Finite part of operator K theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds
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2015, Mathematical Sciences Publishers. All rights reserved. In this paper, we study lower bounds on the Ktheory 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 Ktheory 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, Gromovs 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 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 2k - 1 (k > 2) and the fundamental group of M is finitely embeddable.
