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.
Signature operator on Lipschitz manifoldsWe start recalling fundamental results on Lipschitz manifolds. For further details we refer to [26,27,8,25,30].Definition 2.1. A Lipschitz atlas on a topological manifold M is an atlas such that the map ϕ • ψ −1 is a Lipschitz homeomorphism for any two charts ϕ : U → R n and ψ : V → R n . By definition a Lipschitz manifold structure on M is a maximal Lipschitz atlas.
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.
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Using standard results from higher (secondary) index theory, we prove that the positive scalar curvature bordism groups Pos spin 4n (G × Z) are infinite for any n ≥ 1 and G a group with non-trivial torsion. We construct representatives of each of these classes which are connected and with fundamental group G × Z. We get the same result for Pos spin 4n+2 (G × Z) 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. * PP thanks Ministero Istruzione Università Ricerca for partial support through the PRIN 2015 Spazi di Moduli e Teoria di Lie. TS and VFZ thank the German Science Foundation and its priority program "Geometry at Infinity" for partial support.
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