Mapping the surgery exact sequence for topological manifolds to analysis

In this paper we prove the existence of a natural mapping from the surgery exact sequence for topological manifolds to the analytic surgery exact sequence of N. Higson and J. Roe. This generalizes the fundamental result of Higson and Roe, but in the treatment given by Piazza and Schick, from smooth manifolds to topological manifolds. Crucial to our treatment is the Lipschitz signature operator of Teleman. We also give a generalization to the equivariant setting of the product defined by Siegel in his Ph.D. thesis. Geometric applications are given to stability results for rho classes. We also obtain a proof of the APS delocalised index theorem on odd dimensional manifolds, both for the spin Dirac operator and the signature operator, thus extending to odd dimensions the results of Piazza and Schick. Consequently, we are able to discuss the mapping of the surgery sequence in all dimensions.Comment: 26 pages, accepted in "Journal of Topology and Analysis

Singular spaces, groupoids and metrics of positive scalar curvature

We define and study, under suitable assumptions, the fundamental class, the index class and the rho class of a spin Dirac operator on the regular part of a spin stratified pseudomanifold. More singular structures, such as singular foliations, are also treated. We employ groupoid techniques in a crucial way; however, an effort has been made in order to make this article accessible to readers with only a minimal knowledge of groupoids. Finally, whenever appropriate, a comparison between classical microlocal methods and groupoids methods has been provided.Comment: 50 page

Mapping analytic surgery to homology, higher rho numbers and metrics of positive scalar curvature

Let $\Gamma$ be a f.g. discrete group and let $\tilde M$ be a Galois $\Gamma$-covering of a smooth closed manifold $M$. Let $S_*^\Gamma(\tilde{M})$ be the analytic structure group, appearing in the Higson-Roe analytic surgery sequence $\to S_*^\Gamma(\tilde M)\to K_*(M)\to K_*(C_r^*\Gamma)\to$. We prove that for an arbitrary discrete group $\Gamma$ it is possible to map the whole Higson-Roe sequence to the long exact sequence of even/odd-graded noncommutative de Rham homology $\to H_{[*-1]}(\mathcal{A}\Gamma)\to H^{del}_{[*-1]}(\mathcal{A}\Gamma)\to H^{e}_{[*]}(\mathcal{A}\Gamma)\to$, with $\mathcal{A}\Gamma$ a dense homomorphically closed subalgebra of $C^*_r\Gamma$. Here, $H_{*}^{del}(\mathcal{A}\Gamma)$ is the delocalized homology and $H_{*}^{e}(\mathcal{A}\Gamma)$ is the homology localized at the identity element. Then, under additional assumptions on $\Gamma$, we prove the existence of a pairing between $HC^*_{del}(\mathbb{C}\Gamma)$, the delocalized part of the cyclic cohomology of $\mathbb{C}\Gamma$, and $H^{del}_{*-1}(\mathcal{A}\Gamma)$. This, in particular, gives a pairing between $S^\Gamma_*(\tilde M)$ and $HC^{*-1}_{del}(\mathbb{C}\Gamma)$. We also prove the existence of a pairing between $S^\Gamma_*(\tilde M)$ and the relative cohomology $H^{[*-1]}(M\to B\Gamma)$. Both these parings 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 $\rho(\tilde D)\in S_*^\Gamma(\tilde M)$ of an invertible $\Gamma$-equivariant Dirac type operator on $\tilde M$. Finally, we provide a precise study for the behavior of all previous K-theoretic and homological objects and of the higher rho numbers under the action of the diffeomorphism group of $M$. Then, we establish new results on the moduli space of metrics of positive scalar curvature when $M$ is spin.Comment: 103 pages. Changes from the first version: the title has been modified; imprecisions have been corrected; more details are given; several new sections with many geometric applications have been adde

Positive Scalar Curvature due to the Cokernel of the Classifying Map

This paper contributes to the classification of positive scalar curvature metrics up to bordism and up to concordance. Let $M$ be a closed spin manifold of dimension $\ge 5$ which admits a metric with positive scalar curvature. We give lower bounds on the rank of the group of psc metrics over $M$ up to bordism in terms of the corank of the canonical map $KO_*(M)\to KO_*(B\pi_1(M))$, provided the rational analytic Novikov conjecture is true for $\pi_1(M)$

On positive scalar curvature bordism

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.Comment: 7 pages. v2 corrected typos, added references and more details of some proofs and constructions, following suggestions of referees. To appear in Communications in Analysis and Geometr