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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

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(Fn)), 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 4k 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 M is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group S̃(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 2k 1 (k > 2) and the fundamental group of M is finitely embeddable.

geometry of groups, rigidity of manifolds, $K$–theory
Mathematical Subject Classification 2010
Primary: 19K99
Secondary: 20F99, 58D29
Received: 14 April 2014
Accepted: 28 December 2014
Published: 20 October 2015
Proposed: Benson Farb
Seconded: Steve Ferry, Leonid Polterovich
Shmuel Weinberger
Department of Mathematics
University of Chicago
5734 S University Avenue
Chicago, IL 60637-1514
Guoliang Yu
Department of Mathematics
Texas A&M University
College Station, TX 77843-3368
United States
Shanghai Center for Mathematical Sciences
Fudan University