Finite part of operator K –theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds Academic Article uri icon

abstract

  • © 2015, Mathematical Sciences Publishers. All rights reserved. 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 (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 (Mʹ, f), where Mʹ is a compact manifold and f: Mʹ →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.

author list (cited authors)

  • Weinberger, S., & Yu, G.

citation count

  • 12

complete list of authors

  • Weinberger, Shmuel||Yu, Guoliang

publication date

  • October 2015