09

Diffeomorphisms and the topology of positive scalar curvature

One of the fundamental goals of geometric topology is to understand the interplay between an underlying topology and the types of geometry supported by this topology.

For this project, the "underlying topology" is given by a smooth manifold $$M$$, and also the diffeomorphism group of such a manifold is part of this "underlying topology". A lot of our knowledge about the diffeomorphism group uses homotopy theory. One part of our project will be concerned with explicit geometric constructions of this homotopy theoretically given information. We then aim at exploiting this for new (secondary) constructions and applications.

The type of applications we plan to concentrate on is the study of the action of the diffeomorphism group on the space of metrics of positive scalar curvature $${\mathcal R}^+(M)$$ on $$M$$. Such applications raise the following research challenges and questions:

• get appropriate information about $$Diff(M)$$. Classically, this involves unstable homotopy theory, smoothing theory and other tools from algebraic and geometric topology.
• find tools to understand the effect of the action on $${\mathcal R}^+(M)$$, and apply them to the situation at hand. This will involve index theory of Dirac operators and the Gromov-Lawson surgery technique for metrics of positive scalar curvature.
• develop concordance space variants for $${\mathcal R}^+(M)$$ which allow for more flexible constructions and are more appropriate for calculations, and compare them to the true $${\mathcal R}^+(M)$$.
• find explicit geometric constructions. We do this to get a better theoretic understanding of the geometric content of the abstract homotopy theoretic information about $${\mathcal R}^+(M)$$, and also with the hope to arrive at new and additional tools.
• develop rigidity results on the action of $${\mathcal R}^+(M)$$. It is known that in many cases this action factors through a cobordism category. Computation in the kernel of the passage from $$Diff(M)$$ to this cobordism category should show that large subgroups of the homotopy groups of $$Diff(M)$$ act trivially on the homotopy groups of $${\mathcal R}^+(M)$$.
• Finally, we will adress the question to which extent the action of (compactly supported) diffeomorphisms on a non-compact manifold $$M$$ gives rise to non-trivial classes in $${\mathcal R}^+(M)$$, where $${\mathcal R}^+(M)$$ now denotes the space of complete uniformly positively scalar curvature metrics. This will involve an interplay between the flexibility given by "pushing off to infinity" and new tools from coarse index theory.

## Publications

Given a manifold $M$, we completely determine which rational $\kappa$-classes are non-trivial for (fiber homotopy trivial) $M$-bundles over the $k$-sphere, provided that the dimension of $M$ is large compared to $k$. We furthermore study the vector space of these spherical $\kappa$-classes and give an upper and a lower bound on its dimension. The proof is based on the classical approach to studying diffeomorphism groups via block bundles and surgery theory and we make use of ideas developed by Krannich--Kupers--Randal-Williams.

As an application, we show the existence of elements of infinite order in the homotopy groups of the spaces $\mathcal R_{\mathrm{Ric}>0}(M)$ and $\mathcal R_{\mathrm{sec}>0}(M)$ of positive Ricci and positive sectional curvature, provided that $M$ is $\mathrm{Spin}$, has a non-trivial rational Pontryagin class and admits such a metric. This is done by showing that the $\kappa$-class associated to the $\hat{\mathcal A}$-class is spherical for such a manifold.

In the appendix co-authored by Jens Reinhold it is (partially) determined which classes of the rational oriented cobordism ring contain an element that fibers over a sphere of a given dimension.

We construct and study an H-space multiplication on $$\mathcal{R}^+(M)$$ for manifolds M which are nullcobordant in their own tangential 2-type. This is applied to give a rigidity criterion for the action of the diffeomorphism group on $$\mathcal{R}^+(M)$$ via pullback. We also compare this to other known multiplicative structures on $$\mathcal{R}^+(M)$$.

 Journal Transactions of the AMS Link to preprint version Link to published version

We present a rigidity theorem for the action of the mapping class group $$\pi_0(\mathrm{Diff}(M))$$on the space $$\mathcal{R}^+(M)$$ of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional Spin-manifolds.

We construct smooth bundles with base and fiber products of two spheres whose total spaces have nonvanishing A-hat-genus. We then use these bundles to locate nontrivial rational homotopy groups of spaces of Riemannian metrics with lower curvature bounds for all Spin manifolds of dimension 6 or at least 10, which admit such a metric and are a connected sum of some manifold and $$S^n\times S^n$$ or $$S^n\times S^{n+1}$$, respectively. We also construct manifolds M whose spaces of Riemannian metrics of positive scalar curvature have homotopy groups that contain elements of infinite order that lie in the image of the orbit map induced by the push-forward action of the diffeomorphism group of M.

 Journal International Mathematics Research Notices Link to preprint version Link to published version

Let NM be a submanifold embedding of spin manifolds of some codimension k≥1. A classical result of Gromov and Lawson, refined by Hanke, Pape and Schick, states that M does not admit a metric of positive scalar curvature if k=2 and the Dirac operator of N has non-trivial index, provided that suitable conditions are satisfied. In the cases k=1 and k=2, Zeidler and Kubota, respectively, established more systematic results: There exists a transfer KO∗(C∗π1M)→KO∗−k(C∗π1N) which maps the index class of M to the index class of N. The main goal of this article is to construct analogous transfer maps E∗(Bπ1M)→E∗−k(Bπ1N) for different generalized homology theories E and suitable submanifold embeddings. The design criterion is that it is compatible with the transfer E∗(M)→E∗−k(N) induced by the inclusion NM for a chosen orientation on the normal bundle. Under varying restrictions on homotopy groups and the normal bundle, we construct transfers in the following cases in particular: In ordinary homology, it works for all codimensions. This slightly generalizes a result of Engel and simplifies his proof. In complex K-homology, we achieve it for k≤3. For k≤2, we have a transfer on the equivariant KO-homology of the classifying space for proper actions.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

Let Γ be a finitely generated discrete group and let M˜ be a Galois Γ-covering of a smooth compact manifold M. Let u:XBΓ be the associated classifying map. Finally, let SΓ∗(M˜) be the analytic structure group, a K-theory group appearing in the Higson-Roe exact sequence ⋯→SΓ∗(M˜)→K∗(M)→K∗(C∗Γ)→⋯. Under suitable assumptions on Γ we construct two pairings, first between SΓ∗(M˜) and the delocalized part of the cyclic cohomology of CΓ, and secondly between SΓ∗(M˜) and the relative cohomology H∗(MBΓ). Both are compatible with known pairings associated with the other terms in the Higson-Roe sequence. In particular, we define higher rho numbers associated to the rho class ρ(D˜)∈SΓ∗(M˜) of an invertible Γ-equivariant Dirac type operator on M˜. Regarding the first pairing we establish in fact a more general result, valid without additional assumptions on Γ: indeed, we prove that it is possible to map the Higson-Roe analytic surgery sequence to the long exact sequence in noncommutative de Rham homology ⋯−→jH∗−1(AΓ)→ιHdel∗−1(AΓ)→δHe∗(AΓ)−→j∗⋯ with AΓ a dense homomorphically closed subalgebra of CrΓ and Hdel∗(AΓ) and He∗(AΓ) denoting versions of the delocalized homology and the homology localized at the identity element, respectively.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

Gromov and Lawson developed a codimension 2 index obstruction to positive scalar curvature for a closed spin manifold M, later refined by Hanke, Pape and Schick. Kubota has shown that also this obstruction can be obtained from the Rosenberg index of the ambient manifold M which takes values in the K-theory of the maximal C*-algebra of the fundamental group of M, using relative index constructions. In this note, we give a slightly simplified account of Kubota's work and remark that it also applies to the signature operator, thus recovering the homotopy invariance of higher signatures of codimension 2 submanifolds of Higson, Schick, Xie.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups of a cartesian product GxZ are infinite in dimension 4n if n>0 G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group GxZ. We get the same result in dimension 4n+2 (n>0) if G is finite and contains an element which is not conjugate to its inverse. This generalizes the main result of Kazaras, Ruberman, Saveliev, "On positive scalar curvature cobordism and the conformal Laplacian on end-periodic manifolds" to arbitrary even dimensions and arbitrary groups with torsion.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

We show that an enlargeable Riemannian metric on a (possibly nonspin) manifold cannot have uniformly positive scalar curvature. This extends a well-known result of Gromov and Lawson to the nonspin setting. We also prove that every noncompact manifold admits a nonenlargeable metric. In proving the first result, we use the main result of the recent paper by Schoen and Yau on minimal hypersurfaces to obstruct positive scalar curvature in arbitrary dimensions. More concretely, we use this to study nonzero degree maps f from a manifold X to the product of the k-sphere with the n-k dimensional torus, with k=1,2,3. When X is a closed oriented manifold endowed with a metric g of positive scalar curvature and the map f is (possibly area) contracting, we prove inequalities relating the lower bound of the scalar curvature of g and the contracting factor of the map f.

 Journal Proc. AMS Publisher AMS Link to preprint version Link to published version

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

In this paper we prove a strengthening of a theorem of Chang, Weinberger and Yu on obstructions to the existence of positive scalar curvature metrics on compact manifolds with boundary. They construct a relative index for the Dirac operator, which lives in a relative K-theory group, measuring the difference between the fundamental group of the boundary and of the full manifold. Whenever the Riemannian metric has product structure and positive scalar curvature near the boundary, one can define an absolute index of the Dirac operator taking value in the K-theory of the C*-algebra of fundamental group of the full manifold. This index depends on the metric near the boundary. We prove that the relative index of Chang, Weinberger and Yu is the image of this absolute index under the canonical map of K-theory groups. This has the immediate corollary that positive scalar curvature on the whole manifold implies vanishing of the relative index, giving a conceptual and direct proof of the vanishing theorem of Chang, Weinberger, and Yu. To take the fundamental groups of the manifold and its boundary into account requires working with maximal C* completions of the involved *-algebras. A significant part of this paper is devoted to foundational results regarding these completions.

Related project(s):
9Diffeomorphisms and the topology of positive scalar curvature

• 1

## Team Members

Dr. Simone Cecchini
Georg-August Universität Göttingen
simone.cecchini(at)uni-goettingen.de

Prof. Dr. Diarmuid Crowley
Principal investigator
University of Melbourne
dcrowley(at)unimelb.edu.au

Thorsten Hertl
Doctoral student
Georg-August-Universität Göttingen/Universität Augsburg
thorsten.hertl(at)stud.uni-goettingen.de