Finite part of operator K–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds

Weinberger, Shmuel; Yu, Guoliang

doi:10.2140/gt.2015.19.2767

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Finite part of operator $K$–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds

Shmuel Weinberger and Guoliang Yu

Geometry & Topology 19 (2015) 2767–2799

DOI: 10.2140/gt.2015.19.2767

Abstract

In this paper, we study lower bounds on the $K$ –theory of the maximal $C^{*}$ –algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator $K$ –theory and give a lower bound that is valid for a large class of groups, called the finitely embeddable groups. The class of finitely embeddable groups includes all residually finite groups, amenable groups, Gromov’s monster groups, virtually torsion-free groups (eg $Out (F_{n})$ ), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. We apply this result to measure the degree of nonrigidity for any compact oriented manifold $M$ with dimension $4 k - 1$ $(k > 1)$ . In this case, we derive a lower bound on the rank of the structure group $S (M)$ , which is roughly defined to be the abelian group of all pairs $(M^{'}, f)$ , where $M^{'}$ is a compact manifold and $f : M^{'} \to M$ is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group $\tilde{S} (M)$ , which measures the size of the collections of compact manifolds homotopic equivalent to but not homeomorphic to $M$ by any homeomorphism at all (not necessary homeomorphism in the homotopy equivalence class). For a compact Riemannian manifold $M$ with dimension greater than or equal to $5$ and positive scalar curvature metric, there is an abelian group $P (M)$ that measures the size of the space of all positive scalar curvature metrics on $M$ . We obtain a lower bound on the rank of the abelian group $P (M)$ when the compact smooth spin manifold $M$ has dimension $2 k - 1$ $(k > 2)$ and the fundamental group of $M$ is finitely embeddable.

Keywords

geometry of groups, rigidity of manifolds, $K$–theory

Mathematical Subject Classification 2010

Primary: 19K99

Secondary: 20F99, 58D29

References

Publication

Received: 14 April 2014

Accepted: 28 December 2014

Published: 20 October 2015

Proposed: Benson Farb

Seconded: Steve Ferry, Leonid Polterovich

Authors

Shmuel Weinberger
Department of Mathematics University of Chicago 5734 S University Avenue Chicago, IL 60637-1514 USA

Guoliang Yu
Department of Mathematics Texas A&M University College Station, TX 77843-3368 United States	Shanghai Center for Mathematical Sciences Fudan University Shanghai China

Volume 19, issue 5 (2015)