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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 boundthat is valid for a large class of groups, called the "finitely embeddablegroups". The class of finitely embeddable groups includes all residually finitegroups, amenable groups, Gromov's monster groups, virtually torsion free groups(e.g. $Out(F_n)$), and any group of analytic diffeomorphisms of an analyticconnected manifold fixing a given point. It is an open question if everycountable group is finitely embeddable. We apply this result to measure thedegree 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 dimensiongreater than or equal to 5 and positive scalar curvature metric, there is anabelian group $P(M)$ that measures the size of the space of all positive scalarcurvature metrics on $M$. We obtain a lower bound on the rank of the abeliangroup $P(M)$ when the compact smooth spin manifold $M$ has dimension $2k-1$$(k>2)$ and the fundamental group of $M$ is finitely embeddable.

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