Journal of Noncommutative Geometry


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Volume 13, Issue 2, 2019, pp. 617–666
DOI: 10.4171/JNCG/335

Published online: 2019-07-17

Rough index theory on spaces of polynomial growth and contractibility

Alexander Engel[1]

(1) Universit├Ąt Regensburg, Germany

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 $K$-theory of the uniform Roe algebra.

As an application we will discuss non-vanishing of rough index classes of Dirac operators over such manifolds, and we will furthermore get higher-codimensional 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.

Keywords: Novikov conjecture, uniform Roe algebra, uniformly finite homology

Engel Alexander: Rough index theory on spaces of polynomial growth and contractibility. J. Noncommut. Geom. 13 (2019), 617-666. doi: 10.4171/JNCG/335