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

الأصل · EN

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(Fₙ)), 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.