–4) η p D q (cid:16) η p D q (cid:0) dim p Ker p D p mod Z q . Lemma 7–5 [33, Lemma 3] η p D q only depends on D and D , and η p D q onlydepends on D .Now suppose that B additionally is a Riemannian spin c -manifold, equipped with aspin c -connection p ∇ TB on the spinor bundle S B . Let E be a Z { Z -graded vectorbundle over B . We think of r E s as defining an element of K (cid:1) n p B q , for some even n . If A is a superconnection on E and s P R (cid:0) , let A s denote the result of multiplying eachfactor of u in A by s .Let A (cid:16) t A p s qu s ¥ be a smooth 1-parameter family of superconnections on E suchthat • There are a δ ¡ A on E such that for s P p , δ q , wehave A p s q (cid:16) p A q s . • There are a ∆ ¡ A on E such that for s P p ∆ , ,we have A p s q (cid:16) p A s .Suppose that A is invertible. For z P C , Re p z q ¡¡
0, define r η p A qp z q P Ω p B ; R q(cid:1) n (cid:1) { Image p d q by(7–6) r η p A qp z q (cid:16) u (cid:1) n { R u » 8 z s str (cid:2) u (cid:1) dA p s q ds e (cid:1) u (cid:1) A p s q (cid:10) ds . Lemma 7–7 [33, Lemma 4] r η p A qp z q extends to a meromorphic vector-valued func-tion on C with simple poles. Its residue at zero vanishes in Ω p B ; R q(cid:1) n (cid:1) { Image p d q .Define the eta form of A by(7–8) r η p A q (cid:16) r η p A qp q . As in Lemma 7–5, r η p A q only depends on A and A .Given a superconnection A on E , let A denote the associated first-order differentialoperator [9, Section 3.3]. It is the essentially self-adjoint operator on C E b S B q obtained by replacing the Grassmann variables in A by Clifford variables and replacing D. S. Freed and J. Lott u by 1. Now given a family A of superconnections as above and a parameter ǫ ¡ D p ǫ q by(7–9) D p ǫ qp s q (cid:16) A p s q ǫ (cid:1) . (In the fiber bundle situation, this corresponds to multiplying the fiber lengths by afactor of ǫ . The paper [11] instead expands the base, but the two approaches areequivalent.) Let η p D p ǫ qq be the corresponding eta invariant. Then a generalization of[11, (A.1.7)] says that(7–10) lim ǫ Ñ η p D p ǫ qq (cid:16) u dim p B q(cid:0) n (cid:0) » B Todd p p ∇ TB q ^ r η p A q p mod Z q . We continue with the setup of Section 3, namely a family of Dirac-type operators,except that we no longer assume that Ker p D V q forms a smooth vector bundle on B .In order to deal with this more general situation, we will use a perturbation argument,following the approach of [33, Section 5]. For this, we need to assume that B iscompact.We first recall a technical lemma of Mischenko-Fomenko, along with its proof. Lemma 7–11 ([39]) Suppose that B is compact. Then there are finite-dimensionalvector subbundles L (cid:8) € H (cid:8) and complementary closed subbundles K (cid:8) € H (cid:8) ,i.e.(7–12) H (cid:8) (cid:16) K (cid:8) ` L (cid:8) , sothat D V (cid:0) P Hom p H (cid:0) , H (cid:1)q isblock diagonal asa map(7–13) D V (cid:0) : K (cid:0) ` L (cid:0) Ñ K (cid:1) ` L (cid:1) and D V (cid:0) restricts to an isomorphism between K (cid:0) and K (cid:1) . (Note that K (cid:8) may not beorthogonal to L (cid:8) .) Proof
This is proved in [39, Lemma 2.2]. For completeness, we sketch the argument.One first finds finite-dimensional vector subbundles L H (cid:8) so that the projectedmap p L D V (cid:0)Ñ H (cid:1) Ñ p L is an isomorphism. With respect to the orthogonaldecomposition H (cid:8) (cid:16) p L L , write(7–14) D V (cid:0) (cid:16) (cid:2) A BC D (cid:10) ifferential index theorem where A : p L L is an isomorphism. Set K (cid:0) (cid:16) p L , (7–15) L (cid:0) (cid:16) Image pp(cid:1) A (cid:1) B (cid:0) I q : L H (cid:0)q , K (cid:1) (cid:16) Image pp I (cid:0) CA (cid:1) q : p L H (cid:1)q , L (cid:1) (cid:16) L . This proves the lemma.Let i (cid:1) : L (cid:1) Ñ H (cid:1) be the inclusion map and let p (cid:0) : H (cid:0) Ñ L (cid:0) be the projectionmap coming from (7–12). Put r H (cid:8) (cid:16) H (cid:8) ` L (cid:9) . Given α P C , define r D V (cid:0)p α q P Hom p r H (cid:0) , r H (cid:1)q by the matrix(7–16) r D V (cid:0)p α q (cid:16) (cid:2) D V (cid:0) α i (cid:1) α p (cid:0) (cid:10) . That is,(7–17) r D V (cid:0)p α qp h (cid:0) ` l (cid:1)q (cid:16) p D V (cid:0) h (cid:0) (cid:0) α i (cid:1) l (cid:1) ` α p (cid:0) h (cid:0)q . Lemma 7–18 If α (cid:24) r D V (cid:0)p α q isinvertible. Proof
Suppose that r D V (cid:0)p α qp h (cid:0) ` l (cid:1)q (cid:16)
0. As p (cid:0) h (cid:0) (cid:16)
0, we know that h (cid:0) P K (cid:0) .Then D V (cid:0) h (cid:0) P K (cid:1) . As D V (cid:0) h (cid:0) (cid:0) α i (cid:1) l (cid:1) (cid:16)
0, we conclude that D V (cid:0) h (cid:0) (cid:16) l (cid:1) (cid:16) D V (cid:0) is injective on K (cid:0) , it follows that h (cid:0) (cid:16)
0. Hence r D V (cid:0)p α q is injective.Now suppose that h l H (cid:1) . With respect to (7–12), write h k l . Put h (cid:0) (cid:16) (cid:1) D V (cid:0)(cid:7)(cid:7) K (cid:0)(cid:9)(cid:1) k α (cid:1) l , (7–19) l (cid:1) (cid:16) α (cid:1) l α (cid:1) D V (cid:0) l . One can check that r D V (cid:0)p h (cid:0) ` l (cid:1)q (cid:16) h l . Thus r D V (cid:0)p α q is surjective.Define r D V p α q P End p r H q by r D V p α q (cid:16) r D V (cid:0)p α q`(cid:1)r D V (cid:0)p α q(cid:9)(cid:6) , an essentially self-adjointoperator on each fiber r H b . As r D V p α q is a finite-rank perturbation of D V ` I L , and p I (cid:0)p D V q q(cid:1) is compact on each fiber H b , it follows that p I (cid:0)pr D V p α qq q(cid:1) is compacton each fiber r H b . Lemma 7–18 now implies that if α (cid:24) (cid:1)r D V p α q(cid:9) has strictlypositive spectrum.Give L the projected Hermitian inner product h L and projected compatible connection ∇ L from H . Put ∇ r H (cid:8) (cid:16) ∇ H (cid:8) ` ∇ L (cid:9) . Let α : r ,
8q Ñ r , s be a smooth function D. S. Freed and J. Lott for which α p s q (cid:16) s is near 0, and α p s q (cid:16) s ¥
1. We view r L s as defining anelement of K (cid:1) n p B q .For s ¡
0, define a superconnection r A s by(7–20) r A s (cid:16) su { r D V p α p s qq (cid:0) ∇ r H (cid:1) s (cid:1) u (cid:1) { c p T q . Then(7–21) lim s Ñ u (cid:1) n { R u STr (cid:1) e (cid:1) u (cid:1) r A s (cid:9) (cid:16) π (cid:6)p ω p ∇ E qq (cid:1) ω p ∇ L q , while(7–22) lim s Ñ8 u (cid:1) n { R u STr (cid:1) e (cid:1) u (cid:1) r A s (cid:9) (cid:16) . Note that unlike in Subsection 7.1, we do not have to use zeta-function regularizationbecause when s Ñ
0, the H and L factors in r H decouple and so we are reduced to theshort-time asymptotics of the Bismut superconnection (3–7).Put(7–23) ˜ η (cid:16) u (cid:1) n { R u » 8 STr (cid:3) u (cid:1) d r A s ds e (cid:1) u (cid:1) r A s (cid:11) ds P Ω p B ; R q(cid:1) n (cid:1) { Image p d q . It is independent of the particular choice of the function α . Also,(7–24) d ˜ η (cid:16) π (cid:6)p ω p ∇ E qq (cid:1) ω p ∇ L q . Definition 7–25
Given a generator E (cid:16) (cid:0) E , h E , ∇ E , φ (cid:8) for q K p X q , we define theanalytic index(7–26) ind an p E q (cid:16) (cid:0) L , h L , ∇ L , π (cid:6)p φ q (cid:0) r η (cid:8) asan element of q K (cid:1) n p B q ,where L is chosen asin Lemma7–11.Given a generator E of WF q K p X q , we define ind an p E q P q K (cid:1) n p B q by the same formula(7–26). We prove in Corollary 7–33 that this definition is independent of the choiceof L . Lemma 7–27 If E is a generator for WF q K p X q then ω p ind an p E qq (cid:16) π (cid:6)p ω p E qq in Ω p B ; R q(cid:1) n . Proof
We have(7–28) ω p ind an p E qq (cid:16) ω p ∇ L q (cid:0) d p π (cid:6)p φ q (cid:0) r η q (cid:16) π (cid:6)p ω p ∇ E q (cid:0) d φ q (cid:16) π (cid:6)p ω p E qq , which proves the lemma. ifferential index theorem Continuing with the assumptions of the previous subsection, suppose that B is a closedodd-dimensional Riemannian manifold with a spin c -structure. Let p ∇ TB be a spin c -connection on S B . Combining with the Riemannian structure on π and the differentialspin c -structure on π , we obtain a Riemannian metric g TX on X and a spin c -connection p ∇ TX on S X .As in 7.1, given a parameter ǫ ¡
0, we define a family of pseudodifferential operators D p ǫ q (living on X ) by(7–29) D p ǫ qp s q (cid:16) pr A s q ǫ (cid:1) . Then the family D p ǫ q satisfies the formalism of 7.1.To identify the operators D and D corresponding to the family D p ǫ q , let X ǫ denotethe Riemannian structure on X coming from multiplying g TX in the vertical directionby ǫ . If s is near zero then α p s q vanishes and the superconnection r A s of (7–20) justbecomes the direct sum of the Bismut superconnection on H and the connection on Π L , the latter being L with the opposite grading. Therefore, D (cid:16) D X ǫ , E ` D B , Π L is thesum of ordinary Dirac-type operators on X ǫ and B . On the other hand, if s ¡ ∆ then α p s q (cid:16) r D V p α p s qq is L -invertible. From (7–20), D is the Dirac operator on B coupled to the superconnection ǫ (cid:1) r D V p q (cid:0) ∇ H (cid:1) ǫ c p T q . If ǫ is small then the term ǫ (cid:1) r D V p q dominates when computing the spectrum of D , so D is an invertiblefirst-order self-adjoint elliptic pseudodifferential operator on the disjoint union X \ B .Let η p D p ǫ qq be the reduced eta invariant of the rescaled family D p ǫ q . As in Lemma7–5, η p D p ǫ qq only depends on D . It follows that(7–30) η p D p ǫ qq (cid:16) η p D X ǫ , E q (cid:0) η p D B , Π L q (cid:16) η p D X ǫ , E q (cid:1) η p D B , L q . A generalization of [11, Theorem 4.35] says that(7–31) lim ǫ Ñ η p D p ǫ qq (cid:16) u dim p X q(cid:0) » B Todd p p ∇ TB q ^ r η p r A q p mod Z q . We can now go through the proof of Theorem 6–2, using (7–24) and (7–30)+(7–31) inplace of (3–10) and (6–15), respectively, to derive the following result.
Theorem 7–32
Supposethat π : X Ñ B isasmoothfiberbundlewithcompactfibersof even dimension. Suppose that π is equipped with a Riemannian structure and adifferential spin c -structure. Assume that X is compact. Then for all E P q K p X q , wehave ind an p E q (cid:16) ind top p E q . D. S. Freed and J. Lott
Corollary 7–33 (1) Thehomomorphism ind top : q K p X q Ñ q K (cid:1) n p B q isindependentof the choice ofembedding ι .(2) Theassignment E Ñ ind an p E q factorsthroughahomomorphism ind an : q K p X q Ñq K (cid:1) n p B q .(3) The map ind an : q K p X q Ñ q K (cid:1) n p B q is independent of the choice of finite-dimensional vector subbundle L (cid:8) .(4) If D V hasvectorbundlekernelthentheanalyticindexdefinedinDefinition3–11equals the analytic index defined inDefinition 7–25.Aside from its intrinsic interest, the next proposition will be used in Section 8. Proposition 7–34
The following diagrams commute :(7–35) 0 Ω p X ; R q(cid:1) Ω p X ; R q(cid:1) K π (cid:6) j q K p X q ind an c K p X ; Z q ind an Ω p B ; R q(cid:1) n (cid:1) Ω p B ; R q(cid:1) n (cid:1) K j q K (cid:1) n p B q c K (cid:1) n p B ; Z q K (cid:1) p X ; R { Z q ind an j q K p X q ind an ω Ω p X ; R q K π (cid:6) K (cid:1) n (cid:1) p B ; R { Z q j q K (cid:1) n p B q ω Ω p B ; R q(cid:1) nK . Proof
The commuting of (7–35) follows immediately from the definition of ind an .The left-hand square of (7–36) commutes from the definition of ind an : K (cid:1) p X ; R { Z q Ñ K (cid:1) n (cid:1) p B ; R { Z q in [33]. The right-hand square of (7–36) commutes from Lemma7–27. We use the limit theorem in the appendix to define the index maps for proper submer-sions and extend Theorem 7–32. ifferential index theorem Suppose that π : X Ñ B is a proper submersion of relative dimension n , with n even. Suppose that π is equipped with a Riemannian structure and a differential spin c -structure. Let B € B € (cid:4) (cid:4) (cid:4) be an exhaustion of B by compact codimension-zerosubmanifolds-with-boundary. From Theorem A–2, there is an isomorphism(7–37) q K (cid:1) n p B q (cid:21) lim Ý i q K (cid:1) n p B i q . Put X i (cid:16) π (cid:1) p B i q . Given E P q K p X q , we can define ind top p E (cid:7)(cid:7) X i q P q K (cid:1) n p B i q as inSection 5, after making a choice of embedding ι : X i Ñ S N (cid:2) B i . Clearly if B j € B i thenind top p E (cid:7)(cid:7) X j q , as defined using the restriction of ι to X j , is the restriction of ind top p E (cid:7)(cid:7) X i q to B j . Using the fact from Corollary 7–33 that ind top p E (cid:7)(cid:7) X j q is independent of thechoice of embedding, it follows that we have defined a topological index ind top p E q in q K (cid:1) n p B q (cid:21) lim Ý i q K (cid:1) n p B i q .Similarly, we can define ind an p E (cid:7)(cid:7) X i q P q K (cid:1) n p B i q as in the earlier part of this section,after making a choice of the finite-dimensional vector subbundle L (cid:8) over B i . Clearly if B j € B i then ind an p E (cid:7)(cid:7) X j q , as defined using the restriction of L (cid:8) to B j , is the restrictionof ind an p E (cid:7)(cid:7) X i q to B j . Using the fact from Corollary 7–33 that ind an p E (cid:7)(cid:7) X j q is independentof the choice of vector subbundle, it follows that we have defined an analytic indexind an p E q in q K (cid:1) n p B q (cid:21) lim Ý i q K (cid:1) n p B i q . Theorem 7–38
Suppose that π : X Ñ B is a proper submersion with even relativedimension. Suppose that π isequipped withaRiemannian structure andadifferentialspin c -structure. Thenfor all E P q K p X q ,wehave ind an p E q (cid:16) ind top p E q . Proof
By Theorem 7–32, we know that for each i , ind an p E (cid:7)(cid:7) X i q (cid:16) ind top p E (cid:7)(cid:7) X i q in q K (cid:1) n p B i q . Along with (7–37), the theorem follows. In this section we illustrate how our main index theorem relates to other work in thegeometric index theory of Dirac operators. We first treat the determinant line bundle,using the holonomy theorem of [12] to show that the determinant of the analytic push-forward is the determinant bundle. (For a different approach to this question see [32,Chapter 9]). As a consequence, our main theorem gives a “topological” constructionof the determinant line bundle, equipped with its connection, up to isomorphism. D. S. Freed and J. Lott
Next, in Subsection 8.2 we remark that when specialized to R { Z -valued K-theory, ourtheorem implies that the topological pushforward constructed in Section 4 coincideswith the topological pushforward constructed from generalized cohomology theory.The Chern character map, from topological K-theory to rational cohomology, has adifferential refinement, going from differential K-theory to rational differential coho-mology. In Subsection 8.3 we apply this refined Chern character map to our maintheorem and recover the Riemann-Roch formula of [15].Finally, under certain assumptions, there are geometric invariants of families of Diracoperators which live in higher-degree integral differential cohomology [35]. In Subsec-tion 8.4 we point out that our index theorem computes them in terms of the topologicalpushforward. There is a map Det from q K p X q to isomorphism classes of line bundles on X , equippedwith a Hermitian metric and a compatible connection. (The latter group may beidentified with the integral differential cohomology group q H p X q .) Given a generator E (cid:16) p E , h E , ∇ E , φ q for q K p X q , its image Det p E q is represented by the line bundle Λ max p E q , equipped with the Hermitian metric h Λ max p E q and the connection(8–1) ∇ Det p E q (cid:16) ∇ Λ max p E q (cid:1) π i φ p q . Here φ p q P Ω p X q{ Image p d q denotes u times the component of φ P Ω p X ; R q(cid:1) { Image p d q in u (cid:1) Ω p X q{ Image p d q . Note that changing a particular representative p φ P Ω p X q for φ by an exact form df amounts to acting on the connection ∇ Λ max p E q (cid:1) π i p φ by a gaugetransformation g (cid:16) e π if .Suppose that π : X Ñ B is a compact fiber bundle with fibers of even dimension n , endowed with a Riemannian structure and a differential spin c structure. If E is a generator for q K p X q of the form p E , h E , ∇ E , q then there is a correspondingdeterminant line bundle Det an on B , which is equipped with a Hermitian metric h an (due to Quillen [41]) and a compatible connection ∇ an (due to Bismut-Freed [12]);see [9, Chapter 9.7]. The construction is analytic; for example, the construction of h an uses ζ -functions built from the spectrum of the fiberwise Dirac-type operators D V . Proposition 8–2
After using periodicity to shift ind an p E q P q K (cid:1) n p B q into q K p B q , wehave that Det p ind an p E qq isthe inverse of Det an . ifferential index theorem Proof
Suppose first that Ker p D V q is a Z { Z -graded vector bundle on B . UsingDefinition 3–11 for ind an p E q , it follows that(8–3) Det p ind an p E qq (cid:16) Λ max p Ker p D V q(cid:0)q b (cid:0) Λ max p Ker p D V q(cid:1)q(cid:8)(cid:1) is the inverse of the determinant line bundle. The connection ∇ Det p ind an p E qq is(8–4) ∇ Det p ind an p E qq (cid:16) ∇ Λ max p Ker p D V q(cid:0)qbp Λ max p Ker p D V qqq(cid:1) (cid:1) π i r η p q . From (3–9),(8–5) r η p q (cid:16) (cid:1) π i » 8 STr (cid:1) D V r ∇ H , D V s e (cid:1) s p D V q (cid:9) s ds . In this case of vector bundle kernel, h an differs from the L -metric h L by a factorinvolving the Ray-Singer analytic torsion. Let T P End p Det an q be multiplication by b h L h an , so that T is an isometric isomorphism from (cid:1) Det an , h L (cid:9) to p Det an , h an q . Then(8–6) T (cid:6) p Det an , h an , ∇ an q (cid:16) (cid:1) Det p ind an p E qq(cid:1) , h Det p ind an p E qq(cid:1) , ∇ Det p ind an p E qq(cid:1) (cid:9) ;see [9, Proof of Proposition 9.45].If one does not assume vector bundle kernel then a direct proof of the proposition istrickier for the following reason. The usual construction of the determinant line bundleproceeds by making spectral cuts over suitable open subsets of B , constructing a linebundle with Hermitian metric and compatible connection over each open set, and thenshowing that these local constructions are compatible on overlaps. As Det p ind an p E qq is only defined up to isomorphism, it cannot be directly recovered from its restrictionsto the elements of an open cover of B . For this reason, a direct comparison ofDet p ind an p E qq and Det an is somewhat involved. Instead, we will just compare theirholonomies.Without loss of generality, we can assume that B is connected. Let Æ P B be a basepoint.Let PB denote the smooth maps c : r , s Ñ B with c p q (cid:16) Æ . Let Ω B denote theelements of PB with c p q (cid:16) Æ . A unitary connection on a line bundle over B gives riseto a homomorphism H : Ω B Ñ U p q , the holonomy map. Given H , we can constructa line bundle L Ñ B as L (cid:16) p PB (cid:2) C q{ (cid:18) , where p c , z q (cid:18) p c , z q if c p q (cid:16) c p q and z (cid:16) H p c (cid:1) (cid:4) c q z . There is an evident notion of parallel transport on L Ñ B anda corresponding unitary connection. There is also a unique Hermitian inner producton L Ñ B , up to overall scaling, with which the unitary connection is compatible.Hence, given two Hermitian line bundles on B with compatible connections, if theirholonomies are the same then they are isomorphic. D. S. Freed and J. Lott
To compare the holonomies of the connections on Det p ind an p E qq and Det an around aclosed curve, we pull the fiber bundle structure back to the curve and thereby reduce tothe case when B (cid:16) S . From Definition 7–25, the holonomy around S of ∇ Det p ind an p E qq is e π i ³ S r η p q times the holonomy around S of ∇ Λ max L (cid:0)bp Λ max L (cid:1)q(cid:1) . From (7–30) and(7–31),(8–7) e π i ³ S r η p q (cid:16) e π i p lim ǫ Ñ η p D X ǫ, E q(cid:1) η p D S , L qq . By a standard computation, the holonomy around S of ∇ Λ max L (cid:0)bp Λ max L (cid:1)q(cid:1) equals e π i η p D S , L q . Thus the holonomy around S of ∇ Det p ind an p E qq is e π i lim ǫ Ñ η p D X ǫ, E q . Onthe other hand, the holonomy around S of ∇ Det an is e (cid:1) π i lim ǫ Ñ η p D X ǫ, E q [12, Theorem3.16]. This proves the proposition.As a consequence of Theorem 7–38 and Proposition 8–2, the determinant line bundlewith its Hermitian metric and compatible connection can be constructed up to isomor-phism without using any spectral analysis. This was also derived in [32, Chapter 9],though with a different model of differential K -theory. R { Z -index theory Under the assumptions of Theorem 7–38, there is a topological index ind top : K (cid:1) p X ; R { Z q Ñ K (cid:1) p B ; R { Z q which can be constructed from a general procedure in generalized coho-mology theory. Proposition 8–8
Thefollowing diagram commutes :(8–9) 0 K (cid:1) p X ; R { Z q ind top j q K p X q ind top ω Ω p X ; R q K π (cid:6) K (cid:1) n (cid:1) p B ; R { Z q j q K (cid:1) n p B q ω Ω p B ; R q(cid:1) nK . Proof
The right-hand square commutes by Lemma 5–35. The left-hand square com-mutes from the fact that the diagram (7–36) commutes, along with the facts that theanalytic and topological indices agree in differential K-theory (Theorem 7–38), and in R { Z -valued K-theory [33]. Remark 8–10
In a similar vein, one might think that Theorem 7–38, along with thecommuting of the right-hand squares in (7–35) and (8–9), gives a new and purely ifferential index theorem analytic proof of the Atiyah-Singer families index theorem [7]. However, such is notthe case. The proof of Theorem 7–38 uses Proposition 2–31(2), whose proof uses [4,Theorem (5.3)], whose proof in turn uses the Atiyah-Singer families index theorem [4,Section 8]. For a subring Λ € R we define R Λ (cid:16) Λ r u , u (cid:1) s , analogous to (2–1). There is a notionof differential cohomology q H p X ; R Λ q(cid:13) , the generalized differential cohomology theoryattached to ordinary cohomology with coefficients in the graded ring R Λ . It fits intoexact sequences(8–11) 0 ÝÑ H (cid:13)(cid:1) p X ; p R { Λ qr u , u (cid:1) sq i ÝÝÑ q H (cid:13)p X ; R Λ q ω ÝÝÑ Ω p X ; R q(cid:13) Λ ÝÑ , (8–12) 0 ÝÑ Ω p X ; R q(cid:13)(cid:1) Ω p X ; R q(cid:13)(cid:1) Λ j ÝÝÑ q H p X ; R Λ q(cid:13) c ÝÝÑ H (cid:13)p X ; R Λ q ÝÑ , where Ω p X ; R q(cid:13) Λ denotes the closed R -valued forms on X with periods in R Λ (cid:16) Λ r u , u (cid:1) s .The differential cohomology theory q H p X ; R Λ q(cid:13) is essentially the same as the Cheeger-Simons theory of differential characters [16]. Namely, let C k p X q and Z k p X q denote thegroups of smooth singular k -chains and k -cycles in X , respectively. Then an elementof q H k p X ; R Λ q l corresponds to a homomorphism F : Z k (cid:1) p X q Ñ u l (cid:1) k (cid:4) p R { Λ q withthe property that there is some α P Ω p X ; R q l so that for all c P C k p X q , we have F pB c q (cid:16) ³ c α mod u l (cid:1) k (cid:4) Λ .Given a proper submersion π : X Ñ B of relative dimension n which is oriented inordinary cohomology, there is an “integration over the fiber” map ³ X { B : q H p X ; R Λ q(cid:13) Ñq H p B ; R Λ q(cid:13)(cid:1) n [28, Section 3.4]. In short, if F is a differential character on X and z P Z (cid:6)p B q then the evaluation of ³ X { B on z is F p π (cid:1) p z qq P p R { Λ qr u , u (cid:1) s ; see also[14].There is a Chern character q ch : q K (cid:13)p X q Ñ q H p X ; R Q q(cid:13) . When acting on generators of q K p X q of the form E (cid:16) p E , h E , ∇ E , q , the Chern character q ch p E q was defined in [16,§ 2] as a differential character. It then suffices to additionally define q ch on Image p j q ,where j is the map in (2–19). For this, we may note that there is a natural map of thedomain of j in (2–19) to the domain of j in (8–12), since Ω p X ; R q(cid:13)(cid:1) K € Ω p X ; R q(cid:13)(cid:1) Q .Or, in the language of differential characters, we define the evaluation of q ch p j p φ qq on D. S. Freed and J. Lott a cycle z P Z (cid:6)p X q to be ³ z φ mod R Q . The Chern character on differential K-theorywas also considered in [15, Section 6].In the proof of the next proposition we will make use of the Chern character(8–13) ch (cid:6) : K (cid:13)p X ; Z q Ñ H p X ; R Q q(cid:13) on the K -homology of a space X . Recall from the proof of Theorem 6–2 that every K -homology class can be written as u k f (cid:6)r M s for some integer k and some continuous map f : M Ñ X of a spin c manifold M into X , where r M s P K q p M ; Z q is the fundamentalclass. Let Todd p M q P H p M ; R Q q be the Todd class and let Todd p M q_ P H p M ; R Q q q be its Poincar´e dual. Then(8–14) ch (cid:6)(cid:0) u k f (cid:6)r M s(cid:8) (cid:16) u k f (cid:6)(cid:0) Todd p M q_(cid:8) P H p X ; R Q q q (cid:0) k . The Chern character maps on cohomology and homology are compatible with thenatural pairings, meaning that for a P K ℓ p X ; Z q and α P K ℓ p X ; Z q , we have(8–15) x a , α y (cid:16) x ch (cid:6)p a q , ch p α qy . The Chern character on homology is an isomorphism after tensoring K (cid:13)p X ; Z q with R Q .Recall from [16, § 2] that there are characteristic classes in q H p X ; R Q q(cid:13) . In particular,if W is a real vector bundle with a spin c -structure and p ∇ W is a spin c -connection on theassociated spinor bundle then there is a refined Todd class ~ Todd (cid:1) p ∇ W (cid:9) P q H p X ; R Q q . Proposition 8–16
Let π : X Ñ B be a proper submersion of relative dimension n ,with n even. Suppose that π has a Riemannian structure and a differential spin c -structure. Then for all E P q K p X q ,(8–17) q ch p ind top p E qq (cid:16) » X { B ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p E q P q H p B ; R Q q(cid:1) n . Proof
Put(8–18) ∆ (cid:16) q ch p ind top p E qq (cid:1) » X { B ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p E q . From Lemma 5–35, we have ω p ∆ q (cid:16)
0. Then (8–11) implies that ∆ (cid:16) i p U q for someunique U P H p B ; p R { Q qr u , u (cid:1) sq(cid:1) n (cid:1) . Using the universal coefficient theorem andthe fact that R { Q is divisible, to show that U vanishes it suffices to prove the vanishingof its pairings with H (cid:6)p B ; R Q q . We use the fact that the Chern character (8–13) issurjective after tensoring K (cid:6)p X ; Z q with R Q to argue, as in the proof of Theorem 6–2,that we can reduce to the case when B is a closed odd-dimensional spin c -manifold andwe are evaluating on the Chern character of its fundamental K -homology class. ifferential index theorem We equip B with a Riemannian metric and a unitary connection ∇ L B on the charac-teristic line bundle L B . In the rest of this proof, we work modulo R Q (cid:16) Q r u , u (cid:1) s .From (8–15) and [16, § 9],(8–19) x ch (cid:6)r B s , U y (cid:17) η p B ; ind top p E qq (cid:1) » B » X { B π (cid:6) ~ Todd (cid:1) p ∇ TB (cid:9) Y ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p E q . From (6–5)-(6–8), η p B , ind top p E qq (cid:17) u (cid:1) dim p X q(cid:0) η p D X , E q (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q(cid:0) (8–20) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) ^ Todd (cid:1) p ∇ T V X (cid:9) ^ φ (cid:17) » X ~ Todd p ∇ TX q Y q ch p ∇ E q (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q(cid:0)» X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) ^ Todd (cid:1) p ∇ T V X (cid:9) ^ φ. Here X has the induced Riemannian metric from its embedding in S N (cid:2) B . Thus x ch (cid:6)r B s , U y (cid:17) » X ~ Todd p ∇ TX q Y q ch p ∇ E q (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q(cid:0) (8–21) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) ^ Todd (cid:1) p ∇ T V X (cid:9) ^ φ (cid:1)» X ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p E q(cid:17) » X ~ Todd p ∇ TX q Y q ch p ∇ E q (cid:1) » X π (cid:6) Todd (cid:1) p ∇ TB (cid:9) Todd (cid:1) p ∇ ν (cid:9) ^ r C ^ ω p ∇ E q(cid:1)» X π (cid:6) ~ Todd (cid:1) p ∇ TB (cid:9) Y ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p ∇ E q . As in the proof of Theorem 6–2, we can deform to the case T H X (cid:16) p T H p S N (cid:2) B qq(cid:7)(cid:7) X without changing the right-hand side of (8–21). In this case,(8–22) ~ Todd (cid:1) p ∇ TX (cid:9) (cid:16) π (cid:6) ~ Todd (cid:1) p ∇ TB (cid:9) Y (cid:1) p ∇ T V X (cid:9) D. S. Freed and J. Lott and Lemma 5–6 says that r C (cid:16)
0. The proposition follows.
Corollary 8–23
Let π : X Ñ B beapropersubmersionofrelativedimension n ,with n even. Suppose that π has a Riemannian structure and a differential spin c -structure.Thenfor all E P q K p X q ,(8–24) q ch p ind an p E qq (cid:16) » X { B ~ Todd (cid:1) p ∇ T V X (cid:9) Y q ch p E q P q H p B ; R Q q(cid:1) n . Corollary 8–23 was proven by different means in [15, Section 6.4].
In general, the image of q ch lies in the rational differential cohomology group q H p X ; R Q q(cid:13) but not in the integral differential cohomology group q H p X ; Z r u , u (cid:1) sq(cid:13) . However, insome special cases one gets integral differential cohomology classes. Recall that thereis a filtration of the usual K-theory K (cid:13)p X ; Z q (cid:16) K (cid:13)p qp X ; Z q (cid:129) K (cid:13)p qp X ; Z q (cid:129) . . . , where K (cid:13)p i qp X ; Z q consists of the elements x of K (cid:13)p X ; Z q with the property that for any finitesimplicial complex Y of dimension less than i and any continuous map f : Y Ñ X ,the pullback f (cid:6) x vanishes in K (cid:13)p Y ; Z q [2, Section 1].Let H p i qp X ; Λ r u , u (cid:1) sq(cid:13) denote the subgroup of H p X ; Λ r u , u (cid:1) sq(cid:13) consisting of termsof X -degree equal to i , and similarly for q H p i qp X ; Λ r u , u (cid:1) sq(cid:13) . Given r E s P K (cid:13)p i qp X ; Z q ,one can refine the component of ch pr E sq P H p X ; Q r u , u (cid:1) sq(cid:13) in H p i qp X ; Q r u , u (cid:1) sq(cid:13) to an integer class ch p i qpr E sq P H p i qp X ; Z r u , u (cid:1) sq(cid:13) . Similarly, if r E s P q K (cid:13)p X q and c pr E sq P K (cid:13)p i qp X ; Z q then one can refine the component of q ch pr E sq P q H p X ; Q r u , u (cid:1) sq(cid:13) in q H p i qp X ; R Q q(cid:13) to an element q ch p i qpr E sq P q H p i qp X ; Z r u , u (cid:1) sq(cid:13) . In general there ismore than one such refinement, but if X is p i (cid:1) q -connected then there is a canonicalchoice.Under the hypotheses of Theorem 7–38, suppose in addition that B is p k (cid:1) q -connected and the Atiyah-Singer index ind top pr E sq (cid:16) ind an pr E sq P K (cid:1) n p B ; Z q lies inthe subset K (cid:1) n p k qp B ; Z q € K (cid:1) n p B ; Z q . Then an explicit cocycle in a certain integralDeligne cohomology group is constructed in [35, Section 4]. More precisely, thecocycle is a 2 k -cocycle for the Cech-cohomology of the complex of sheaves(8–25) Z ÝÑ Ω ÝÑ . . . ÝÑ Ω k (cid:1) on B . From the viewpoint of the present paper, the cocycle constructed in [35, Section 4]represents q ch p k qp ind an p E qq P q H p k qp B ; Z r u , u (cid:1) sq(cid:1) n . Then Theorem 7–38 implies thatthe same integral differential cohomology class can be computed as q ch p k qp ind top p E qq . ifferential index theorem In this section we extend the index-theoretic results of the previous sections to the caseof odd differential K-theory classes. Because some of the arguments in the section aresimilar to what was already done in the even case, we state some results without proof.In Subsection 9.1 we give a model for odd differential K-theory whose generatorsconsist of a Hermitian vector bundle, a compatible connection, a unitary automorphismand an even differential form. We then construct suspension and desuspension mapsbetween the even and odd differential K-theory classes. These are used to prove thatthe odd groups defined here are isomorphic to those defined from the general theoryin [28].In Subsection 9.2 we define the analytic and topological indices in the case of an odddifferential K-theory class on the total space of a fiber bundle with even-dimensionalfibers. The definition uses suspension and desuspension to reduce to the even case.It is likely that the topological index map agrees with that constructed using a moretopological model [32]. We do not attempt to relate the odd analytic pushforward withthe odd Bismut superconnection.In Subsection 9.3 we define the analytic and topological indices in the case of an evendifferential K-theory class on the total space of a fiber bundle with odd-dimensionalfibers. The definition uses the trick, taken from [12, Proof of Theorem 2.10], ofmultiplying both the base and fiber by a circle and then tensoring with the Poincar´eline bundle on the ensuing torus.In Subsection 9.4 we look at the result of applying a determinant map to the analyticand topological index, to obtain a map from B to S . We show that this map is givenby the reduced eta-invariants of the fibers X b . In particular if Z is a closed Riemannianspin c -manifold of odd dimension n , and π : Z Ñ pt is the map to a point then for any E P q K p Z q , the topological and analytic indices ind topodd p E q , ind anodd p E q P q K (cid:1) n p pt q equalthe reduced eta-invariant η p Z , E q . This is a version of the main theorem in [32].In Subsection 9.5 we indicate the relationship between the odd differential K-theoryindex and the index gerbe of [35]. Let X be a smooth manifold. We can describe K (cid:1) p X ; Z q as an abelian group in termsof generators and relations. The generators are complex vector bundles G over X D. S. Freed and J. Lott equipped with Hermitian metrics h G and unitary automorphisms U G . The relationsare that(1) p G , h G , U G q (cid:16) p G , h G , U G q (cid:0) p G , h G , U G q whenever there is a shortexact sequence of Hermitian vector bundles(9–1) 0 ÝÑ G ÝÑ G ÝÑ G ÝÑ G U G G U G G U G G G G p G , h G , U G (cid:5) U G q (cid:16) p G , h G , U G q (cid:0) p G , h G , U G q .Given a generator p G , h G , U G q for K (cid:1) p X ; Z q , let ∇ G be a unitary connection on G .Put(9–3) A p t q (cid:16) p (cid:1) t q ∇ G (cid:0) tU G (cid:4) ∇ G (cid:4) p U G q(cid:1) and(9–4) ω p ∇ G , U G q (cid:16) » R u tr (cid:1) e (cid:1) u (cid:1) p dt B t (cid:0) A p t qq (cid:9) P Ω p X ; R q(cid:1) . Then ω p ∇ G , U G q is a closed form whose de Rham cohomology class ch p G , U G q P H p X ; R q(cid:1) is independent of ∇ G . The assignment p G , U G q Ñ ch p G , U G q factorsthrough a map ch : K (cid:1) p X ; Z q ÝÑ H p X ; R q(cid:1) which becomes an isomorphism aftertensoring the left-hand side with R .We can represent K (cid:1) p X ; Z q using Z { Z -graded vector bundles. A generator of K (cid:1) p X ; Z q is then a Z { Z -graded complex vector bundle G (cid:16) G (cid:0) ` G (cid:1) on X ,equipped with a Hermitian metric h G (cid:16) h G (cid:0) ` h G (cid:1) and a unitary automorphism U G (cid:8) P Aut p G (cid:8)q . Choosing compatible connections ∇ G (cid:8) , and putting(9–5) A (cid:8)p t q (cid:16) p (cid:1) t q ∇ G (cid:8) (cid:0) tU G (cid:8) (cid:4) ∇ G (cid:8) (cid:4) p U G q(cid:1) (cid:8) , we put(9–6) ω p ∇ G , U G q (cid:16) » R u str (cid:1) e (cid:1) u (cid:1) p dt B t (cid:0) tA p t qq (cid:9) P Ω p X ; R q(cid:1) . ifferential index theorem If ∇ G and ∇ G are two metric-compatible connections on a Hermitian vector bundle G with a unitary automorphism U G then there is an explicit form CS p ∇ G , ∇ G , U G q P Ω p X ; R q(cid:1) { Image p d q so that(9–7) dCS p ∇ G , ∇ G , U G q (cid:16) ω p ∇ G , U G q (cid:1) ω p ∇ G , U G q . More generally, if we have a short exact sequence (9–1) of Hermitian vector bundles,unitary automorphisms U G i P Aut p G i q , a commutative diagram (9–2) and metric-compatible connections t ∇ G i u i (cid:16) then there is an explicit form CS p ∇ G , ∇ G , ∇ G , U G , U G , U G q P Ω p X ; R q(cid:1) { Image p d q so that(9–8) dCS p ∇ G , ∇ G , ∇ G , U G , U G , U G q (cid:16) ω p ∇ G , U G q (cid:1) ω p ∇ G , U G q (cid:1) ω p ∇ G , U G q . To construct CS p ∇ G , ∇ G , ∇ G , U G , U G , U G q , put W (cid:16) r , s (cid:2) X and let p : W Ñ X be the projection map. Put F (cid:16) p (cid:6) G , h F (cid:16) p (cid:6) h G and U F (cid:16) p (cid:6) U G . Let ∇ F be a unitary connection on F which equals p (cid:6) ∇ G near t u (cid:2) X and which equals p (cid:6)p ∇ G ` ∇ G q near t u (cid:2) X . Then(9–9) CS p ∇ G , ∇ G , ∇ G q (cid:16) » ω p ∇ F , U F q P Ω p X ; R q(cid:1) { Image p d q . Also, if G is a Hermitian vector bundle with a unitary connection ∇ G and twounitary automorphisms U G , U G then there is an explicit form CS p ∇ G , U G , U G q P Ω p X ; R q(cid:1) { Image p d q so that(9–10) dCS p ∇ G , U G , U G q (cid:16) ω p ∇ G , U G (cid:5) U G q (cid:1) ω p ∇ G , U G q (cid:1) ω p ∇ G , U G q . To construct CS p ∇ G , U G , U G q , let ∆ € A be the simplex(9–11) ∆ (cid:16) tp t , t q P A : t ¥ , t ¥ , t (cid:0) t ¤ u , with the orientation induced from the canonical orientation of the affine plane A . Put(9–12) A p t , t q (cid:16) ∇ G (cid:0) t U G (cid:4) ∇ G (cid:4) p U G q(cid:1) (cid:0) t p U G (cid:5) U G q (cid:4) ∇ G (cid:4) p U G (cid:5) U G q(cid:1) . Then(9–13) CS p ∇ G , U G , U G q (cid:16) » ∆ R u tr (cid:1) e (cid:1) u (cid:1) p dt B t (cid:0) dt B t (cid:0) A p t , t qq (cid:9) . Definition 9–14
The differential K-theory group q K (cid:1) p X q is the abelian group de-fined by the following generators and relations. The generators are quintuples G (cid:16)p G , h G , ∇ G , U G , φ q where • G isa complex vector bundle on X . D. S. Freed and J. Lott • h G isaHermitian metric on G . • ∇ G is an h G -compatible connection on G . • U G isa unitary automorphism of G . • φ P Ω p X ; R q(cid:1) { Image p d q .Therelations are(1) G (cid:16) G (cid:0) G whenever there is a short exact sequence (9–1) of Hermitianvector bundles, along with a commuting diagram (9–2) and φ (cid:16) φ (cid:0) φ (cid:1) CS p ∇ G , ∇ G , ∇ G , U G , U G , U G q .(2) p G , h G , ∇ G , U G (cid:5) U G , (cid:1) CS p ∇ G , U G , U G qq (cid:16) p G , h G , ∇ G , U G , q(cid:0)p G , h G , ∇ G , U G , q .By a generator of q K (cid:1) p X q we mean a quadruple G (cid:16) p G , h G , ∇ G , U G , φ q as above.There is a homomorphism ω : q K (cid:1) p X q Ñ Ω p X ; R q(cid:1) given on generators by ω p G q (cid:16) ω p ∇ G , U G q (cid:0) d φ .There is a similar model of q K r p X q for any odd r . A generator is a quintuple G (cid:16)p G , h G , ∇ G , U G , φ q as above with only a change in degree: φ P Ω p X ; R q r (cid:1) { Image p d q .Then ω p G q (cid:16) u p r (cid:0) q{ ω p ∇ G , U G q (cid:0) d φ P Ω p X ; R q r . Also, the exact sequences (2–18)and (2–19) hold in odd degrees.There is a suspension map S : q K (cid:1) p X q Ñ q K p S (cid:2) X q given on generators as follows.Let G (cid:16) p G , h G , ∇ G , U G , φ q be a generator for q K (cid:1) p X q . Let p : r , s (cid:2) X Ñ X bethe projection map. Put F (cid:16) p (cid:6) G , h F (cid:16) p (cid:6) h G and(9–15) ∇ F (cid:16) dt B t (cid:0) p (cid:1) t q ∇ G (cid:0) tU G (cid:4) ∇ G (cid:4) p U G q(cid:1) . Let p E , h E , ∇ E q be the Hermitian vector bundle with connection on S (cid:2) X obtainedby gluing F (cid:7)(cid:7)t u(cid:2) X with F (cid:7)(cid:7)t u(cid:2) X using the automorphism U G . Put(9–16) Φ (cid:16) dt ^ φ P Ω p S (cid:2) X ; R q(cid:1) { Image p d q . Then S p G q (cid:16) p E , h E , ∇ E , Φ q P q K p S (cid:2) X q . Equivalently, S is multiplication by acertain element of q K p S q .There is also a suspension map S : q K p X q Ñ q K p S (cid:2) X q given on generators asfollows. Let E (cid:16) p E , h E , ∇ E , Φ q be a generator for q K p X q . Let p : S (cid:2) X Ñ S and p : S (cid:2) X Ñ X be the projection maps. Let L be the trivial complex line bundle over S , equipped with the product Hermitian metric h L , the product connection ∇ L andthe automorphism U L which multiplies the fiber L e π it over e π it P S by e π it . Put G (cid:16) p (cid:6) L b p (cid:6) E , h G (cid:16) p (cid:6) h L b p (cid:6) h E , ∇ G (cid:16) p (cid:6) ∇ L b I (cid:0) I b p (cid:6) ∇ E , U G (cid:16) p (cid:6) U L and(9–17) φ (cid:16) p (cid:6) dt ^ p (cid:6) Φ P Ω p S (cid:2) X ; R q { Image p d q . ifferential index theorem Then S p E q (cid:16) p G , h G , ∇ G , U G , φ q P q K p S (cid:2) X q . Again, S is multiplication by anelement of q K p S q .The double suspension S : q K p X q Ñ q K p T (cid:2) X q can be described explicitly asfollows. Let p : T (cid:2) X Ñ T and p : T (cid:2) X Ñ X be the projection maps.Consider the trivial complex line bundle M on r , s (cid:2) r , s with product Hermitianmetric and connection dt B t (cid:0) dt B t (cid:1) π it dt . Define the hermitian line bundle P Ñ T by making identifications of M along the boundary of r , s (cid:2) r , s ; it is the“Poincar´e” line bundle on the torus. Let h P and ∇ P be the ensuing Hermitian metricand compatible connection on P . Given a generator E (cid:16) p E , h E , ∇ E , φ q for q K p X q , itsdouble suspension is(9–18) S p E q (cid:16) (cid:0) p (cid:6) P b p (cid:6) E , p (cid:6) h P b p (cid:6) h E , p (cid:6) ∇ P b I (cid:0) I b p (cid:6) ∇ E , dt ^ dt ^ φ (cid:8) P q K p T (cid:2) X q . Finally, there is a desuspension map D : q K p S (cid:2) X q Ñ q K (cid:1) p X q given on generators asfollows. Let E (cid:16) p E , h E , ∇ E , Φ q be a generator for q K p S (cid:2) X q . Picking a basepoint Æ P S , define A : X Ñ S (cid:2) X by A p x q (cid:16) pÆ , x q . Put G (cid:16) A (cid:6) E , h G (cid:16) A (cid:6) h E and ∇ G (cid:16) A (cid:6) ∇ E . Let U G P Aut p G q be the map given by parallel transport around thecircle fibers on S (cid:2) X , starting from tÆu (cid:2) X . Put(9–19) φ (cid:16) » S Φ P Ω p X ; R q(cid:1) { Image p d q . Then(9–20) D p E q (cid:16) p G , h G , ∇ G , U G , φ q P q K (cid:1) p X q . It is independent of the choice of basepoint
Æ P S . Also, D (cid:5) S is the identity on q K (cid:1) p X q .There are analogous suspension and desuspension maps in other degrees.We now show how to use the suspension and desuspension maps to relate Defini-tion 9–14 to the notion of odd differential K-theory groups from [28]. Proposition 9–21
Theodddifferential K -groupsdefinedhereareisomorphictothosedefined by the theory ofgeneralized differential cohomology. Proof
Temporarily denote the geometrically defined groups in Definition 2–14 andDefinition 9–14 as q L (cid:13)p X q . Then, looking at degree (cid:1) q L (cid:1) p X q S q L p S (cid:2) X q (cid:21) q K p S (cid:2) X q D ³ S q K (cid:1) p X q D. S. Freed and J. Lott defines a homomorphism we claim is an isomorphism. (See Remark 2–15 for theisomorphism in the middle of (9–22).) This follows from the 5-lemma applied to thediagram(9–23) 0 Ω p X ; R q(cid:1) Ω p X ; R q(cid:1) K j q L (cid:1) p X q c K (cid:1) p X ; Z q Ω p X ; R q(cid:1) Ω p X ; R q(cid:1) K j q K (cid:1) p X q c K (cid:1) p X ; Z q q L p S (cid:2) X q (cid:21) q K p S (cid:2) X q , the desuspension mapcorresponds to integration over the circle. Suppose that π : X Ñ B is a proper submersion of relative dimension n , with n even. Suppose that π is equipped with a Riemannian structure and a differentialspin c -structure. The product submersion π : S (cid:2) X Ñ S (cid:2) B inherits a productRiemannian structure and differential spin c -structure, so we have the analytic andtopological indices ind an , ind top : q K p S (cid:2) X q Ñ q K (cid:1) n p S (cid:2) B q .Define the analytic index(9–24) ind anodd : q K (cid:1) p X q Ñ q K (cid:1) n (cid:1) p B q by ind anodd (cid:16) D (cid:5) ind an (cid:5) S . Define the topological index ind topodd : q K (cid:1) p X q Ñ q K (cid:1) n (cid:1) p B q by(9–25) ind topodd (cid:16) D (cid:5) ind top (cid:5) S . As an immediate consequence of Theorem 7–38, ind topodd (cid:16) ind anodd . Suppose that π : X Ñ B is a proper submersion of relative dimension n , with n odd. Suppose that π is equipped with a Riemannian structure and a differential spin c -structure. ifferential index theorem Following [12, Pf. of Theorem 2.10], we construct a new submersion π : T (cid:2) X Ñ S (cid:2) B by multiplying the base and fibers by S . Given a ¡
0, we endow π with a product Riemannian structure and differential spin c -structure, so that the circlefibers have length a . Then we have analytic and topological indices ind an , ind top : q K p T (cid:2) X q Ñ q K (cid:1) n (cid:0) p S (cid:2) B q .Define the analytic index ind anodd : q K p X q Ñ q K (cid:1) n p B q by(9–26) ind anodd (cid:16) D (cid:5) ind an (cid:5) S . Define the topological index ind topodd : q K p X q Ñ q K (cid:1) n p B q by(9–27) ind topodd (cid:16) D (cid:5) ind top (cid:5) S . As an immediate consequence of Theorem 7–38, ind topodd (cid:16) ind anodd . Lemma 9–28
The indices ind topodd and ind anodd are independent of the choice of fibercircle length a . Proof
Given a , a ¡ E P q K p X q , let ind topodd , p E q and ind topodd , p E q denote theindices in q K (cid:1) n p B q as computed using circle fibers of length a and a , respectively.Consider the fiber bundle π : r , s (cid:2) X Ñ r , s (cid:2) B , equipped with the productRiemannian structure and differential spin c -structure. Let a : r , s Ñ R (cid:0) be a smoothfunction so that a p t q is a near t (cid:16)
0, and a p t q is a near t (cid:16)
1. Given t P r , s , letthe circle fiber length over t t u (cid:2) B in the fiber bundle r , s (cid:2) T (cid:2) X Ñ r , s (cid:2) B be a p t q . Let ind topodd , r , s(cid:2) B p E q P q K (cid:1) n pr , s (cid:2) B q be the topological index of E ascomputed using the fiber bundle π . Using Lemmas 2–22 and 5–35,(9–29)ind topodd , (cid:1) ind topodd , (cid:16) j p»r , s ω p ind topodd , r , s(cid:2) B qq (cid:16) j (cid:3)»r , s Todd (cid:1) p ∇ T V pr , s(cid:2) T (cid:2) X q(cid:9) ^ ω p E q(cid:11) . One can check that p ∇ T V pr , s(cid:2) T (cid:2) X q pulls back from a connection p ∇ T V p T (cid:2) X q on thevertical tangent bundle of π : T (cid:2) X Ñ S (cid:2) B . Thus(9–30) »r , s Todd (cid:1) p ∇ T V pr , s(cid:2) T (cid:2) X q(cid:9) ^ ω p E q (cid:16) . This proves the lemma. D. S. Freed and J. Lott
There is a homomorphism Det from q K (cid:1) p X q to the space of smooth maps r X , S s .Namely, given a generator E (cid:16) p E , h E , ∇ E , U E , φ q of q K (cid:1) p X q , its image Det p E q Pr X , S s sends x P X to e π i φ p qp x q det p U E p x qq , where φ p q P Ω p X q denotes u times thecomponent of φ P Ω p X ; R q(cid:1) { Image p d q in u (cid:1) Ω p X q{ Image p d q (cid:16) u (cid:1) Ω p X q . Proposition 9–31
Let π : X Ñ B be a proper submersion of relative dimension n ,with n odd. Supposethat π isequipped withaRiemannianstructure andadifferentialspin c -structure. Given E P q K p X q , after using periodicity to shift ind an p E q P q K (cid:1) n p B q into q K (cid:1) p B q ,themap Det p ind anodd p E qq sends b P B to u n (cid:0) η (cid:1) X b , E (cid:7)(cid:7) X b (cid:9) P R { Z ,where X b (cid:16) π (cid:1) p b q . Proof
From (9–26), we want to apply Det (cid:5) D to p ind an (cid:5) S qp E q and evaluate the resultat b . Using (9–18), S p E q is a certain element of q K p T (cid:2) X q and then p ind an (cid:5) S qp E q isits analytic index in q K (cid:1) n (cid:0) p S (cid:2) B q . In order to apply Det (cid:5) D to this, and then computethe result at b , it suffices to just use the restriction of p ind an (cid:5) S qp E q to S (cid:2) t b u . Afterdoing so, the proof of Proposition 8–2 implies the result of applying Det (cid:5) D is(9–32) lim ǫ Ñ η (cid:2) D p T (cid:2) X b q ǫ , p (cid:6) P b p (cid:6) E (cid:7)(cid:7) Xb (cid:10) (cid:0) u n (cid:0) » S » S (cid:2) X b Todd (cid:1) p ∇ T V p T (cid:2) X b q(cid:9) ^ p (cid:6) φ. For any ǫ ¡
0, separation of variables gives(9–33) η (cid:2) D p T (cid:2) X b q ǫ , p (cid:6) P b p (cid:6) E (cid:7)(cid:7) Xb (cid:10) (cid:16) Index p D T , P q (cid:4) η (cid:2) D X b , E (cid:7)(cid:7) Xb (cid:10) (cid:16) η (cid:2) D X b , E (cid:7)(cid:7) Xb (cid:10) . Then the evaluation of Det p ind anodd p E qq at b is(9–34) η (cid:2) D X b , E (cid:7)(cid:7) Xb (cid:10) (cid:0) u n (cid:0) » X b Todd (cid:1) p ∇ TX b (cid:9) ^ φ (cid:16) u n (cid:0) η p X b , E (cid:7)(cid:7) X b q . This proves the proposition.
Corollary 9–35 ([32]) Suppose that Z is a closed Riemannian spin c -manifold ofodd dimension n with a spin c -connection p ∇ TZ . Let π : Z Ñ pt be the mapping to apoint. Given E (cid:16) p E , h E , ∇ E , φ q P q K p Z q ,wehave(9–36) ind anodd p E q (cid:16) ind topodd p E q (cid:16) η p Z , E q in q K (cid:1) n p pt q (cid:16) u (cid:1) n (cid:0) (cid:4) p R { Z q . ifferential index theorem Let π : X Ñ B be a proper submersion of relative dimension n , with n odd. Supposethat π is equipped with a Riemannian structure and a differential spin c -structure. If E isa Hermitian vector bundle on X with compatible connection ∇ E then one can constructan index gerbe [35], which is an abelian gerbe-with-connection. Isomorphism classes ofsuch gerbes-with-connection are in bijection with the differential cohomology group q H p X ; Z q . On the other hand, taking E (cid:16) p E , h E , ∇ E , q P q K p X q , the componentof q ch p ind anodd p E qq (cid:16) q ch p ind topodd p E qq P q H p B ; Q r u , u (cid:1) sq(cid:13) in q H p B ; Q q comes from anelement of q H p B ; Z q . Presumably this is the class of the index gerbe. It should bepossible to prove this by comparing the holonomies around surfaces in B , along thelines of the proof of Proposition 8–2. Appendix: Limits in differential K -theory Let X be a topological space and let X € X € (cid:4) (cid:4) (cid:4) be an increasing sequence ofcompact subspaces whose union is X . Milnor [38] proved that for every cohomologytheory h and every integer q , there is an exact sequence(A–1) 0 ÝÑ lim Ý i h q p X i q ÝÑ h q p X q ÝÑ lim Ý i h q p X i q ÝÑ , in which the quotient is the (inverse) limit and the kernel is its first derived functor;see [25, § 3.F], [36, § 19.4] for modern expositions and [5, § 4] for the specific case of K -theory. In this appendix we prove that the differential cohomology of the union ofcompact manifolds is isomorphic to the inverse limit of the differential cohomologies:there is no lim Ý term. So as to not introduce new notation, we present the argument fordifferential K -theory, which is the case of interest for this paper. We remark that forintegral differential cohomology the theorem is immediate if we use the isomorphismwith Cheeger-Simons differential characters, as a differential character is determinedby its restriction to compact submanifolds. The universal coefficient theorem for K -theory [45] plays an analogous role in the following proof. Theorem A–2
Let X be a smooth manifold and X € X € (cid:4) (cid:4) (cid:4) an increasing se-quence ofcompact codimension-zero submanifolds-with-boundary whoseunion is X .Thenfor any integer q ,the restriction maps induce anisomorphism(A–3) q K q p X q ÝÑ lim Ý i q K q p X i q . D. S. Freed and J. Lott
The proof is based on the exact sequence (2–18) and the following lemmas.
Lemma A–4
Therestriction mapsinduce an isomorphism(A–5) Ω p X ; R q qK ÝÑ lim Ý i Ω p X i ; R q qK . Proof
We first note that the universal coefficient theorem for K -theory implies that K q p X q{ torsion (cid:21) Hom (cid:0) K q p X q , Z (cid:8) . Representing K -homology classes by spin c -manifolds, as in the proof of Theorem 6–2, we see that the cohomology class ofa closed differential form ω P Ω p X ; R q q is in the image of the Chern character ifand only if for every closed Riemannian spin c -manifold W and every smooth map f : W Ñ X , we have(A–6) » W Todd p W q ^ f (cid:6) ω P u q (cid:1) dim p W q (cid:4) Z . For each such choice of W and f , we know that f p W q € X i for some i .Returning to the map (A–5), a differential form is determined pointwise, so the mapis injective. For the same reason, an element t ω i u in the inverse limit on the right-hand side glues to a global differential form ω , and d ω (cid:16) d is local. Since ω (cid:7)(cid:7) X i P Ω p X i ; R q qK for each i , the previous paragraph implies that ω P Ω p X ; R q qK . Thisproves the lemma. Lemma A–7
Therestriction mapsinduce an isomorphism(A–8) K q (cid:1) p X ; R { Z q ÝÑ lim Ý i K q (cid:1) p X i ; R { Z q . Proof
The Ext term in the universal coefficient theorem for K -theory vanishes since R { Z is divisible. Applying the universal coefficient theorem twice and the fact thathomology commutes with colimits, we obtain K q (cid:1) p X ; R { Z q (cid:21) Hom (cid:0) K q (cid:1) p X q , R { Z (cid:8)(cid:21) Hom (cid:0) lim ÝÑ i K q (cid:1) p X i q , R { Z (cid:8)(cid:21) lim Ý i Hom (cid:0) K q (cid:1) p X i q , R { Z (cid:8)(cid:21) lim Ý i K q (cid:1) p X i ; R { Z q . (A–9) ifferential index theorem Proof of Theorem A–2
Milnor’s exact sequence (A–1) and Lemma A–7 imply that(A–10) lim Ý i K q (cid:1) p X i ; R { Z q (cid:16) . Thus the limit of the short exact sequence (2–18) is a short exact sequence [44, § 3.5].The restriction maps then fit into a commutative diagram(A–11) 0 K q (cid:1) p X ; R { Z q q K q p X q Ω p X ; R q qK
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Math. 12, p. 305-323 (1975)Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX78712-0257DepartmentofMathematics,UniversityofCalifornia,Berkeley,970EvansHall-Algebras”,Izv. Akad. Nauk SSSR, Ser. Mat. 43, p. 831-859 (1979)[40] M. Ortiz, “Differential equivariant K-theory”, http://arxiv.org/abs/0905.0476 (2009)[41] D. Quillen, “Determinants of Cauchy-Riemann operators over a Riemann surface”,Funk. Anal. iprilozen 19, p. 37-41 (1985)[42] D. Quillen, “Superconnections and the Chern character”, Topology 24, p. 89-95 (1985)[43] J. Simons and D. Sullivan, “Structured vector bundles define differential K-theory”,http://arxiv.org/abs/0810.4935 (2008)[44] C. A. Weibel, An introduction to homological algebra. Cambridge Studies in AdvancedMathematics, 38, Cambridge University Press, Cambridge (1994)[45] Z. Yosimura, “Universal coefficient sequences for cohomology theories of CW-spectra”, Osaka J. Math. 12, p. 305-323 (1975)Department of Mathematics, University of Texas, 1 University Station C1200, Austin, TX78712-0257DepartmentofMathematics,UniversityofCalifornia,Berkeley,970EvansHall