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

Abstract

In this paper, we study lower bounds on the $K$–theory of the maximal $C^{\ast }$–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 $Out\left(F_n\right)$), 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 $4k-1$ $\left(k>1\right)$. In this case, we derive a lower bound on the rank of the structure group $S\left(M\right)$, which is roughly defined to be the abelian group of all pairs $\left(M^{\prime },f\right)$, where $M^{\prime }$ is a compact manifold and $f:M^{\prime }\to M$ is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group $\tilde{S}\left(M\right)$, 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 $P\left(M\right)$ 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 $P\left(M\right)$ when the compact smooth spin manifold $M$ has dimension $2k-1$ $\left(k>2\right)$ and the fundamental group of $M$ is finitely embeddable.