Abstract
In this paper, we study lower bounds on the –theory of the maximal –algebra of a discrete group based on the amount of torsion it contains. We call this the finite part of the operator –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 ), 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 with dimension . In this case, we derive a lower bound on the rank of the structure group , which is roughly defined to be the abelian group of all pairs , where is a compact manifold and is a homotopy equivalence. In many interesting cases, we obtain a lower bound on the reduced structure group , which measures the size of the collections of compact manifolds homotopic equivalent to but not homeomorphic to by any homeomorphism at all (not necessary homeomorphism in the homotopy equivalence class). For a compact Riemannian manifold with dimension greater than or equal to and positive scalar curvature metric, there is an abelian group that measures the size of the space of all positive scalar curvature metrics on . We obtain a lower bound on the rank of the abelian group when the compact smooth spin manifold has dimension and the fundamental group of is finitely embeddable.
Citation
Shmuel Weinberger. Guoliang Yu. "Finite part of operator $K$–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds." Geom. Topol. 19 (5) 2767 - 2799, 2015. https://doi.org/10.2140/gt.2015.19.2767
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