C*-Algebraic Higher Signatures and an Invariance Theorem in Codimension Two
aa r X i v : . [ m a t h . K T ] O c t C*-Algebraic Higher Signatures and anInvariance Theorem in Codimension Two
Nigel Higson, Thomas Schick, and Zhizhang Xie
Abstract
We revisit the construction of signature classes in C ∗ -algebra K -theory,and develop a variation that allows us to prove equality of signatureclasses in some situations involving homotopy equivalences of non-compact manifolds that are only defined outside of a compact set. Asan application, we prove a counterpart for signature classes of a codi-mension two vanishing theorem for the index of the Dirac operatoron spin manifolds (the latter is due to Hanke, Pape and Schick, and isa development of well-known work of Gromov and Lawson). Let X be a connected, smooth, closed and oriented manifold of dimension d , and let π be a discrete group. Associated to each π -principal bundle e X over X there is a signature class Sgn π ( e X/X ) ∈ K d ( C ∗ r ( π )) in the topological K -theory of the reduced group C ∗ -algebra of π . The sig-nature class can be defined either index-theoretically or combinatorially,using a triangulation. Its main property, which is easiest to see from thecombinatorial point of view, is that if h : Y → X is a homotopy equiva-lence of smooth, closed, oriented manifolds that is covered by a map of π -principal bundles e Y → e X , then(1.1) Sgn π ( e X/X ) =
Sgn π ( e Y/Y ) ∈ K d (cid:0) C ∗ r ( π ) (cid:1) . This is the point of departure for the C ∗ -algebraic attack on Novikov’shigher signature conjecture (see [23, 24] for surveys).1his note has two purposes. The first is to simplify some of the ma-chinery involved in the combinatorial approach to signature classes in C ∗ -algebra K -theory. The main novelty is a technique to construct signatureclasses that uses only finitely-generated and projective modules (in part byappropriately adjusting the C ∗ -algebras that are involved).The second purpose is to prove a codimension-two invariance resultfor the signature that is analogous to a recent theorem of Hanke, Pape andSchick [7] about positive scalar curvature (which is in turn developed fromwell-known work of Gromov and Lawson [5, Theorem 7.5]). To prove thetheorem we shall use generalizations of the signature within the contextof coarse geometry. These have previously been studied closely (see forexample [22]), but our new approach to the signature has some definiteadvantages in this context (indeed it was designed with coarse geometryin mind). Here is the result. Let h : N − → M be a smooth, orientation-preserving homotopy equivalence between twosmooth, closed, oriented manifolds of dimension d + . Let X be a smooth,closed, oriented submanifold of M of codimension (and hence dimension d ), and assume that h is transverse to X , so that the inverse image Y = h − [ X ] is a smooth, closed, oriented submanifold of N of codimension . Suppose e X is the universal cover of X . Then it pulls back along the map h to give a π ( X ) -covering space e Y of Y . Assume that (i) π ( X ) → π ( M ) is injective; (ii) π ( X ) → π ( M ) is surjective; and (iii) the normal bundle of X in M is trivializable.Then (1.2) (cid:0) Sgn π ( X ) ( e X/X ) −
Sgn π ( X ) ( e Y/Y ) (cid:1) = in K d ( C ∗ r ( π ( X ))) . Let us make some remarks about this theorem. First, we do not knowwhether the factor of in (1.2) is necessary, but in view of what is known2bout L -theory concerning splitting obstructions and “change of decora-tion,” where non-trivial -torsion phenomena do occur, we suspect thatthere are situations where the factor of in (1.2) is indeed necessary.In surgery theory (symmetric) signatures are defined in the L -theorygroups of the group ring Z [ π ] . There are comparison maps from these L -groups to our K -groups (at least after inversion of ; see [16] for a thoroughdiscussion of this issue) and the comparison maps take the respective sig-natures to each other (this correspondence is developed systematically in[9, 10, 11]). We leave it as an open question to lift Theorem 1.1 to an equal-ity of L -theoretic higher signatures.Finally, in view of the strong Novikov conjecture, one might expect thatall interesting homotopy invariants of a closed oriented manifold M withuniversal cover f M can be derived from the signature classSgn π ( M ) ( f M/M ) ∈ K ∗ ( C ∗ r ( π ( M ))) . Theorem 1.1 shows that Sgn π ( X ) ( X ) ∈ K ∗ ( C ∗ r ( π ( X ))) is a new homotopyinvariant of the ambient manifold M , under the conditions of the theorem.It is an interesting question to clarify the relationship between this newinvariant and the signature class of f M/M .Here is a brief outline of the paper. In Section 2 we shall summarize thekey properties of the signature class, including a new invariance propertythat is required for the proof of Theorem 1.1. In Section 3 we shall presentthe proof of Theorem 1.1. As in the work of Hanke, Pape and Schick, theproof involves coarse-geometric ideas, mostly in the form of the partitionedmanifold index theorem of Roe [20]. With our new approach to the signa-ture, the proof of the positive scalar curvature result can be adapted withvery few changes. In the remaining sections we shall give our variationon the construction of the signature class, using as a starting point the ap-proach of Higson and Roe in [9, 10].
In this section we shall review some features of the C ∗ -algebraic signature,and then introduce a new property of the signature that we shall use in theproof of Theorem 1.1. We shall not give the definition of the signature inthis section, but we shall say more later, in Section 4. Let π be a discrete group and let X be a smooth, closed and orientedmanifold of dimension d (without boundary). Mishchenko, Kasparov and3thers have associated to every principal π -bundle e X over X a signature class in the topological K -theory of the reduced C ∗ -algebra of the group π :(2.1) Sgn π ( e X/X ) ∈ K d ( C ∗ r ( π )) . The first and most important property of the signature class is that itis an oriented homotopy invariant : given a commuting diagram of principal π -fibrations e Y / / (cid:15) (cid:15) e X (cid:15) (cid:15) Y h / / X in which the bottom map is an orientation-preserving homotopy equiva-lence, we have Sgn π ( e Y/Y ) =
Sgn π ( e X/X ) ∈ K d ( C ∗ r ( π )) . Assume that X is in addition connected, and let e X be a universal cov-ering manifold. In this case we shall abbreviate our notation and writeSgn π ( X ) ( X ) ∈ K d ( C ∗ ( π ( X ))) , omitting mention of the cover. If d is congruent to zero, mod four, thenwe can recover the usual signature of the base manifold X from the C ∗ -algebraic signature Sgn π ( X ) ( X ) by applying the homomorphismTrace π : K d ( C ∗ r ( π )) − → R associated to the canonical trace on C ∗ r ( π ) as a consequence of Atiyah’s L -index theorem. But the C ∗ -algebraic signature usually contains much moreinformation. It is conjectured to determine the so-called higher signaturesof X [18, 17, 15]. This is the Strong Novikov conjecture , and it is known ina great many cases, for example for all fundamental groups of complete,nonpositively curved manifolds [14], for all groups of finite asymptotic di-mension [28], and for all linear groups [6].
Let us return to general covers. The signature class is functorial, in thesense that if α : π → π is an injective group homomorphism, and if the π - It is not necessary to keep track of basepoints in what follows since inner automor-phisms of π act trivially on K ∗ ( C ∗ r ( π )) . π -principal bundles e X and e X are linked by a commuting diagram e X / / (cid:15) (cid:15) e X (cid:15) (cid:15) X X in which the top map is π -equivariant, where π acts on e X via α , then α ∗ : Sgn π ( e X /X ) → Sgn π ( e X /X ) ∈ K d ( C ∗ r ( π )) . (We recall here that the reduced group C ∗ -algebra is functorial for injectivegroup homomorphisms, and therefore so is its K -theory. ) The signature is a bordism invariant in the obvious sense: if W is a com-pact oriented smooth manifold with boundary, and if f W is a principal π -bundle over W , then the signature class associated to the restriction of f W to ∂W is zero; see [10, Section 4.2]. A locally compact space Z is continuously controlled over R if it is equippedwith a proper and continuous map c : Z − → R . If Z is a smooth and oriented manifold of dimension d + that is continu-ously controlled over R , and if the control map c : Z → R is smooth, thenthe inverse image X = c − [ a ] ⊆ Z of any regular value a ∈ R is a smooth, closed, oriented manifold. Givena π -principal bundle e Z over Z , we can restrict it to X and then form thesignature class (2.1). Thanks to the bordism invariance of the signatureclass, the K -theory class we obtain is independent of the choice of regularvalue a ∈ R . Indeed it depends only on the homotopy class of the propermap c . Definition.
We shall call the signature class of X above the transverse signa-ture class of the continuously controlled manifold Z , and denote it bySgn c,π ( e Z/Z ) :=
Sgn π ( e X/X ) ∈ K d ( C ∗ r ( π )) . It is also possible to define a signature class in the K -theory of the full group C ∗ -algebra,and this version is functorial for arbitrary group homomorphisms. But since it is in certainrespects more awkward to manipulate with we shall mostly confine our attention in thispaper to the signature class for the reduced C ∗ -algebra. .7. To prove the main theorem of the paper we shall need to invoke aninvariance property of the transverse signature class of the following sort.Suppose given a morphism h : Z − → W of smooth, oriented manifolds that are continuously controlled over R , andsuppose that h is, in suitable sense, a controlled homotopy equivalence.The conclusion is that if Z and W are equipped with π -principal bundlesthat are compatible with the equivalence, then (up to a factor of that weshall make precise in a moment) the associated transverse signature classesof Z and W are equal.Actually we shall need to consider situations where h will be only de-fined on the complement of a compact set; see Paragraph 2.9. But for nowwe shall ignore this additional complication. Definition.
Let Z and W be locally compact spaces that are continuouslycontrolled over R , with control maps c Z and c W . A continuous and propermap f : Z → W is continuously controlled over R if the symmetric difference c − [ ∞ ) △ f − c − [ ∞ ) is a compact subset of Z .Thus the map f : Z → W is continuously controlled over R if it is com-patible with the decompositions of Z and W into positive and negativeparts (according to the values of the control maps), modulo compact sets. Definition.
Let Z and W be complete Riemannian manifolds, both withbounded geometry [1, Section 1].(a) A boundedly controlled homotopy equivalence from Z to W is a pair ofsmooth maps f : Z − → W and g : W − → Z, together with a pair of smooth homotopies H Z : Z × [
0, 1 ] − → Z and H W : W × [
0, 1 ] − → W between the compositions of f with g and the identity maps on Z and W , for which all the maps have bounded derivatives, meaning that, forexample, sup X ∈ TZ, || X || ≤ || Df ( X ) || < ∞ .6b) If Z and W are continuously controlled over R , then the above bound-edly controlled homotopy equivalence is in addition continuously con-trolled over R if all the above maps are continuously controlled over R .(c) The above boundedly controlled homotopy equivalence is compatible with given π -principal bundles over Z and W if all the above maps liftto bundle maps.The following result is a small extension of Roe’s partitioned manifoldindex theorem [20]. We shall prove it in Section 4.8. Let Z and W be ( d + ) -dimensional, complete oriented Rieman-nian manifolds with bounded geometry that are continuously controlled over R .If e Z and f W are principal π -bundles over Z and W , and if there is a boundedlycontrolled orientation-preserving homotopy equivalence between Z and W that iscontinuously controlled over R and compatible with e Z and f W , then ε (cid:0) Sgn c,π ( e Z/Z ) −
Sgn c,π ( f W/W ) (cid:1) = in K d ( C ∗ r ( π )) , where ε = (cid:14) if d is even if d is odd. As we already noted, we shall actually need a slightly stronger versionof Theorem 2.8. It involves the following concept:
Definition.
Let Z and W be locally compact Hausdorff spaces. An eventualmap from Z to W is a continuous and proper map Z − → W, where Z is a closed subset of Z whose complement has compact closure.Two eventual maps from Z to W are equivalent if they agree on the comple-ment of some compact set in Z .There is an obvious category of locally compact spaces and equivalenceclasses of eventual maps. Let us denote the morphisms as follows: Z W . Definition.
Two eventual morphisms f , f : Z W homotopic if there is an eventual morphism g : Z × [
0, 1 ] W for whichthe compositions Z ⇒ Z × [
0, 1 ] g W with the inclusions of Z as Z × { } and Z × { } equal to f and f , respectively,in the category of locally compact spaces and eventual maps.Homotopy is an equivalence relation on morphisms that is compatiblewith composition, and so we obtain a notion of eventual homotopy equiva-lence . It is clear how to define the concept of continuously controlled even-tual morphisms between locally compact spaces that are continuously con-trolled over R , as well as eventual homotopy equivalences that are con-tinuously controlled over R . And we can speak of boundedly controlled eventual homotopy equivalences between bounded geometry Riemannianmanifolds Z and W , and compatibility of these with π -principal bundles on Z and W . An orientation of a manifold Z of dimension d is a class in H lf d ( Z ) , thehomology of locally finite chains, that maps to a generator in each group H lf d ( Z, Z \ { z } ) . We shall call the image of the orientation class under the map H lf d ( Z ) − → lim − → K H lf d ( Z, K ) , where the direct limit is over the compact subsets of Z , the associated even-tual orientation class . The direct limit is functorial for eventual morphisms,and so we can speak of an eventual homotopy equivalence being orientation-preserving .The invariance property that we shall use to prove Theorem 1.1 is asfollows: Let Z and W be ( d + ) -dimensional, connected complete orientedRiemannian manifolds with bounded geometry that are continuously controlledover R . If e Z and f W are principal π -bundles over Z and W , and if there is aboundedly controlled eventual homotopy equivalence between Z and W that iscontinuously controlled over R , orientation-preserving, and compatible with e Z and f W , then ε (cid:0) Sgn c,π ( e Z/Z ) −
Sgn c,π ( f W/W ) (cid:1) = in K d ( C ∗ r ( π )) , where ε = (cid:14) if d is even if d is odd.
8e shall prove this result in Section 5.
Finally, we shall need a formula for the signature of a product mani-fold. Let X be a smooth, closed and oriented manifold of dimension p , and e X → X a π -principal bundle over X . Similarly, let Y be a smooth, closed andoriented manifold of dimension q , and e Y → Y a σ -principal bundle over Y .There is a natural product map − ⊗ − : K p ( C ∗ r ( π )) ⊗ K q ( C ∗ r ( σ )) − → K p + q ( C ∗ r ( π × σ )) and we have the following product formula:Sgn π × σ ( e X × e Y/X × Y ) = ε ( X,Y ) Sgn π ( e X/X ) ⊗ Sgn σ ( e Y/Y ) where ε ( X, Y ) = (cid:14) if one of X or Y is even-dimensional if both X and Y are odd-dimensional.This will be proved in Section 4.5. In this section, we shall prove Theorem using the properties of the con-trolled signature that were listed above.Let h : N → M be an orientation-preserving, smooth homotopy equiva-lence between smooth, closed, oriented manifolds of dimension d + . Let X be a smooth, closed, oriented submanifold of M of codimension (andhence dimension d ), and assume that the smooth map h : N → M is trans-verse to X , so that the inverse image Y = h − [ X ] is a smooth, closed, oriented, codimension-two submanifold of N . Assumein addition that(i) π ( X ) → π ( M ) is injective;(ii) π ( X ) → π ( M ) is surjective; and(iii) the normal bundle of X in M is trivializable.9e fix a base point in X and assume that X is connected. With assump-tion (i) above, we shall view π ( X ) as a subgroup of π ( M ) . Let p : c M − → M be the covering of M corresponding to the subgroup π ( X ) ⊂ π ( M ) . Inother words, c M is the quotient space of the universal covering of M by thesubgroup π ( X ) . In particular, we have the following lemma. The inverse image of X under the projection from c M to M is a disjointunion of copies of coverings of X . The component of the base point is homeomorphicto X , with homeomorphism given by the restriction of the covering projection. Fix the base point copy of X in the inverse image p − ( X ) . We will usethe same notation, X for this lift of the submanifold X ⊆ M : X ⊆ p − ( X ) . Denote by D ( X ) a closed tubular neighborhood of X in c M and consider thesmooth manifold c M \ ˚ D ( X ) (the circle denotes the interior) with boundary ∂ (cid:0)c M \ ˚ D ( X ) (cid:1) ∼ = X × S . The following lemma is contained in the proof of [7, Theorem 4.3].
Under the assumptions (i) and (ii) made above, the inclusion X × S ֒ → c M \ ˚ D ( X ) induces a split injection π ( X × S ) → π ( c M \ ˚ D ( X )) on fundamental groups. Glue two copies of c M \ ˚ D ( X ) along X × S so as to form the space W = c M \ ˚ D ( X ) ∪ X × S c M \ ˚ D ( X ) . By applying the Seifert-van Kampen theorem, we obtain the following lemma,as in the proof of [7, Theorem 4.3]. 10 .3 Lemma.
The map π ( X × S ) − → π ( W ) that is induced from the inclusion X × S ֒ → W is a split injection. Pull back the covering p : c M − → M to N along the map h , as indicated in the following diagram: b N b h / / (cid:15) (cid:15) c M p (cid:15) (cid:15) N h / / M. We denote by Y = b h − ( X ) ⊂ b N, the inverse image under b h of our chosen copy of X in c M . The normal bundle of Y in N , or equivalently in b N , is trivial.Proof. By transversality, the normal bundle of Y in N is the pullback of thenormal bundle of X in M , and hence is trivial by our assumption (iii).Now let us prove Theorem 1.1, assuming Theorem 2.12 on the eventualhomotopy invariance of the controlled signature (the rest of the paper willbe devoted to proving Theorem 2.12). Proof of Theorem . Form the space Z = b N \ ˚ D ( Y ) ∪ Y × S b N \ ˚ D ( Y ) . Here ˚ D ( Y ) is the inverse image of ˚ D ( X ) under b h , which is an open diskbundle of Y in b N , with boundary Y × S thanks to Lemma 3.4.The map b h : b N → c M induces a canonical continuous map(3.1) ϕ : Z − → W. Although it was constructed using a process that started from a ho-motopy equivalence, the map ϕ in (3.1) is not a homotopy equivalence ingeneral. However, we still get an eventual homotopy equivalence.11 .5 Lemma. The map ϕ is a boundedly controlled eventual homotopy equivalencethat is continuously controlled over R , with respect to the control maps c W : W − → R and c Z : Z − → R , where c W ( w ) = (cid:14) d ( w, X × S ) if w is in the first copy M \ ˚ D ( X ) in W , − d ( w, X × S ) if w is in the second copy M \ ˚ D ( X ) in W , and where c Z is constructed correspondingly. Here d ( w, X × S ) is the distancebetween w and X × S in W .Proof. The map h , its homotopy inverse g : M → N , and the homotopies H : N × [
0, 1 ] → M and H : M × [
0, 1 ] → N between the compositions andthe identity maps all lift to maps ^ g, ^ h, ^ H , ^ H on ^ M , ^ N . In general, only ^ h will map ˚ D ( Y ) ⊂ ^ N to ˚ D ( X ) ⊂ ^ M and therefore give rise to an honest map φ : Z → W . But the other maps at least give rise to eventual maps whichestablish the required eventual homotopy equivalence.We lift the Riemannian metrics from the compact manifolds N, M to ^ N, ^ M and obtain metrics on W and Z by smoothing these metrics in a com-pact neighborhood of X × S or Y × S , respectively. Therefore, we obtainmetrics of bounded geometry. All our maps are lifted from the compactmanifolds M, N and therefore are boundedly controlled. By construction,they also preserve the decomposition into positive and negative parts ac-cording to the control maps c W , c Z (up to a compact deviation). That is,they are continuously controlled over R .Let r : π ( W ) → π ( X × S ) be the splitting of the inclusion homomor-phism ι : π ( X × S ) → π ( W ) . Let f W be the covering space of W corre-sponding to the subgroup ker ( r ) , that is, the quotient space of the universalcover of W by ker ( r ) . Let e Z be the covering space of Z that is the pullbackof f W → W along the map ϕ . It follows from Theorem that ε (cid:16) Sgn c,π ( f W/W ) −
Sgn c,π ( e Z/Z ) (cid:17) = in K d ( C ∗ r ( π )) .where π = π ( X × S ) = π ( X ) × Z and ε = (cid:14) if d is even if d is odd.12quivalently, we have ε (cid:16) Sgn π ( ^ X × S /X × S ) − Sgn π ( ^ Y × S /Y × S ) (cid:17) = in K d ( C ∗ r ( π )) .Here ^ X × S is the universal cover of X × S , which pulls back to give a cov-ering space ^ Y × S over Y × S . It is a basic computation of higher signaturesthat Sgn Z ( S ) is a generator of K ( C ∗ r ( Z )) ∼ = Z . Moreover, we have for anarbitrary group Γ the K ¨unneth isomorphism given by the product map K p ( C ∗ r ( Γ )) ⊗ K ( C ∗ r ( Z )) ⊕ K p − ( C ∗ r ( Γ )) ⊗ K ( C ∗ r ( Z )) ∼ = − → K p ( C ∗ r ( Γ × Z )) . The theorem now follows from the product formula of signature operators(cf. Section ).We conclude this section with a discussion of counterexamples to somewould-be strengthenings of the theorem, as well as some further comments.Techniques from surgery theory, and in particular Wall’s π - π theorem[27, Theorem 3.3], show that conditions (i) and (ii) in Theorem 1.1 are bothnecessary. Concerning condition (i), let us start by noting that the n -torus T n embeds into the ( n + ) -sphere S n + with trivial normal bundle (indeedit maps into every ( n + ) -dimensional manifold with trivial normal bundle,but one embedding into the sphere will suffice). Let f : V → W be a degree-one normal map between simply connected, closed, oriented -manifoldswith distinct signatures Sgn ( V ) = Sgn ( W ) ∈ Z . Consider the degree-onenormal map of pairs f × Id : (cid:0) V × B n + , V × S n + (cid:1) − → (cid:0) W × B n + , W × S n + (cid:1) . According to Wall’s π - π theorem this is normally cobordant to a homotopyequivalence of pairs; let us write the homotopy equivalence on boundariesas h : N ≃ − → W × S n + . Consider the codimension-two submanifold (with trivial normal bundle) W × T n ⊆ W × S n + . Its signature class in the free abelian group K ( C ∗ r ( Z n )) is Sgn ( W ) · Sgn Z n ( T n ) where Sgn Z n ( T n ) has infinite order. However the signature class of thetransverse inverse image M = h − ( W × T n ) is equal toSgn Z n ( M ) = Sgn ( V ) · Sgn Z n ( T n ) ∈ K ( C ∗ r ( Z n )) . f × id S n + and h restricts to a normalbordism between f × id T n and h | M . Therefore (even twice) the signatureclasses of the two submanifolds of codimension are distinct.As for condition (ii), start with any degree-one normal map f : W → X such that (cid:16) Sgn π ( X ) ( f W/W ) −
Sgn π ( X ) ( e X/X ) (cid:17) = (here e X is the universal cover of X , and f W is the pullback to W along f ). ByWall’s theorem again, the map f × Id : W × S − → X × S is normally cobordant to an orientation-preserving homotopy equivalence h : N ≃ − → X × S . Let Y be the transverse inverse image of the codimension-two submanifold X × { pt } ⊆ X × S under the map h . It follows from cobordism invariance ofthe signature class thatSgn π ( X ) ( e Y/Y ) =
Sgn π ( X ) ( f W/W ) , and therefore that (cid:16) Sgn π ( X ) ( e Y/Y ) −
Sgn π ( X ) ( e X/X ) (cid:17) = as required.On the other hand, it seems to be difficult to determine whether or notcondition (iii) is necessary. This question is also open for the companionresult about positive scalar curvature; compare in particular [5].Finally, as we mentioned in the introduction, we do not know if thefactor of that appears in Theorem 1.1 is really necessary, although wesuspect it is. Let p : M → Σ be any bundle of oriented closed manifoldsover an oriented closed surface Σ = S with fiber X = p − ( pt ) . If h : N → M is a homotopy equivalence, then, whether or not the composition N h − → M p − → Σ is homotopic to a fiber bundle projection, and whether or not the inducedmap from the “fiber” Y = ( p ◦ h ) − ( pt ) to X is cobordant to a homotopy14quivalence, Theorem 1.1 guarantees that the C ∗ -algebraic higher signa-tures of Y and X coincide in K ∗ ( C ∗ r ( π ( X ))) , after multiplication with . Inthe related L -theory context, examples can be constructed where the fac-tor of is really necessary, and perhaps the same is true here. Moreoverit seems possible that many nontrivial examples of the theorem can be ob-tained from this construction. In this section we shall review the construction of the C ∗ -algebraic signa-ture class. One route towards the definition of the signature class goes viaindex theory and the signature operator. See for example [2, Section 7] foran introduction. Although we shall make use of the index theory approach,we shall mostly follow a different route, adapted from [9, 10], which makesit easier to handle the invariance properties of the signature class that weneed. Hilbert-Poincar´e complexes were introduced in [9] to adapt the standardsymmetric signature constructions in L -theory to the context of C ∗ -algebra K -theory. We shall review the main definitions here; see [9] for more details. Let A be a unital C ∗ -algebra. An d -dimensional Hilbert-Poincar´e complex over A is a complex of finitely generated (and thereforealso projective) Hilbert A -modules and bounded, adjointable differentials, E ← − E ← − · · · b ← − E d , together with bounded adjointable operators D : E p → E d − p such that(i) if v ∈ E p , then D ∗ v = (− ) ( d − p ) p Dv ;(ii) if v ∈ E p , then Db ∗ v + (− ) p bDv = ; and(iii) D is a homology isomorphism from the dual complex E d b ∗ ← − E d − ∗ ← − · · · b ∗ ← − E to the complex ( E, b ) . 15o each d -dimensional Hilbert-Poincar´e complex over A there is associ-ated a signature in C ∗ -algebra K -theory:Sgn ( E, D ) ∈ K d ( A ) . See [9, Section 3] for the construction. If A is the C ∗ -algebra of complexnumbers, and d ≡ mod , then the signature identifies with the signa-ture of the quadratic form induced from D on middle-dimensional homol-ogy.The signature is a homotopy invariant [9, Section 4] and indeed a bor-dism invariant [9, Section 7]. These concepts will be illustrated in the fol-lowing subsections. Let Σ be a finite-dimensional simplicial complex and denote by C [ Σ p ] thevector space of finitely-supported, complex-valued functions on the set of p -simplices in Σ . After assigning orientations to simplices we obtain in theusual way a chain complex(4.1) C [ Σ ] b ← − C [ Σ ] b ← − · · · . If Σ is, in addition, locally finite (meaning that each vertex is contained inonly finitely many simplices) then there is also an adjoint complex(4.2) C [ Σ ] b ∗ − → C [ Σ ] b ∗ − → · · · , in which the differentials are adjoint to those of (4.1) with respect to thestandard inner product for which the delta functions on simplices are or-thonormal.If Σ is furthermore an oriented combinatorial manifold of dimension d with fundamental cycle C , then there is a duality operator(4.3) D : C [ Σ p ] − → C [ Σ d − p ] defined by the usual formula D ( ξ ) = ξ ∩ C (compare [10, Section 3], wherethe duality operator is called P ). There is also an adjoint operator D ∗ : C [ Σ d − p ] − → C [ Σ p ] . We are, as yet, in a purely algebraic, and not C ∗ -algebraic, context, but theoperator D satisfies all the relations given in Definition 4.1, except for the16rst. As for the first, the operators D and (− ) ( d − p ) p D ∗ are chain homo-topic, and if we replace D by the average ( D + (− ) ( d − p ) p D ∗ ) , which we shall do from now on, then all the relations are satisfied.In order to pass from complexes of vector spaces to complexes of Hilbertmodules we shall assume the following: A simplicial complex Σ is of bounded geometry if there isa positive integer k such that any vertex of Σ lies in at most k differentsimplices of Σ . Let P be a proper metric space and let S be a set. A function c : S − → P is a proper control map if for every r > 0 , there is a bound N < ∞ such that if B ⊆ P is any subset of diameter less than r , then c − [ B ] has cardinality lessthan N . Let P be a proper metric space, let S and T be sets, and let c S : S − → P and c T : T − → P be proper control maps. Suppose in addition that a discrete group π acts on P properly through isometries, and on the sets S and T , and suppose thatthe control maps are equivariant. A linear map A : C [ S ] − → C [ T ] is boundedly geometrically controlled over P if(i) the matrix coefficients of A with respect to the bases given by thepoints of S and T are uniformly bounded; and(ii) there is a constant K > 0 such that the ( t, s ) -matrix coefficient of A iszero whenever d ( c ( t ) , c ( s )) > K .We shall denote by C P,π [ T, S ] the linear space of equivariant, boundedly ge-ometrically controlled linear operators from C [ S ] to C [ T ] .Every boundedly geometrically controlled linear operator is adjointable,and the space C P,π [ S, S ] is a ∗ -algebra by composition and adjoint of opera-tors. 17 .5 Definition. We shall denote by C ∗ P,π ( S, S ) the C ∗ -algebra completion of C P,π [ S, S ] in the operator norm on ℓ ( S ) . The norm above is related to the norm in reduced group C ∗ -algebras. There is also a norm on C P,π [ S, S ] appropriate to full group C ∗ -algebras, namely k A k = sup ρ k ρ ( A ) : H → H k , where the supremum is over all representations of the ∗ -algebra C P,π [ S, S ] as bounded operators on Hilbert space. See [4], where among many otherthings it is shown that the supremum above is finite.Now the space C P,π [ T, S ] in Definition 4.4 is a right module over C P,π [ S, S ] by composition of operators. If there is an equivariant, injective map T → S making the diagram T / / c T (cid:15) (cid:15) S c S (cid:15) (cid:15) P P commute, then C P,π [ T, S ] is finitely generated and projective over C P,π [ S, S ] .Proof. Define p : C [ S ] → C [ S ] sending t ∈ T to t and s ∈ S \ T to . Then p ∈ C P,π [ S, S ] is an idempotent with image precisely C P,π [ T, S ] . So the imageis projective and generated by the element p .Assume that the hypothesis in the lemma holds (we shall always as-sume this in what follows). The induced module C ∗ P,π ( T, S ) = C P,π [ T, S ] ⊗ C P,π [ S,S ] C ∗ P,π ( S, S ) is then a finitely generated and projective module over C ∗ P,π ( S, S ) . It isisomorphic to the completion of C P,π [ T, S ] in the norm associated to the C P,π [ S, S ] -valued, and hence C ∗ P,π ( S, S ) -valued, inner product on C P,π [ T, S ] defined by h A, B i = A ∗ B. So it is a finitely generated and projective Hilbert module over C ∗ P,π ( S, S ) .Let us return to the bounded geometry, combinatorial manifold Σ , whichwe shall now assume is equipped with an action of π . Assume that the sets Σ p of p -simplices admit proper control maps to some P so that there is auniform bound on the distance between the image of any simplex and the18mage of any of its vertices. Then the differentials in (4.1) and the dualityoperator (4.3) are boundedly geometrically controlled, and we obtain fromthem a complex(4.4) C ∗ P,π ( Σ , S ) b ← − C ∗ P,π ( Σ , S ) b ← − · · · b d ← − C ∗ P,π ( Σ d , S ) , of finitely generated and projective Hilbert modules over C ∗ P,π ( S, S ) for any S that satisfies the hypothesis of Lemma 4.7 for all T = Σ p . The procedurein Section 4.1 therefore gives us a signature(4.5) Sgn P,π ( Σ, S ) ∈ K d ( C ∗ P,π ( S, S )) . If S is a free π -set with finitely many π -orbits, and if we regard theunderlying set of π as a π -space by left multiplication, then the bimodule C P,π [ π, S ] is a Morita equivalence C P,π [ S, S ] ∼ Morita C P,π [ π, π ] , and upon completion we obtain a C ∗ -algebra Morita equivalence(4.6) C ∗ P,π ( S, S ) ∼ Morita C ∗ P,π ( π, π ) . But the action of π on the vector space C [ π ] by right translations gives anisomorphism of ∗ -algebras C [ π ] ∼ = − → C P,π [ π, π ] and then by completion an isomorphism(4.7) C ∗ r ( π ) ∼ = − → C ∗ P,π ( π, π ) . Putting (4.6) and (4.7) together we obtain a canonical isomorphism(4.8) K ∗ ( C ∗ P,π ( S, S )) ∼ = K ∗ ( C ∗ r ( π )) . Now let X be a smooth, closed and oriented manifold of dimension d ,and let e X be a π -principal bundle over X . Fix any triangulation of X and liftit to a triangulation Σ of e X . In addition, fix a Riemannian metric on X andlift it to a Riemannian metric on e X . Let P be the underlying proper metricspace. If we choose S to be the the disjoint union of all Σ p , then we obtain asignature as in (4.5) above. 19 .8 Definition. We denote bySgn π ( e X/X ) ∈ K d ( C ∗ r ( π )) the signature class associated to (4.5) under the K -theory isomorphism (4.8). By using the C ∗ -algebra norm in Remark 4.6 we would insteadobtain a signature class in K d ( C ∗ ( π )) , where C ∗ ( π ) is the maximal group C ∗ -algebra.The general invariance properties of the signature of a Hilbert-Poincar´ecomplex imply that the signature is an oriented homotopy invariant, as in(2.2) and a bordism invariant, as in (2.5). Compare [10, Sections 3&4]. Let P be a proper metric space. A standard P -module is a separable Hilbertspace that is equipped with a nondegenerate representation of the C ∗ -alg-ebra C ( P ) . It is unique up to finite-propagation unitary isomorphism. Asa result, the C ∗ -algebra C ∗ ( P ) generated by finite-propagation, locally com-pact operators is unique up to inner (in the multiplier C ∗ -algebra) isomor-phism, and its topological K -theory is therefore unique up to canonical iso-morphism. For all this see for example [12, Section 4].If a discrete group π acts properly and isometrically on P , then all thesedefinitions and constructions can be made equivariantly, and we shall de-note by C ∗ π ( P ) the C ∗ -algebra generated by the π -equivariant, finite-prop-agation, locally compact operators.If S is a π -set that is equipped with an equivariant proper control mapto P , as in Definition 4.3, then ℓ ( S ) carries a π -covariant representation of C ( P ) via the control map, and there is an isometric, equivariant, finite-propagation inclusion of ℓ ( S ) into any π -covariant standard P -module; see[12, Section 4] again. This induces an inclusion of C ∗ P,π ( S, S ) into C ∗ π ( P ) , anda canonical homomorphism(4.9) K ∗ ( C ∗ P,π ( S, S )) − → K ∗ ( C ∗ π ( P )) . If the quotient space
P/π is compact, then this is an isomor-phism , and, as we noted, in this case the left-hand side of (4.9) is canonicallyisomorphic to K ∗ ( C ∗ r ( π )) . But in general the right-hand side can be quitedifferent from either K ∗ ( C ∗ P,π ( S, S )) or K ∗ ( C ∗ r ( π )) .20 .11 Definition. Let W be a complete, bounded geometry, oriented Rie-mannian manifold, and let f W be a π -principal bundle over W . We shalldenote by Sgn π ( f W/W ) ∈ K ∗ ( C ∗ π ( f W )) the signature class that is obtained from a π -invariant, bounded geometrytriangulation Σ of f W by applying the map (4.9) to the class (4.5).This is the same signature class as the one considered in [10]. Onceagain, the general invariance properties of the signature of a Hilbert-Poincar´ecomplex immediately provide a homotopy invariance result: The signature class
Sgn π ( f W/W ) is independent of the triangu-lation. In fact the signature class is a boundedly controlled oriented homotopyinvariant. Let P be an oriented, complete Riemannian manifold. Wedenote by D P the signature operator on P ; see for example [10, Section 5].If W is an oriented, complete Riemannian manifold, and if f W is a π -principal bundle over W , then the operator D e W is a π -equivariant, first-order elliptic differential operator on f W , and as explained in [2, 21, 22] ithas an equivariant analytic indexInd π ( D e W ) ∈ K d ( C ∗ π ( f W )) . ([10, Theorems 5.5 and 5.11]) . If W is a complete bounded geom-etry Riemannian manifold of dimension d , and if f W is a π -principal bundle over W , then Sgn π ( f W/W ) =
Ind π ( D e W ) ∈ K d ( C ∗ π ( f W )) . The signature operator point of view on the signature class makes it easyto verify the product formula in Paragraph 2.13.If one of X or Y is even-dimensional, then the signature operator on X × Y is the product of the signature operators on X and Y ; if both X and Y are odd-dimensional, then the signature operator on X × Y is the directsum of two copies of the product of the signature operators on X and Y . The21roduct formula for the signature class follows from the following commu-tative diagram: K p ( Bπ ) ⊗ K q ( Bσ ) / / (cid:15) (cid:15) K p + q ( Bπ × Bσ ) (cid:15) (cid:15) K p ( C ∗ r ( π )) ⊗ K q ( C ∗ r ( σ )) / / K p + q ( C ∗ r ( π × σ )) where the horizontal arrows are the products in K -homology and K -theory,and the vertical maps are index maps (or in other words assembly maps);compare [3]. The argument uses the fact that the K -homology class of theproduct of two elliptic operators is equal to the product of their correspond-ing K -homology classes. The signature operator of S is the operator i ddθ , which is well known torepresent a generator of K ( B Z ) ∼ = Z , and whose index in K ( C ∗ r ( Z )) is wellknown to be a generator of K ( C ∗ r ( Z )) ∼ = Z . In this section we shall describe a minor extension of Roe’s partitionedmanifold index theorem [20, 8]. Compare also [29]. We shall consider onlythe signature operator, but the argument applies to any Dirac-type opera-tor. Let A be a C ∗ -algebra. If I and J are closed, two-sided ideals in A with I + J = A , then there is a Mayer-Vietoris sequence in K -theory: K ( I ∩ J ) / / K ( I ) ⊕ K ( J ) / / K ( A ) ∂ (cid:15) (cid:15) K ( A ) ∂ O O K ( I ) ⊕ K ( J ) o o K ( I ∩ J ) . o o See for example [12, Section 3]. The boundary maps may be described asfollows. There is an isomorphism I/ ( I ∩ J ) ∼ = − → A/J, and the boundary maps are the compositions K ∗ ( A ) − → K ∗ ( A/J ) ∼ = K ∗ ( I/ ( I ∩ J )) − → K ∗ − ( I ∩ J ) , K -theory long exact se-quence for the ideal I ∩ J ⊆ I . The same map is obtained, up to sign, if I and J are switched.Now let W be a complete Riemannian manifold that is continuouslycontrolled over R , and let f W be a principal π -bundle over W . Let X ⊆ W bethe transverse inverse image of ∈ R and let e X be the restriction of f W to X .Denote by I the ideal in A = C ∗ π ( f W ) generated by all finite propagation,locally compact operators that are supported in c − [ a, ∞ ) for some a ∈ R ,and denote by J the ideal in A generated by all finite propagation, locallycompact operators that are supported on c − (− ∞ , a ] for some a ∈ R . Thesum I + J is equal to A .The intersection I ∩ J is generated by all operators that are supportedin the inverse images under the control map of compact subsets of R . Anyfinite propagation isometry from a standard e X module into a standard f W -module induces an inclusion of C ∗ π ( e X ) into the intersection, and this in-clusion induces a canonical isomorphism in K -theory; see [12, Section 5].Following Remark 4.10, since X is compact we therefore obtain canonicalisomorphisms K ∗ ( C ∗ r ( π )) ∼ = − → K ∗ ( C ∗ π ( e X )) ∼ = − → K ∗ ( I ∩ J ) . Let W be an oriented, complete Riemannian manifold of dimen-sion d + that is continuously controlled over R , and let f W be a π -principal bundleover W . The Mayer-Vietoris boundary homomorphism ∂ : K d + ( C ∗ π ( f W )) − → K d ( C ∗ r ( π )) maps the index class of the signature operator D e W to the index class of the signa-ture operator D e X if d is even, and to twice the index class of D e X if d is odd.Proof. The argument of [20] reduces the theorem to the case of a productmanifold, that is, W = X × R with the control map c : X × R → R given bythe projection to R (see also [8]). The case of the product manifold X × R follows from a direct computation, as in [8]. Once again, the reason for thefactor of , in the case when d is odd, is that signature operator on X × R is the direct sum of two copies of the product of the signature operators of X and R in this case. Assume that W has bounded geometry. The Mayer-Vietorisboundary homomorphism ∂ : K d + ( C ∗ π ( f W )) − → K d ( C ∗ r ( π )) aps Sgn π ( f W/W ) to the transverse signature class Sgn c,π ( f W/W ) , if d is even,and it maps it to twice the transverse signature class of W , if d is odd. We can now prove Theorem 2.8. Let Z and W be as in the statement.The boundedly controlled homotopy equivalence between them identifiesSgn π ( f W/W ) with Sgn π ( e Z/Z ) . Since the equivalence is continuously con-trolled over R there is a commuting diagram K d + ( C ∗ π ( f W )) (cid:15) (cid:15) / / K d ( C ∗ r ( π )) K d + ( C ∗ π ( e Z )) / / K d ( C ∗ r ( π )) involving Mayer-Vietoris boundary homomorphisms. Thanks to Roe’s par-titioned manifold index theorem, in the form of Corollary 4.16, the hori-zontal maps send the signature classes Sgn π ( f W/W ) and Sgn π ( e Z/Z ) to ε times the transverse signature classes Sgn c,π ( f W/W ) and Sgn c,π ( e Z/Z ) , re-spectively. So the two transverse signature classes, times ε , are equal. It remains to prove Theorem 2.12. This is what we shall do here, and it is atthis point that we shall make proper use of the complex C ∗ π ( Σ • , S ) that weintroduced in Section 4.2. Let P be proper metric space and assume that a discrete group π acts on P properly and through isometries. Let
S, T be sets equipped with π -actions, and let c S : S − → P and c T : T − → P be π -equivariant proper control maps. A boundedly geometrically con-trolled linear operator A : C [ S ] → C [ T ] is π -compactly supported if there exists24 closed subset F ⊆ P whose quotient by π is compact such that A t,s = unless c S ( s ) ∈ F and c T ( t ) ∈ F . Define C ∗ P,π, cpt ( T, S ) ⊆ C ∗ P,π ( T, S ) to be the operator norm-closure of the linear space of π -compactly sup-ported, boundedly geometrically controlled operators.When S = T the space C ∗ P,π, cpt ( S, S ) is a closed two-sided ideal of the C ∗ -algebra C ∗ P,π ( S, S ) . We define the quotient C ∗ -algebra C ∗ P,π, evtl ( S, S ) := C ∗ P,π ( S, S ) /C ∗ P,π, cpt ( S, S ) and the quotient space C ∗ P,π, evtl ( T, S ) := C ∗ P,π ( T, S ) /C ∗ P,π, cpt ( T, S ) . Under the assumption T ⊆ S the space C ∗ P,π, evtl ( T, S ) is a finitely generatedand projective Hilbert C ∗ P,π, evtl ( S, S ) -module.Now suppose W is a ( d + ) -dimensional, connected, complete, orientedRiemannian manifold of bounded geometry and that P = f W is a principal π -space over W . Choose a triangulation of bounded geometry of the base(compare [9, 5.8, 5.9]), and lift to a π -invariant triangulation Σ of f W . Asbefore, let S be the disjoint union of all the Σ p . We have the followingHilbert-Poincar´e complex over C ∗ e W,π, evtl ( S, S ) :(5.1) C ∗ e W,π, evtl ( Σ , S ) b ← − C ∗ e W,π, evtl ( Σ , S ) b ← − · · · b ← − C ∗ e W,π, evtl ( Σ d + , S ) , with duality operator obtained from cap product with the fundamental cy-cle as before. The eventual signature class of f W → W is defined to beSgn π, evtl ( f W/W ) :=
Sgn π, evtl ( Σ, S ) ∈ K d + ( C ∗ e W,π, evtl ( S, S )) . The eventual signature class is invariant under oriented eventualhomotopy equivalences which are boundedly controlled and compatible with theprincipal π -bundles. roof. This is a special case of [9, Theorem 4.3] once we have shown thatour eventual homotopy equivalence f : Z W induces an algebraic homo-topy equivalence of Hilbert-Poincar´e complexes in the sense of [9, Defini-tion 4.1]. For this, we observe that the map f and its eventual homotopyinverse g : W Z , although only defined outside a compact subset, de-fine in the usual way chain maps f ∗ , g ∗ between the C ∗ e W,π, evtl ( S, S ) -Hilbertchain complexes of the triangulations, as compactly supported morphismsare divided out. For the construction of f ∗ , we need bounded control of themaps and bounded geometry of the triangulations. Similarly, the eventualhomotopies can be used in the standard way to obtain chain homotopiesbetween the identity and the composition of these induced maps, showingthat f ∗ is a homology isomorphism. Finally, f being orientation-preserving,it maps the (locally finite) fundamental cycle of Z , considered as a cyclerelative to a suitable compact subset, to a cycle homologous to the fun-damental cycle of W , again relative to a suitable compact subset. As theduality operator of the Hilbert-Poincar´e chain complex C ∗ e W,π, evtl ( Σ • , S ) , ob-tained from cap product with the fundamental cycle, is determined alreadyby such a relative fundamental cycle, the induced map f ∗ intertwines thetwo duality operators in the sense of [9, Definition 4.1]. Therefore [9, Theo-rem 4.3] implies the assertion. Theorem 2.12 is an almost immediate corollary of Theorem above. Wejust need the following supplementary computation:
Assume that W is noncompact. The kernel of the homomorphism K ∗ ( C ∗ e W,π ( S, S )) − → K ∗ ( C ∗ e W,π, evtl ( S, S )) induced from the quotient map from C ∗ e W,π ( S, S ) to C ∗ e W,π, evtl ( S, S ) is included inthe kernel of the homomorphism K ∗ ( C ∗ e W,π ( S, S )) − → K ∗ ( C ∗ π ( f W )) . Proof.
Since W is noncompact it contains a ray R that goes to infinity. Let e R ⊂ f W be its inverse image under the projection f W → W . We have the ideal C ∗ π ( e R ⊂ f W ) generated by operators supported in bounded neighborhoodsof e R . Because every π -compactly supported operator is also supported in a26ounded neighborhood of e R we get a commutative diagram(5.2) C ∗ e W,π, cpt ( S, S ) −−−− → C ∗ e W,π ( S, S ) −−−− → C ∗ e W,π, evtl ( S, S ) y y C ∗ π ( e R ⊂ f W ) −−−− → C ∗ π ( f W ) . We have a canonical isomorphism K ∗ ( C ∗ π ( e R )) ∼ = − → K ∗ ( C ∗ π ( e R ⊂ f W )) , see [25,Proposition 2.9], and a standard Eilenberg swindle shows that(5.3) K ∗ ( C ∗ π ( e R ⊂ f W )) = See [12, Section 7, Proposition 1] or [25, Proposition 2.6]. Now we applyK-theory to the diagram (5.2). The long exact K-theory sequence of the firstline of (5.2) shows that the kernel we have to consider is equal to the imageof the map induced by the first arrow of this line. Naturality and commu-tativity implies by (5.3) that this image is mapped to zero in K ∗ ( C ∗ π ( f W )) , asclaimed. Proof of Theorem 2.12. If W is compact, then so is Z , and the transverse sig-natures of both are zero. Otherwise, from Theorem we have thatSgn π, evtl ( f W/W ) =
Sgn π, evtl ( e Z/Z ) ∈ K d + ( C ∗ e W,π, evtl ( S, S )) , and from Lemma 5.5 we conclude from this thatSgn π ( f W/W ) =
Sgn π ( e Z/Z ) ∈ K d + ( C ∗ π ( f W )) . The theorem now follows from Theorem 2.8, the appropriate version ofRoe’s partitioned manifold index theorem.
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