Finite part of operator K-theory for groups finitely embeddable into Hilbert space and the degree of non-rigidity of manifolds

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 (e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analytic connected manifold fixing a given point. It is an open question if every countable group is finitely embeddable. We apply this result to measure the degree of non-rigidity for any compact oriented manifold $M$ with dimension $4k-1$ $(k>1)$. We derive a lower bound on the rank of the structure group $S(M)$ in this case. 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.

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