Rough index theory on spaces of polynomial growth and contractibility

TL;DR

This paper develops a rough index theory for polynomially contractible manifolds with bounded geometry, linking coarse cohomology with K-theory, and explores implications for Dirac operators and scalar curvature obstructions.

Contribution

It introduces a new framework connecting rough cohomology and K-theory on polynomial growth manifolds, with applications to index theory and geometric obstructions.

Findings

Coarse and rough cohomology classes pair continuously with K-theory.

Non-vanishing of rough index classes of Dirac operators established.

Higher-codimensional index obstructions to positive scalar curvature derived.

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 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.