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Paolo Piazza and Thomas Schick
Abstract

In this paper, we study the space of metrics of positive scalar curvature using methods from coarse geometry.

Given a closed spin manifold with fundamental group , Stephan Stolz introduced the positive scalar curvature exact sequence, in analogy to Wall’s surgery exact sequence in topology. It calculates a structure group of metrics of positive scalar curvature on (the object we want to understand) in terms of spin-bordism of (the classifying space of ) and a further group .

Higson and Roe introduced a K-theory exact sequence in coarse geometry which contains the Baum-Connes assembly map , with canonically associated to . The K-theory groups in question are the home of interesting index invariants and secondary invariants, in particular the rho-class of a metric of positive scalar curvature.

One of our main results is the construction of a map from the Stolz exact sequence to the Higson-Roe exact sequence (commuting with all arrows), using coarse index theory throughout. This theorem complements the results of Higson and Roe in [10, 11, 12] where they show that it is indeed possible to map the surgery exact sequence in topology to their sequence .

Our main tool is an index theorem of Atiyah-Patodi-Singer type, which is of independent interest. Here, assume that is a compact spin manifold with boundary, with a Riemannian metric which is of positive scalar curvature when restricted to the boundary (and ). Because the Dirac operator on the boundary is invertible, one constructs an APS-index . This can be pushed forward to (corresponding to the “delocalized part” of the index). We then prove a delocalized APS-index theorem, equating this class to the rho-class of the boundary .

As a companion to this, we prove a secondary partitioned manifold index theorem. Given a (non-compact) spin manifold with positive scalar curvature metric , with a free and discrete isometric action by a group and a -invariant cocompact partitioning hypersurface , one can use a “partitioned manifold construction” in order to obtain the partitioned manifold -class . Assume in addition that has a tubular neighborhood where the metric is a product . Then we prove the partitioned manifold -class theorem . We use this secondary partitioned manifold index theorem to distinguish isotopy classes of positive scalar curvature on .

Rho-classes, index theory and Stolz’ psc sequence]Rho-classes, index theory and Stolz’ positive scalar curvature sequence

1 Introduction and main results

1.1 Basics on coarse geometry and coarse index theory

We start by recalling the basic constructions of coarse geometry, their associated -algebras and K-theory as used in the paper. We shall freely use concepts and results from [8] and [32].

Definition 1.1

Let be a complete Riemannian manifold of positive dimension, the compactly supported continuous functions with values in and its sup-norm closure, the continuous functions vanishing at infinity.

Let be a Hermitean vector bundle. We consider and . These are so-called adequate -modules, which means that is a Hilbert space with a -homomorphism , given here by pointwise multiplication, and if then it does not act as compact operator, and that is dense in . We have a canonical isometry mapping into the first direct summand of . Using this, we map an operator on to the operator on . We will implicitly do this throughout the paper and this way consider the operators on as operators on , without explicitly mentioning it.

  • is defined to be the algebra of bounded operators on with the following properties:

    1. has finite propagation, which means that there is an such that for each and for each , .

    2. is pseudo-local: for each , the commutator is compact.

    is defined to be the norm closure of .

  • is defined to be the subalgebra of of operators which are in addition locally compact, i.e.  and are compact for each . is the -closure of . This is the Roe algebra of .

The definition generalizes to an arbitrary proper metric space ; then has to be replaced by an abstract adequate -module.

There are natural functoriality properties that we recall from [8, 14, 32]:

Definition 1.2

A map between proper metric spaces is a coarse map if for each there is such that the image under of every -ball is contained in an -ball and, moreover, the inverse image of every bounded set is bounded.

We will not prove the following functoriality results of Higson and Roe, but we recall, after the Proposition, the relevant construction which we are going to use.

Proposition 1.3

If is a continuous coarse map and are adequate or -modules, respectively, then induces a non-canonical, but with suitable choices functorial homomorphism which maps to . The induced map in K-theory is canonical.

Definition 1.4

Applying Proposition 1.3 to we observe that and depend only mildly on the adequate module , and that their K-theory is independent of this choice. We follow the custom of [32] and drop from the notation, writing simply or instead of , .

For the construction of of Proposition 1.3, we need the following concepts:

Definition 1.5

Let and two adequate modules and let be a coarse map. We say that an isometric embedding covers in the -sense if is the norm-limit of linear maps satisfying the following condition:

(1.5)

Given a coarse map it is always possible to find such a . Then the map , from the bounded operators on to the bounded operators of , sends to and we define . The induced map in K-theory is independent of the choice of , see [14, Lemma 3]. Moreover, by [7] the functor is a coarse homotopy invariant.

Regarding we have the following.

Definition 1.6

Let be a continuous coarse map, let , be two adequate modules. We shall say that an isometry covers in the -sensein [38, Definition 2.4], the same property is denoted “ covers topologically” if is the norm-limit of bounded maps satisfying the following two conditions:

  • there is an such that if , for and ;

  • is compact for each .

For such a one proves that sends into and induces therefore a morphism . As for , one proves that the induced map in K-theory does not depend on the choice of . See again [14, Lemma 3].

Up to tensoring with , see [8, Lemma 7.7], it is always possible to find an isometry satisfying the required two properties, which is the reason why we included this tensor product with in the definition of .

By [8, Lemma 7.8], is invariant under continuous coarse homotopy.


To be able to use standard techniques from the K-theory of -algebras, given a subspace we replace by an ideal of as follows:

Definition 1.7

Let be a proper metric space and a closed subset. Define as the closure of those operators such that there is an satisfying whenever with . Define we deviate here from the notation employed by Roe, e.g.  for in [32, Definition 3.10]. Our notations and definitions agree with those used in [38]. as the closure of those such that

  1. there is satisfying whenever with and

  2. and are compact.

Then and are ideals in .

We now describe equivariant versions of the constructions made so far. Assume therefore in addition that a discrete group acts freely and isometrically on the manifold and the Hermitean bundle . It then also acts by unitaries on and we define

  1. to be the norm closure of the -invariant part , and its ideal as the norm closure of . If is a -invariant subspace, we define in the corresponding way the ideals and .

  2. The construction generalizes to an arbitrary proper metric space with proper isometric -action, using a -adequateadequate requires a little bit of extra care, compare [32, Definition 5.13]: replacing by will do -module with compatible unitary -action.

  3. As indicated in the notation, one has suitable independence on , along the way with the obvious generalization of functoriality to -equivariant maps.

  4. If the quotient is a finite complex, then ; see [32, Lemmas 5.14, 5.15].

Lemma 1.8

(compare [14, Lemma 1], [38, Proposition 3.8]) Given a closed -subspace of a proper metric -space , the inclusion induces K-theory isomorphisms , .

Directly from the above results and the short exact sequence

we obtain the Higson-Roe surgery sequence for a manifold with quotient compacts:

(1.8)

We will also be interested in a universal version of this sequence. First we give a definition:

Definition 1.9

Let be a discrete group. Define

Here, is any contractible CW-complex with free cellular -action, a universal space for free actions.

By coarse invariance of and [32, Lemma 5.14], there is a canonical isomorphism for any free cocompact -space . Therefore the definition of is along canonical isomorphisms and we get canonically . Once this definition is given, we obtain immediately the (universal) Higson-Roe surgery sequence:

(1.9)

which can be rewritten as

(1.9)

It is proved by Roe in [33] that the homomorphism appearing in (1.9) is precisely equal to the assembly map. This implies:

(1.9)

If is a proper complete metric space with a free cocompact isometric -action, there is a universal -map with range in a -finite subcomplex ( is automatically coarse), and any two such maps are (coarsely continuously) -homotopic. We therefore get canonical induced maps

Moreover, for the map is a canonical isomorphism.

More generally, if is a complete metric space with free -action, is -invariant and is compact then is the limit of as , where is the closed -neighborhood of , again a -compact metric space. We get a compatible system of universal maps to , all with image in finite subcomplexes, and an induced compatible system of maps in K-theory, giving rise to the maps

whose composition is the universal map for .

1.2 Index and -classes

We now recall Roe’s method of applying -techniques to the Dirac operator to efficiently define primary and secondary invariants for spin manifolds in the context of coarse geometry.

Let be an arbitrary complete spin manifold with free isometric action by of dimension . Fix an odd continuous chopping function , i.e. . With the Dirac operator we now consider . Roe proves, using finite propagation speed of the wave operator and ellipticity, that this is an element in , compare [30, Proposition 2.3].

Proposition 1.10

Assume that is a -invariant closed subset and the scalar curvature is uniformly positive outside . Then is an involution modulo .

In particular, if we have uniformly positive scalar curvature, then is an involution in .

For the other extreme, without any further curvature assumption, is an involution modulo .

This important proposition is at the heart of the method. It is stated by Roe [32, Proposition 3.11] but without a full proof. A complete proof is given independently in Pape’s thesis [26, Theorem 1.4.28], or by Roe in [34, Lemma 2.3].

Recall that, given an involution in a -algebra , it defines in a canonical way the element . If is odd, in the situation of Proposition 1.10 we obtain the corresponding class .

If is even, we have to use the additional -invariant grading of the spinor bundle . The operator and, because is an odd function, are odd with respect to this decomposition so that we obtain the positive part . We choose a measurable bundle isometry and obtain the induced isometry covering in the -senseIn [32], it is only required that covers . However, as pointed out by Ulrich Bunke, to make sure that one needs the stronger assumption. In a previous version of the paper (the version published in Journal of Topology) we were only requiring that covers in the -sense. But only if we are as specific as in the new version, the K-theory class we introduce is indeed well-defined. This follows because now the possible choices form a contractible set, and by homotopy invariance of K-theory everything is well defined. For more details on this issue compare [27, Section 2.2].

Then is a unitary in and represents .

Definition 1.11

Let be a complete Riemannian spin manifold of dimension with isometric free action of . Define

Here is the boundary map of the long exact sequence of the extension .

Observe that, if we have uniformly positive scalar curvature outside of , we have a canonical lift to

If we have uniformly positive scalar curvature throughout, we define a secondary invariant, the -class of the metric , as

(1.11)

Finally, if is compact, there is the canonical map to of Definition 1.9 and we define , the -class of , as the image of under this map:

(1.11)
Remark 1.12

It is important to point out that in contrast to the -class , the -class vanishes for groups without torsion, at least for those for which the Baum-Connes conjecture holds. See the fundamental remark appearing in (1.9). This means we expect to be different from zero only for groups with torsion.
Basic non-trivial examples of for with torsion are considered in [13].

Notice that the -class is well defined whenever the Dirac operator is -invertible; we denote it in this more general case. In fact, we will sometime employ this notation also for the spin Dirac operator associated to a positive scalar curvature metric.

1.3 Delocalized APS-index theorem

Geometric set-up 1.13

Let now be a -dimensional Riemannian spin manifold with boundary, complete as metric spacei.e. every Cauchy sequence converges. We denote its boundary , and we assume always that we have product structures near the boundary. We assume that the scalar curvature of is uniformly positive, and that acts freely, isometrically and cocompactly on and therefore also on . We denote the quotient of by the action of as , a compact Riemannian manifold with boundary. Associated to these data is with extended product structure on the cylinder. This defines a complete Riemannian metric on and we then have uniformly positive scalar curvature outside .

The considerations of the previous subsection apply now to the pair and we obtain therefore a class and thus a class

(1.13)

Here we use the canonical inclusion which induces an isomorphism in K-theory by Lemma 1.8 below.

Let us remark here that, under the canonical isomorphism , this index class corresponds to any of the other APS-indices for manifolds with boundary defined in this context, e.g. using the Mishchenko-Fomenko approach and the b-calculus or using APS-boundary conditions, compare Section 2.

The passage from to corresponds to the passage to the delocalized part of the index information (we will explain this later). This delocalized part we can compute by a K-theoretic version of the APS-index theorem:

Theorem 1.14 (Delocalized APS-index theorem)

Let be an even dimensional Riemannian spin-manifold with boundary such that has positive scalar curvature. Assume that acts freely isometrically and is compact. Then

(1.14)

Here, we use induced by the inclusion and the inclusion.

Corollary 1.15

By functoriality, using the canonical -map of Definition 1.9, we have . If we define in , then the last equation reads

(1.15)

This gives immediately bordism invariance of the -classes:

Corollary 1.16

Let and be two odd-dimensional free cocompact spin -manifolds of positive scalar curvature. Assume that they are bordant as manifolds with positive scalar curvature, i.e. that there is a Riemannian spin manifold with free cocompact -action such that , has positive scalar curvature and restricts to on . Then

  • Proof.   The rho-class is additive for disjoint union and changes sign if one reverses the spin structure. Because and have uniformly positive scalar curvature, ; thus . The assertion now follows directly from Corollary 1.15.

Remark 1.17

Notice that bordism invariance holds only for -classes; indeed, we need a common -theory group where we can compare the two invariants. Precisely because of this last observation, the following variant of Corollary 1.16 holds:
Let and be two free cocompact spin -manifolds of positive scalar curvature endowed with -equivariant reference maps to a Hausdorff topological -space with compact quotient . Assume that there exists a Riemannian spin manifold as in Corollary 1.16 endowed with a -equivariant reference map such that . Then, defining , we have the following identity:

(1.17)
  • Proof.   Denote by the inclusion and similarly for . Let and be the natural inclusions . Then, from Theorem 1.14 we get

    We now apply . Since and and since , with the on the right hand side going from to , we see that

    Since the left hand side vanishes (recall that on is of positive scalar curvature) this is precisely what we wanted to prove.

Remark 1.18

We are convinced that the theorem also is correct if is odd. In the present paper we only deal with the even case, By using -linear Dirac operators and an appropriate setup for -linear (also called -multigraded) cycles for K-theory, we expect that our method should generalize to all dimensions and also to the refined invariants in real K-theory one can get that way. We plan to address the details in future work.

Remark 1.19

As we shall see, Theorem 1.14 has a surprisingly intricate proof. A different approach for proving it would be to develop a theory for the Calderon projector associated to a Dirac-type operator on a Galois covering with boundary. In this direction, recall the classical formula for the APS numeric index in terms of the Calderon projection and the APS projection : . If one were able to extend this formula to the APS-index class, then the theorem would follow provided one could establish, in addition, that the image of the class of the Calderon projector in vanishes. It would be very interesting to work out this alternative approach to Theorem 1.14, which seems to be, however, quite an intricate question. A first step in this direction is carried out in [1], where the Calderon projector for -module coefficients is constructed.

Example 1.20

The morphisms induce (canonical) isomorphisms in K-theory by Lemma 1.8 and because is a coarse equivalence, as is compact. Consequently we can map also to and compare its image there to .

It turns out that in general these two objects are different, so that a corresponding sharpening of Theorem 1.14 is not possible. Indeed, an additional secondary term, a rho-class of a bordism, shows up. This secondary class appears naturally when one gives a proof of bordism invariance of the rho-index using suitable exact sequences of K-theory of Roe algebras and the principle that “boundary of Dirac is Dirac”. We plan to work this out in a sequel publication.

Explicitly, take with , with the standard metrics (slightly modified to have product structure near the boundary, but clearly with positive scalar curvature as long as ).

Because of overall positive scalar curvature, vanishes, and so does its image in .

On the other hand, the Dirac operator on represents the fundamental class, a non-trivial element in . By the commutativity of the diagram (1.28), another main theorem of this paper, has to be non-trivial, being mapped to a non-trivial element in . Observe that this is a purely topological phenomenon, having nothing to do with analysis.

1.4 Secondary index theorem for -classes on partitioned manifolds

In this section, we formulate a partitioned manifold secondary index theorem, for the -class on a manifold of uniformly positive scalar curvature.

For this aim, let be a (non-compact) Riemannian spin manifold of dimension with isometric free -action and assume that there is a -invariant two-sided hypersurface such that is compact. We get a decomposition .

Let us quickly recall the primary partitioned manifold index theorem. The classical case is , then we obtain . The partition allows to construct a map (showing up in a corresponding Mayer-Vietoris sequence as in Section 3.3) to . The partitioned manifold theorem of Roe [30] then simply states that the image of under this map is . The corresponding statement for non-trivial and even is covered in [45].

We now treat the same question for the secondary rho class of manifolds with uniformly positive scalar curvature. Indeed, let us first give a direct definition of the partitioned manifold rho-class, similar to the definition of the partitioned manifold index as given by Higson [5].

Definition 1.21

Assume, in the above situation, that has dimension and uniformly positive scalar curvature. Then we constructed . Consider the image of under the -Mayer-Vietoris boundary map for the decomposition of into and along (discussed in Section 3.3): . We set

(1.21)

and we call it the partitioned manifold -class associated to the partitioned manifold . We shall be mainly concerned with a universal version of this class: we consider the canonical map and we set

(1.21)

We call this secondary invariant the partitioned manifold -class associated to .

Theorem 1.22

Let be a connected spin manifold partitioned by a hypersurface into . Let act freely on . Let be even. Assume that the metric on has uniformly positive scalar curvature and that the metric on a tubular neighborhood of the hypersurface has product structure, so that the induced metric also has positive scalar curvature. Assume, finally, that is compact. Then

Remark 1.23

We are convinced that the assertion of Theorem 1.22 also holds if is odd. As detailed in Remark 1.18, with appropriate new multigrading input our method might carry over. Again, we hope to work this out in the future.

Corollary 1.24

Let be as in Theorem 1.22 with two -equivariant metrics of uniformly positive scalar curvature (and in the same coarse equivalence class) which are of product type near . If are connected by a path of uniformly positive -equivariant metrics in the same coarse metric class (not necessarily product near ) then .

  • Proof.   We simply have to observe that we get a homotopy between and in and then apply homotopy invariance of K-theory.

As an application of this Corollary, assume that , which is assumed to be compact, has two metrics , with -invariant lifts , that have the property that . Of course, this implies that the two metrics are not concordant on . Stabilize by taking the product with (with the standard metric). We can now conclude that even with the extra room on we can not deform to through metrics of uniformly positive scalar curvature. This follows directly from the Corollary.

The strategy of proof for this -version of the partitioned manifold index theorem is the same as the classical one:

  1. we prove it with an explicit calculation for the product case;

  2. we prove that the partitioned manifold rho-class depends only on a small neighborhood of the hypersurface.

Remark 1.25

In the situation of Theorem 1.22, both and are defined in . However, our method does not give any information about equality of these classes, only about their images in . This is in contrast to Theorem 1.14, where the equality is established in .

On the other hand, we also don’t have an example where the partitioned manifold -class does not coincide with the -class of the cross section. It is an interesting challenge to either find such examples, or to improve the partitioned manifold secondary index theorem. The latter would be important in particular in light of applications like the stabilization problem we just discussed: if on are positive scalar curvature metrics which are not concordant, is the same true for and on ?

1.5 Mapping the positive scalar curvature sequence to analysis

The Stolz exact sequence is the companion for positive scalar curvature of the surgery exact sequence in the classification of high dimensional manifolds. The latter connects the structure set, consisting of all manifold structures in a given homotopy type, with the generalized homology theory given by the L-theory spectrum and the algebraic L-groups of the fundamental group.

Similarly, Stolz’ sequence connects the “structure set” (compare Definition 1.26), which contains the equivalence classes of metrics of positive scalar curvature to the generalized homology group and to . The latter indeed is a group which only depends on the fundamental group of . It is similar to the geometric definition of L-groups. Missing until now is an algebraic and computable description of these -groups, in contrast to the -groups of surgery.

In this subsection, we construct a map from the Stolz positive scalar curvature exact sequence to analysis. We give a picture which describes the transformation as directly as possible, using indices defined via coarse geometry.

Definition 1.26

Given a reference space (often ), define

  • as the set of singular bordism classes of -dimensional closed spin manifolds together with a reference map and a positive scalar curvature metric on . A bordism between and consists of a compact manifold with boundary , with , a reference map restricting to and on the boundary and a positive scalar curvature metric on which has product structure near the boundary and restricts to and on the boundary.

  • as the set of bordism classes where is a compact -dimensional spin-manifold, possibly with boundary, with a reference map , and with a positive scalar curvature metric on the boundary when the latter is non-empty. Two triples , are bordant if there is a bordism with positive scalar curvature between the two boundaries, call it , such that

    is the boundary of a spin manifold . The reference maps to have to extend over . By the surgery method for the construction of positive scalar curvature metrics, this set actually depends only on the fundamental group of if is connected, compare [35, Section 5].

  • is the usual singular spin bordism group of .

Proposition 1.27

As a direct consequence of the definitions we get a long exact sequence, the Stolz exact sequence

with the obvious boundary or forgetful maps.

Theorem 1.28

For a compact space with fundamental group and universal covering , there exists a well defined and commutative diagram, if is odd,