Slowly increasing cohomology for discrete metric spaces with polynomial growth

Abstract

The authors introduce a kind of slowly increasing cohomology HS* (X) for a discrete metric space X with polynomial growth, and construct a character map from the slowly increasing cohomology HS* (X) into HC*_cont(S(X)), the continuous cyclic cohomology of the smooth subalgebra S(X) of the uniform Roe algebra B*(X). As an application, it is shown that the fundamental cocycle, associated with a uniformly contractible complete Riemannian manifold M with polynomial volume growth and polynomial contractibility radius growth, is slowly increasing.

Keywords

Slowly increasing cohomology / Polynomial growth / Uniform Roe algebra

References

[1] Connes A., Moscovici H.. Cyclic cohomology and Novicov conjecture and hyperbolic groups. Topology, 1990, 29(3): 345-388

[2] Yu G.. Cyclic cohomology and higher indices for noncompact complete manifolds. J. Func. Anal., 1995, 133: 442-473

[3] Roe J.. Coarse cohomology and index theory for complete Riemanniam manifold. Mem. Amer. Math. Soc., 1983, 104(497): 90

[4] Chen X., Wei S.. Spectral invariant subalgebras of reduced crossed product C*-algebras. J. Func. Anal., 2003, 197: 228-246

[5] Connes A.. Noncommutative differential geometry. Inst. Hautes Etudes Sci. Publ. Math., 1985, 62: 41-144

[6] Gromov M.. Asymptotic invariants of infinite groups, Geometric Group Theory, Vol. 2, Sussex, 1991, 1993, Cambridge: Cambridge Univ. Press 1-295

[7] Roe J.. An index theorem on open manifold. J. Diff. Geom., 1988, 27: 87-113