Manifolds with fibered singularities

Positive scalar curvature on manifolds with fibered singularities

Abstract.

A (compact) manifold with fibered -singularities is a (possibly) singular pseudomanifold with two strata: an open nonsingular stratum (a smooth open manifold) and a closed stratum (a closed manifold of positive codimension), such that a tubular neighborhood of is a fiber bundle with fibers each looking like the cone on a fixed closed manifold . We discuss what it means for such an with fibered -singularities to admit an appropriate Riemannian metric of positive scalar curvature, and we give necessary and sufficient conditions (the necessary conditions based on suitable versions of index theory, the sufficient conditions based on surgery methods and homotopy theory) for this to happen when the singularity type is either or , and and the boundary of the tubular neighborhood of the singular stratum are simply connected and carry spin structures. Along the way, we prove some results of perhaps independent interest, concerning metrics on spin manifolds with positive “twisted scalar curvature,” where the twisting comes from the curvature of the spin line bundle.

Key words and phrases:
positive scalar curvature, pseudomanifold, singularity, bordism, transfer, -theory, index
2010 Mathematics Subject Classification:
Primary 53C21; Secondary 58J22, 53C27, 19L41, 55N22
JR Partially supported by U.S. NSF grant number DMS-1607162.

Boris Botvinnik]botvinn@uoregon.edu Jonathan Rosenberg]jmr@math.umd.edu

1. Introduction

1.1. Manifolds with fibered singularities

In the paper [7], one of us studied the problem of when a spin manifold with Baas-type singularities admits a metric of positive scalar curvature. This was done when the singularities are a combination of the types , , and the Bott manifold of dimension (a geometric generator of Bott periodicity in -homology). We will use the abbreviation psc-metric for metric of positive scalar curvature.

This paper considers a similar problem of existence of positive scalar curvature metrics for spin manifolds with fibered -singularities, where is a closed manifold. We take to be a compact Lie group, either a cyclic group for some , or . In a sequel paper, we will study the case where is a compact semisimple Lie group, such as or .

In more detail, let be a closed manifold, and be a compact manifold with boundary . We assume that the boundary is a total space of a smooth fiber bundle with the fiber . We denote by the total space of the associated fiber bundle , where the manifold is replaced by the cone fiberwise. The notation is a reminder that this will be a tubular neighborhood of the stratum . The associated singular space has a singular stratum whose normal bundle has fibers homeomorphic to the cone on , but where the normal bundle is not necessarily a trivial bundle. We call a manifold with fibered -singularity. Clearly the original manifold with the given smooth fibration uniquely determines the singular space .

The situation we study fits nicely into a more general framework. The compact metrizable space is taken to be a Thom-Mather stratified space (see [20]) of depth one. That means that is the union of a smooth closed manifold (the singular stratum) and an open smooth manifold (the regular stratum) . There is an open neighborhood of in , with a continuous retraction and a continuous map such that , where is a fiber bundle over with fiber , the cone over , as above. The original manifold , called a resolution of , can be identified with , so that the boundary is the total space of a fibration over with typical fiber . Clearly the interior of can be identified with the regular stratum . Conversely, given a compact manifold with fibered boundary , we obtain a Thom-Mather stratified space with two strata by collapsing the fibers.

1.2. Riemannian metrics on manifolds with fibered singularities

There are (at least) two natural definitions of a Riemannian metric on such a singular manifold . The first possibility, which one can call a cylindrical metric, the definition used in [7], is a Riemannian metric in the usual sense on the nonsingular manifold , which is a product metric on a collar neighborhood on the boundary , such that the compact Lie group acts by isometries of the boundary and the map is a Riemannian submersion. The curvature of such a metric is defined as usual on the (nonsingular) manifold . Such a metric also determines a Riemannian metric on the “Bockstein” manifold .

Assume has a psc-metric as above. Since the metric is assumed to be a product metric in a collar neighborhood of the boundary , the restriction is a -invariant psc-metric. Thus the question of existence of a cylindrical psc-metric comes down to whether or not there exists a -invariant psc-metric on that extends to . When or , the boundary always has a -invariant psc-metric by [18], for which the quotient map is a Riemannian submersion, so the only question is whether or not this metric on extends to . By contrast, when , the map is a covering map, and has a -invariant psc-metric if and only if has a psc-metric, for which there is a well-developed obstruction theory (e.g., [24, 25, 26, 9]). When , the same is true; i.e., has an -invariant psc-metric if and only if has a psc-metric, but this now a hard theorem of Bérard-Bergery [5, Theorem C].

The problem with the notion of cylindrical metric is that it doesn’t really take into account the local structure near . A second possible definition is what one could call a conical metric on . In this point of view, the primary object of study is the singular manifold , not the manifold with non-empty boundary. A conical metric is again an ordinary Riemannian metric on the nonsingular part of (which is diffeomorphic to the interior of ), but we require its local behavior near the singular stratum to look like , where is a translation-invariant (standard) metric on , is a metric on the singular stratum , and is the distance to the singular stratum. Note that such a metric on the nonsingular stratum is necessarily incomplete. We assume that near the boundary of the tubular neighborhood , the metric transitions to a cylindrical psc-metric on , and thus existence of a conical psc-metric is a stronger requirement than existence of a cylindrical psc-metric. When or , the cone on is homeomorphic to or , and so a conical metric near the singular stratum locally looks like an ordinary rotationally invariant Riemannian metric on a vector bundle over , of fiber dimension , resp., , and actually extends to a smooth Riemannian metric on . Here is our main question:

Question: When does admit a cylindrical or conical psc-metric?

Remark 1.1.

In general, existence of cylindrical and of conical psc-metrics are two different questions. However, they are the same provided , since then the term drops out of the definition of conical metric. When , there are some cases where existence of a cylindrical psc-metric implies existence of a conical psc-metric, but for the most part we will focus on the latter, which has greater geometric significance.

1.3. Main results

We consider two cases: when or .

The case of

We denote by the bordism theory of spin manifolds with fibered -singularities, and by the spectrum which represents this bordism theory.

In more detail, the group consists of equivalence classes of maps , where is the singular space associated to an -dimensional spin manifold with fibered -singularities (with given -fold regular covering map preserving the spin structure on induced from the spin structure on ). Two such maps and are said to be equivalent if there is a spin bordism between and as spin manifolds with boundary, with a -fold regular covering map given by a free action of on , such that the restrictions and coincide with the corresponding maps and , and there is a map restricting to and on and . In particular, the manifold gives spin bordism of regular closed spin manifolds .

The groups are closely related to the regular spin bordism groups; indeed, there is an exact triangle

(1)

Here is the homomorphism which considers a regular spin manifold as a manifold with empty singularity, the degree homomorphism takes a -fibered manifold to the map classifying the -folded regular covering , and finally, is a standard transfer.

We denote by the spectrum representing real -theory, and by the map of spectra corresponding to the index map . The transfer map in gives the map of spectra ; we denote by the cofiber of the map . Then there is a natural map which makes the following diagram of spectra commute:

(2)

It turns out that the map is still not quite the right “index map”, since index theory doesn’t detect all of , and the classes of the obvious psc-manifolds with fibered -singularities defined by even-dimensional disks with a free -action on the boundary sphere will map nontrivially under . However, by composing with the real assembly map and its splitting we get the index homomorphism

(See Definition 2.5 and the comments just before it for more details.) The image contains the torsion (all of order ) in the -theory of the real group ring of the cyclic group . Here is our first main result on the existence of psc-metrics:

Theorem A.

Let be a spin manifold with fibered -singularities, of dimension . Assume that is non-empty, and both and are connected and simply connected, and the action of on preserves the spin structure. Then admits a metric of positive scalar curvature if and only if vanishes in the group .

Outline.

Here is an outline of the proof. In order to prove necessity of vanishing of , we use -algebraic index theory to show that is indeed an obstruction to the existence of a psc-metric on a spin manifold with fibered -singularities. On the other hand, we show also that the existence of a psc-metric on depends only on the corresponding bordism class . To prove that vanishing of the index is sufficient, we analyze the spectra and . In particular, we construct the cofiber sequences

(3)

where and denote and with coefficients respectively. Moreover, we show that the map is consistent with these decompositions.

To finish the proof, we use the transfer map from [28], where . Recall that is the isometry group of the standard metric on , and takes a map to the total space of the geometric -bundle induced by .

The diagram (2) allows us to use (3) and the transfer map to show that all elements of the kernel of the index map can be represented by a manifold with fibered -singularities carrying a psc-metric. ∎

The case

This case in some ways is similar to the case of -singularities; however, it has new interesting features.

Let be a manifold with fibered -singularities, i.e., comes with a free -action on the boundary . Let be the smooth -fiber bundle given by this action. Since the cone on is , the corresponding pseudomanifold is actually a smooth manifold, but with a distinguished codimension two submanifold . There are two separate problems to consider: the existence of a cylindrical psc-metric, which is analogous to a psc-metric on a manifold with Baas-type singularities, or the existence of a conical psc-metric, which is about the pseudomanifold , but we focus primarily on the latter.

We assume that is a spin manifold. However, it is worth pointing out that under our definition of fibered -singularities, even if is spin, or may not be. For example, suppose is the unit disk in , and we equip with the usual free action of by scalar multiplication by complex numbers of absolute value . Then is non-spin if is odd, and is non-spin if is even.

In general, the -fiber bundle is given by some classifying map which induces a complex line bundle . We split into the disjoint union of its path-components: . Then for each component , we have two possibilities:

  1. is spin,

  2. is non-spin.

In the case (i), the restriction could be an arbitrary complex line bundle. In the case (ii), has to be in the kernel of the homomorphism since is spin. This means that mod 2 and determines a spin-structure on . Thus the fiber bundle always splits into spin and spin-components:

Thus the Bockstein operator can be described as

(4)

We denote by the bordism theory of spin manifolds with fibered -singularities and by the corresponding spectrum which represents this bordism theory; see section 3 for more details. The Bockstein operator (4) induces a homomorphism

Then we have a transfer map

The transfer takes a pair or (in second case is a spin-structure on ) to the total space of the corresponding -fiber bundle . We show that there is an exact triangle

(5)

where takes a closed spin-manifold to a manifold with empty fibered -singularity. At the level of spectra, we obtain the following cofibration:

Just as in the -case, we consider corresponding -theories and obtain the following commutative diagram of spectra:

(6)

Here and are the corresponding index maps.1 In particular, we have the index homomorphism . Here is the second main result, on the existence of a conical psc-metric on a spin manifold with fibered -singularities:

Theorem B.

Let be a spin manifold with fibered -singularities, of dimension . Assume that is non-empty and connected, and and are simply connected. Then admits a conical metric of positive scalar curvature if and only if vanishes in the group , where is the above index map.

Outline.

Just as in the -case, we show that the map provides an obstruction to the existence of a psc-metric and that the existence of a psc-metric on depends only on the bordism class . This guarantees the necessity in Theorem B. Again we need more information on the spectra and . In particular, we prove the splittings:

(7)

The final step consists of studying the kernels of the index maps and . In the case of , Stolz [28, 29] showed that is the image of a transfer map , where and the transfer amounts to taking the total space of an -bundle. One can do something similar in the (easier) case of , where this time (since splits -locally as a sum of ’s and some Eilenberg-Mac Lane spectra; see [23, §8] and [14, p. 184]), the transfer amounts to taking the total space of -bundles instead of bundles.2 Details of this argument may be found in Section 5. Then (6) allows us to use the splitting (7) and the transfer to show that all elements of the kernel of the index map are realized by - or -bundles and hence can be represented by manifolds with fibered -singularities carrying a psc-metric.

The theorem also holds even when is disconnected, assuming the appropriate indices vanish on each component of . ∎

1.4. Plan of the paper

The organization of the rest of the paper is quite straightforward. Section 2 deals in detail with the case of fibered -singularities, and includes all the details of the proof of Theorem A. Section 3 explains the precise definition of conical psc-metrics in the case of fibered -singularities, and includes all the details of the proof of Theorem B, except for two results of possibly independent interest which are needed for the proof of sufficiency, namely the spin bordism theorem, which is proved in Section 4, and the theorem on -bundles, which is proved in Section 5.

1.5. Acknowledgments

We would like to thank Paolo Piazza for many useful suggestions on the subject of this paper. We expect Paolo to be a coauthor on a subsequent paper dealing with other singularity types . We would also like to thank Bernd Ammann, Bernhard Hanke, and André Neves for organizing an excellent workshop at the Mathematisches Forschungsinstitut Oberwolfach in August 2017 on Analysis, Geometry and Topology of Positive Scalar Curvature Metrics, which led to the present work.

2. The case of fibered -singularities

This section will be about giving necessary and sufficient conditions for a manifold with fibered -singularities to admit a psc-metric, under the additional conditions that and are connected, simply connected, and spin. In particular, we prove Theorem A.

2.1. Some definitions

Recall that a compact closed manifold with fibered -singularities means a compact smooth manifold with boundary, equipped with with a smooth free action of the group on . We denote by the quotient space ; we have a covering map . The associated singular space is , where identifies any two points in having the same image in . We use the notation to emphasize the above -fibered structure. Note that a -manifold in the sense of Sullivan and Baas [31, 3] is a special case.

Just as in the case of Sullivan-Baas singularities, we say that a smooth manifold (with non-empty regular boundary ) is a manifold with fibered -singularities with boundary if the boundary is given a splitting together with free -action on . It is required that so that the -action on restricts to a given free -action on . In particular, we obtain that the -covering restricted to coincides with the -covering , where . We denote by the associated singular space where we identify those points in which belong to the same orbit under -action. Note that then by definition.

Remark 2.1.

We should emphasize that should be thought as a manifold with corners, where its corner is the manifold ; see Fig. 1.

Figure 1. A manifold with fibered singularities and corners

A metric of positive scalar curvature on is a Riemannian psc-metric on , which is a product metric in a collar neighborhood of the boundary , and with the metric on -invariant. If , it defines a psc-metric on a closed manifold with fibered -singularities.

2.2. Bordism theory

Here we set up the bordism theory and the variant of -theory that will be needed in this section. We use a slight modification of the bordism theory of Baas [3]. Below we assume that all manifolds in this section are spin.

Let be a topological space. Then the group consists of equivalence classes of maps , where is the singular space associated to an -dimensional spin manifold with fibered -singularities, with preserving the spin structure on (this is only an issue if is even). Two such maps and are said to be equivalent if there exist a spin manifold with fibered -singularities with boundary and a map restricting to and on and respectively.

We use notation . In particular, the manifold gives a regular spin bordism between closed manifolds: .

Exactly as in [3], it is easy to see that is a homology theory given by a spectrum . There is an obvious natural transformation which is given by considering closed spin manifolds as manifolds with empty fibered -singularity. The Bockstein operator comes together with a map classifying the -covering . This determines the transformation of degree . Then we have a transformation

of degree , given by a transfer: it sends a map to the lift , where is the -fold cover of determined by the composite .

Just as in [3, Theorem 3.2] and [7, §2.1], there is an exact triangle of (unreduced) bordism theories

(8)

Restating the above assertion in slightly different language, we have the following:

Proposition 2.2.

The transfer defines a map of bordism spectra , and is the cofiber of this map.3

Proof.

Clearly, this is just a restatement of the assertion above about the exact triangle (8). To explain the geometry involved, we write out the geometrical proof of the corresponding exact sequence:

That is clear. To see that , observe that if has fibered -singularities and , , then . And bounds as a spin manifold since . To see that , note that given with corresponding covering , then , and this is since bounds as a manifold with fibered -singularities; take with , , .

To get , observe that if has fibered -singularities and bounds as a spin manifold with mapping to , say , then the associated -fold covering has boundary , and is bordant as a spin manifold with fibered -singularities to the closed manifold . To get , observe that if is a spin manifold with specified -fold covering , and if is a spin boundary, say , then is a spin manifold with fibered -singularities and with . Finally, to see that , suppose is a closed manifold that bounds as a manifold with fibered -singularities. Then there is a with decomposed into two pieces: one of them a copy of and the other projecting down to some ; this shows is bordant to in the image of . ∎

2.3. Relevant -theory

We denote by the spectrum representing real -theory. The transfer map has its analog in -theories:

(9)

which is compatible with the transfer map for -fold coverings of spin manifolds. Let be the -theory associated to the cofiber of the transfer map (9).

From this definition it follows that the groups fit into a long exact sequence

(10)

where when we decompose as , is multiplication by on the first summand and on the second. A similar statement holds for spin bordism. Thus we have the following:

Proposition 2.3.

We have exact sequences

(11)

and

(12)

Here and denote spin bordism and real -theory with coefficients. If is odd, then the group vanishes except when is even, when is an extension of by , and consists of odd torsion.

Proof.

Most of this follows immediately from the exact sequences in Proposition 3 and (10).

When is odd, , which is when , otherwise. Since is only nonzero in odd dimensions, the result follows. ∎

Finally, we need to consider the relationship between the spectra and . Let be the usual Atiyah-Hitchin orientation for spin manifolds (corresponding to the -index of the Dirac operator). The following result is a consequence of the naturality of the transfer:

Proposition 2.4.

There is a map of spectra making the following diagram commute:

The last step is setting up the correct index map which will give the obstruction to a psc-metric on a compact manifold with fibered -singularities. For this we need to recall the construction of the assembly map (see for example [26, 4]). This is much simpler in the case we need here of a finite group .4 There is a map of spectra

where denotes the -theory spectrum of the real group ring, viewed as a Banach algebra, which can be defined in several ways. One method, developed by Loday [19] in a slightly different context, is to use the map on classifying spaces induced by the inclusion . An alternative is to think of assembly as an index map. Given a class in , represented, say, by the Dirac operator on a closed spin manifold with coefficients in some auxiliary bundle, together with a map classifying a -covering of ,

where the right-hand side is the index of the lifted -invariant operator on with coefficients in . Since the assembly map for the trivial group is just the identity map, we can peel this off and consider also the reduced assembly map

where is the sum of the simple summands in the group ring corresponding to all irreducible (real) representations except for the trivial representation. When is cyclic of odd order, all representations except for the trivial representation are of complex type, so the groups are torsion-free while is torsion. Hence the map vanishes in homotopy. When is cyclic of even order, it has exactly two irreducible representations of real type, the trivial representation and the sign representation. These are homomorphisms . The remaining representations are of complex type. In this case, it is shown in [26, Theorem 2.5] that is a split surjection onto the torsion of , which consists of in each dimension .

Definition 2.5.

The index obstruction map (which appears in the statement of Theorem A) is the map defined by composing the map of Proposition 2.4 with projection onto the inverse image of the torsion in . (Recall that assembly gives a split surjection from onto the torsion in