Rough index theory on spaces of polynomial growth and contractibility
Abstract
We will show that for a polynomially contractible manifold of bounded geometry and of polynomial volume growth every coarse and rough cohomology class pairs continuously with the Ktheory of the uniform Roe algebra. As an application we will discuss nonvanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get highercodimensional index obstructions to metrics of positive scalar curvature on closed manifolds with virtually nilpotent fundamental groups. We will give a computation of the homology of (a dense, smooth subalgebra of) the uniform Roe algebra of manifolds of polynomial volume growth.
 Publication:

arXiv eprints
 Pub Date:
 May 2015
 DOI:
 10.48550/arXiv.1505.03988
 arXiv:
 arXiv:1505.03988
 Bibcode:
 2015arXiv150503988E
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  KTheory and Homology;
 58J22 (Primary);
 46L80;
 19K56 (Secondary)
 EPrint:
 v4: final version, to appear in J. Noncommut. Geom. v3: added a computation of the homology of (a smooth subalgebra of) the uniform Roe algebra. v2: added as corollaries to the main theorem the multipartitioned manifold index theorem and the highercodimensional index obstructions against pscmetrics, added a proof of the strong Novikov conjecture for virtually nilpotent groups, changed the title