Mapping the surgery exact sequence for topological manifolds to analysis

Abstract

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.

Positive scalar curvature and product formulas for secondary index invariants

“…However, in Siegel's construction the compatibility between the exterior product and the Mayer-Vietoris boundary map appears to be not straightforward. Recently, this approach has also been studied by Zenobi [30] with a focus on the signature operator and secondary invariants associated to homotopy equivalences. Moreover, the product formula can be implemented using the geometric picture of the structure group due to Deeley-Goffeng [3].…”

Adiabatic groupoid and secondary invariants in K-theory

“…So, if we n, we have that the exterior product with y is rationally injective and that if n = 1 it is injective. [56,Proposition 5.19]. Another related result in this context is given by [55,Corollary 5.8] which is in some sense complementary in the assumptions: apart from the fact that Zeidler deals with partial secondary invariants, the difference between the two results is that in this paper we do not assume anything on the nature of the manifold Y , but we have some assumptions on the group Γ (such as for instance K-amenabilty, see [56,Section 5]); whereas Zeidler assume that the manifold Y is hypereuclidean and that the group Λ is any group.…”

The surgery exact sequence, K-theory and the signature operator

“…This program has now been implemented by Vito Felice Zenobi in [31], building on previous work of Paul Siegel, see [21] and [22]. Consequently, the delocalised APS index theorem presented in this article is now established in every dimension.…”

“…A purely functional analytic argument, which applies to Lipschitz manifolds without a pseudodifferential calculus is given in [31].…”

“…By using the signature operator of Hilsum and Teleman it is possible to extend our results to Lipschitz manifolds and thus, by Sullivan's theorem, to the surgery exact sequence in the category TOP of topological manifolds. We refer the reader to Zenobi's [31] where stability results for the topological structure set under products are also discussed.…”