Homotopy groups of the moduli space of metrics of positive scalar curvature
We show by explicit examples that in many degrees in a stable range the homotopy groups of the moduli spaces of Riemannian metrics of positive scalar curvature on closed smooth manifolds can be non-trivial. This is achieved by further developing and then applying a family version of the surgery construction of Gromov-Lawson to certain nonlinear smooth sphere bundles constructed by Hatcher.
As described, this works for all manifolds of suitable dimension and for the quotient of the space of metrics of positive scalar curvature by the (free) action of the subgroup of diffeomorphisms which fix a point and its tangent space.
We also construct special manifolds with positive scalar curvature where the quotient of the space of metrics of positive scalar curvature by the full diffeomorphism group has non-trivial higher homotopy groups.
Let be a closed smooth manifold. In this article we study the topology of the space of metrics of positive scalar curvature and of corresponding moduli spaces. We abbreviate “metric of positive scalar curvature” by “psc-metric”.
It has been known for a long time that there are quite a few obstructions to the existence of psc-metrics. This starts in dimension , where the Gauß-Bonnet theorem tells us that only the sphere and admit such a metric. In general the Lichnerowicz formula in combination with the Atiyah-Singer index theorem implies that if is a spin manifold and admits a psc-metric, then the -genus of is zero. The Gromov-Lawson-Rosenberg conjecture  was an attempt to completely characterize those spin manifolds admitting psc-metrics. It was later disproved in .
In spite of the complicated picture for general manifolds, the existence question has been resolved completely for simply connected manifolds of dimension at least five. Gromov and Lawson proved in  that if is not spin, then there is no obstruction and admits a psc-metric. Assuming that is spin, Stolz  proved that the only obstruction is the -valued index of the Dirac operator on .
If admits a psc-metric, one can go on and investigate the topology of , the space of psc-metrics on equipped with the the -topology. Note that , the diffeomorphism group of , acts on via pull-back, and so it is even more natural to study the moduli space .
In the spin case index theoretic methods were used to show that the spaces and have infinitely many components in many cases, see e.g. the work of Gromov-Lawson  or Lawson-Michelsohn  or, for more refined versions, the papers [6, 24, 26]. If is simply connected, this applies to the case when .
Hitchin observed in his thesis [17, Theorem 4.7] that sometimes, in the spin case, non-zero elements in the homotopy groups of yield, via the action of on , non-zero elements in the homotopy groups of . More precisely, he proves this way that is non-trivial for and is non-trivial for .
Contrasting these positive results, it has been an open problem to decide whether for or for can be non-trivial. Note that, by construction, Hitchin’s elements in , , are mapped to zero in the moduli space . Some experts even raised the suspicion that the components of this moduli space are always contractible.
1.2. Moduli spaces of psc-metrics
In this paper we will construct many examples of non-zero elements in higher homotopy groups of moduli spaces of psc-metrics on closed smooth manifolds . We denote by the space of all Riemannian metrics with the -topology. The group of diffeomorphisms acts from the right on the space by pull-back: . The orbit space of this action is the moduli space of Riemannian metrics on and written . The orbit space of the restricted -action on the subspace of psc-metrics, the moduli space of Riemannian metrics of positive scalar curvature on , is our principal object of interest.
In general the action of the full diffeomorphism group is not free on : For example, if a finite group acts effectively on (i.e. if occurs as a finite subgroup of ), then any metric on can be averaged over , and the resulting metric will be fixed by . Therefore we also consider the moduli spaces with observer as proposed by Akutagawa and Botvinnik .
Let be a connected closed smooth manifold with some basepoint . Let be the subgroup of of those diffeomorphisms which fix and induce the identity on the tangent space . This is the group of diffeomorphisms which preserve an observer based at .
If is a connected smooth closed manifold with a basepoint then acts freely on the space of Riemannian metrics on .
This lemma is well known, compare e.g. [7, Proposition IV.5]. For convenience we recall the proof. Assume is a Riemannian metric on , and . This means that the map is an isometry of . As and are fixed by , so are all geodesics emanating from (pointwise). Since is closed and connected, every point lies on such a geodesic, so is the identity. ∎
In the following we equip and with the -topologies. Let . We call the observer moduli space of Riemannian metrics on . Since the space is contractible and the action of on is proper (see ), Lemma 1.2 implies that the orbit space is homotopy equivalent to the classifying space of the group . In particular one obtains a -principal fiber bundle
This yields isomorphisms of homotopy groups
Now we restrict the action of to the subspace of psc-metrics. Clearly this action is free as well. We call the orbit space
the observer moduli space of psc-metrics. Again we obtain a -principal fiber bundle
The inclusion induces inclusions of moduli spaces and . We collect our observations in the following lemma.
Let be a connected closed manifold and . Then
there is the following commutative diagram of principal -fibrations
the observer moduli space of Riemannian metrics on is homotopy equivalent to the classifying space ;
there is a homotopy fibration
Our main method will be similar, but starting from the fiber sequence (1.3). We will show that certain non-zero elements of can be lifted to . Once such lifts have been constructed, it is immediate that they represent non-zero elements in as their images are non-zero in .
1.3. The results
We start from the particular manifold . Let be a base point. Then the group is homotopy equivalent to the group of diffeomorphisms of the disk which restrict to the identity on the boundary . These groups and their classifying spaces have been studied extensively. In particular the rational homotopy groups are known from algebraic -theory computations and Waldhausen -theory in a stable range.
(Farrell and Hsiang ) Let . Then
Here and in later places the shorthand notation means that for fixed there is an so that the statement is true for all .
Consider the inclusion map and the corresponding homomorphism of homotopy groups:
Here is our first main result.
is an epimorphism for . In particular, the groups are non-trivial for odd and .
Theorem 1.4 is essentially an existence theorem and does not directly lead to a geometric interpretation of the generators of . This was achieved later in the work of Bökstedt  and Igusa [18, 20] based on a construction of certain smooth nonlinear disk and sphere bundles over due to Hatcher. The non-triviality of some of these bundles is detected by the non-vanishing of a higher Franz-Reidemeister torsion invariant.
Recall from [18, 19, 20] that for any closed smooth manifold there are universal higher Franz-Reidemeister torsion classes , where is the subgroup of diffeomorphisms of that act trivially on . Note that . Furthermore, it is well-known that is the subgroup of consisting of orientation preserving diffeomorphisms. In particular, these classes define characteristic classes for smooth fiber bundles over path connected closed smooth manifolds with acting trivially on . (The last condition can be weakened to being a unipotent -module , but this is not needed here).
The relevant class of the Hatcher bundles over with fiber was computed in [13, 18, 20] and shown to be non-zero, if is odd. The generators of appearing in Theorem 1.4 can be represented by classifying maps of these Hatcher bundles in this way. In order to prove Theorem 1.5 we construct families of psc-metrics on these bundles.
Therefore, in Section 2, we will first study how and under which conditions such constructions can be carried out. Assuming that a given smooth bundle admits a fiberwise Morse function, we use the surgery technique developed by Walsh , which generalizes the Gromov-Lawson construction of psc-metrics via handle decompositions [12, 14] to families of Morse functions, in order to construct families of psc-metrics on this bundle, see Theorem 2.10. This is the technical heart of the paper at hand. Compared to  the novel point is the generalization of the relevant steps of this construction to nontrivial fiber bundles.
Then, we will study particular generators of for suitable and , as in Theorem 1.4. To give a better idea how we are going to proceed, recall that the observer moduli space serves as a classifying space of smooth fiber bundles with fiber and structure group . We obtain the universal smooth fiber bundle
In particular, a map representing an element gives rise to a commutative diagram of smooth fiber bundles
This shows that a lift of the class to is nothing but a family of psc-metrics of positive scalar curvature on the bundle from (1.5).
We will explain the precise relationship in Section 3 and show that the construction described in Section 2 applies to Hatcher’s -bundles. Here we make use of a family of Morse functions on these bundles as described by Goette [13, Section 5.b]. This will finish the proof of Theorem 1.5.
Given a closed smooth manifold of dimension , we can take the fiberwise connected sum of the trivial bundle and Hatcher’s exotic -bundle. Using additivity of higher torsion invariants [20, Section 3] we obtain non-trivial elements in for given for any manifold of odd dimension as long as .
If in addition admits a psc-metric, this can be combined with the fiberwise psc-metric on Hatcher’s -bundle constructed earlier to obtain a fiberwise psc-metric on the resulting nontrivial -bundle over . This shows:
Let be a closed smooth manifold admitting a metric of positive scalar curvature. If is odd, then the homotopy groups are non-trivial for .
In order to study the homotopy type of the classical moduli space of psc-metrics it remains to construct examples of manifolds for which the non-zero elements in constructed in Theorem 1.6 is not mapped to zero under the canonical map . This will be done in Section 4 and leads to a proof of the following conclusive result.
For any there exists a closed smooth manifold admitting a metric of positive scalar curvature so that is non-trivial for .
One should mention that the manifolds we construct in Theorem 1.7 do not admit a spin structure and are of odd dimension. In particular, the usual methods to distinguish elements of , which use the index of the Dirac operator, do not apply to these manifolds, and we have no non-trivial lower bound on the number of components of .
Boris Botvinnik would like to thank K. Igusa and D. Burghelea for inspiring discussions on topological and analytical torsion and thank SFB-478 (Geometrische Strukturen in der Mathematik, Münster, Germany) and IHES for financial support and hospitality. Mark Walsh also would like to thank SFB-478 for financial support and hospitality. Thomas Schick was partially supported by the Courant Research Center “Higher order structures in Mathematics” within the German initiative of excellence.
2. The surgery method in twisted families
The aim of this section is to prove a result on the construction of fiberwise metrcis of positive scalar curvature on certain smooth fiber bundles. At first we briefly review the Gromov-Lawson surgery technique  on a single manifold. Here we use the approach developed by Walsh [30, 31].
2.1. Review of the surgery technique on a single manifold
Let be a compact manifold with non-empty boundary and with . We assume that the boundary is the disjoint union of two manifolds and both of which come with collars
where is taken with respect to some fixed reference metric on , see Definition 2.1 below. By a Morse function on we mean a Morse function such that
and the restriction of to the collars (2.0) coincides with the projection onto the second factor
We denote by the set of critical points of .
We say that a Morse function is admissible if all its critical points have indices at most (where ). We note that the last condition is equivalent to the “codimension at least three” requirement for the Gromov-Lawson surgery method. We denote by and the spaces of Morse functions and admissible Morse functions, respectively, which we equip with the -topologies.
Let . A Riemannian metric on is compatible with the Morse function if for every critical point with the positive and negative eigenspaces and of the Hessian are -orthogonal, and , .
We notice that for a given Morse function , the space of compatible metrics is convex. Thus the space of pairs , where , and is a metric compatible with , is homotopy equivalent to the space . We call a pair as above an admissible Morse pair. We emphasize that the metric on has no relation to the psc-metrics we are going to construct.
[30, Theorem 2.5] Let be a smooth compact cobordism with . Assume that is a positive scalar curvature metric on and is an admissible Morse pair on . Then there is a psc-metric on which extends and has a product structure near the boundary.
We will provide here only an outline and refer to [30, Theorem 2.5] for details.
We begin with a few topological observations. For simplicity, we assume for the moment that is an elementary cobordism, i.e. that has a single critical point of index . The general case is obtained by repeating the construction for each critical point. Fix a gradient like vector field for . Intersecting transversely at there is a pair of trajectory disks and , see Fig. 1. Here the lower -dimensional disk is bounded by an embedded -sphere . It consists of the union of segments of integral curves of the gradient vector field beginning at the bounding sphere and ending at . Here and below we use the compatible metric for all gradient vector fields. Similarly, is a -dimensional disk which is bounded by an embedded -sphere . The spheres and are known as trajectory spheres associated to the critical point . As usual, the sphere is embedded into together with its neighborhood .
We denote by the union of all trajectories originating at the neighborhood , and notice that is a disk-shaped neighborhood of , see Fig. 1. A continuous shrinking of the radius of down to zero induces a deformation retraction of onto .
Now we consider the complement , which coincides with the union of all trajectories originating at . By assumption none of these trajectories have critical points. We use the normalized gradient vector field of to specify a diffeomorphism
Now we construct the metric . On the region , we define the metric to be simply where the coordinate comes from the embedding above.
To extend this metric over the region , we have to do more work. Notice that the boundary of decomposes as
Here is of course the tubular neighborhood while the piece is a tubular neighborhood of the outward trajectory sphere .
Without loss of generality assume that . Let and be constants satisfying . The level sets and divide into three regions:
The region is diffeomorphic to . We use again the flow to identify with in a way compatible with the identification of with . Then, on , we define as the product . Moreover, we extend this metric near the part of the boundary, where again is the trajectory coordinate.
We will now define a family of particularly useful psc-metrics on the disk . For a detailed discussion see .
Let and be a smooth function satisfying the following conditions:
when is near ;
Clearly such functions exists, furthermore, the space of functions satisfying (1), (2), (3) for some is convex. Let be the standard radial distance function on , and be the standard metric on (of radius one). Then the metric on is well-defined on . By restricting this metric to , one obtains the metric on . This metric is defined to be a torpedo metric, see Fig. 3.
It is easy to show that the above conditions (1), (2), (3) guarantee that has positive scalar curvature. Moreover it is -symmetric and is a product with the standard metric on the -sphere of radius near the boundary of and is the standard metric on the -sphere of radius near the center of the disk. Also one can show that the scalar curvature of can be bounded below by an arbitrarily large constant by choosing sufficiently small.
The most delicate part of the construction, carried out carefully in , involves the following: Inside the region , which is identified with the product , the metric smoothly passes into a standard product for some appropriately chosen , globally keeping the scalar curvature positive. This is done so that the induced metric on the level set , denoted , is precisely the metric obtained by applying the Gromov-Lawson construction to . Furthermore, near we have . Finally, on , which is identified with in the usual manner, the metric is simply the product . See Fig. 2 for an illustration.
After the choice of the Morse coordinate diffeomorphism with (and of the other parameters like and ), the construction is explicit and depends continuously on the given metric on .
Later on we will need the following additional facts. The next lemma is proved in [30, Section 3].
The “initial” transition consists of an isotopy. In particular, is isotopic to a metric which, on a neighborhood diffeomorphic to of the surgery sphere in , is .
The whole construction is -equivariant.
Lemma 2.6 will be of crucial importance later, when in a non-trivial family we cannot choose globally defined Morse coordinates giving diffeomorphisms to (as the bundle near the critical set is not trivial). We will construct Morse coordinates well defined up to composition with elements of . The equivariance of Lemma 2.6 then implies that our construction, which a priori depends on the choice of these coordinates, is consistent and gives rise to a smooth globally defined family of metrics.
We should emphasize that this construction can be carried out for a tubular neighborhood of arbitrarily small radius and for and chosen arbitrarily close to . Thus the region , on which the metric is not simply a product and is undergoing some kind of transition, can be made arbitrarily small with respect to the background metric . As critical points of a Morse function are isolated, it follows that this construction generalizes easily to Morse functions with more than one critical point. ∎
2.2. Extension to families
There is a number of ways to generalize the surgery procedure to families of manifolds. A construction relevant to our goals leads to families of Morse functions, or maps with fold singularities. We start with a local description.
A map is called a standard map with a fold singularity of index , if there is a so that is given as
Roughly speaking, the composition
with the projection onto the second factor defines a -parameterized family of Morse functions of index on in standard form.
Let be a compact manifold with boundary , . We denote by the group of all diffeomorphisms of which restrict to the identity near the boundary . Then we consider a smooth fiber bundle with fiber , where and . The structure group of this bundle is assumed to be and the base space to be a compact smooth manifold. Assume that the boundary is split into a disjoint union: .
Let , be the restriction of the fiber bundle to the fibers and respectively. Since each element of the structure group restricts to the identity near the boundary, the fiber bundles , are trivialized:
Choose a splitting of the tangent bundle of the total space as , where is the bundle tangent to the fibers , i.e. choose a connection.
Let be a smooth bundle as above. For each in let
be the inclusion of the fiber . Let be a smooth map. The map is said to be an admissible family of Morse functions or admissible with fold singularities with respect to if it satisfies the following conditions:
commutes. Here is projection on the first factor.
The pre-images and coincide with the submanifolds and respectively.
The set of critical points of is contained in and near each critical point of the bundle is equivalent to the trivial bundle so that with respect to these coordinates on and on the map is a standard map with a fold singularity as in Definition 2.7
For each the restriction
is an admissible Morse function as in Subsection 2.1. In particular, its critical points have indices .
We assume in addition that the smooth bundle is a Riemannian submersion , see . Here we denote by and the metrics on and corresponding to the submersion . Now let be an admissible map with fold singularities with respect to as in Definition 2.8. If the restriction of the submersion metric to each fiber , , is compatible with the Morse function , we say that the metric is compatible with the map .
Let be a smooth bundle as above and be an admissible map with fold singularities with respect to . Then the bundle admits the structure of a Riemannian submersion such that the metric is compatible with the map .
One can choose a Riemannian metric on the base , and for each fiber there is a metric compatible with the Morse function . Using convexity of the set of compatible metrics and the local triviality in the definition of a family of Morse functions, we can choose this family to depend continuously on . Then one can choose an integrable distribution (sometimes called connection) to construct a submersion metric which is compatible with the map , see . ∎
Below we assume that the fiber bundle is given the structure of a Riemannian submersion such that the metric is compatible with the map .
Consider the critical set . It follows from the definitions that is a smooth -dimensional submanifold in , and it splits into a disjoint union of path components (“folds”)
Furthermore, it follows that the restriction of the fiber projection
is a local diffeomorphism for each . In particular, is a covering map, and if the base is simply-connected then is a diffeomorphism onto its image.
Since the metric is a submersion metric, the structure group of the vector bundle is reduced to . Furthermore, since the metrics are compatible with the Morse functions , the restriction to a fold splits further orthogonally into the positive and negative eigenspaces of the Hessian of . Thus the metric induces the splitting of the vector bundle
with structure group for each . Here is the main result of this section:
Let be a smooth bundle, where the fiber is a compact manifold with boundary , the structure group is and the base space is a compact smooth simply connected manifold. Let be an admissible map with fold singularities with respect to . In addition, we assume that the fiber bundle is given the structure of a Riemannian submersion such that the metric is compatible with the map . Finally, we assume that we are given a smooth map .
Then there exists a Riemannian metric on such that for each the restriction to the fiber satisfies the following properties:
is a product metric near , ;
has positive scalar curvature on .
We assume that is path-connected. Let , . We denote, as above, where the is a path-connected fold. For a given point