Wrong way maps in uniformly finite homology and homology of groups

TL;DR

The paper develops wrong way maps in uniformly finite homology and group homology, providing tools to study geometric and topological obstructions such as positive scalar curvature and inessentialness.

Contribution

It introduces a new construction of wrong way maps in uniformly finite and group homology under specific geometric conditions.

Findings

Constructed wrong way maps in uniformly finite homology for certain manifolds.

Extended the construction to homology of groups via equivariant methods.

Applied the maps to identify obstructions to positive scalar curvature and inessentialness.

Abstract

Given a non-compact Riemannian manifold M and a submanifold N of codimension q, we will construct under certain assumptions on both M and N a wrong way map in uniformly finite homology. Using an equivariant version of the construction and applying it to universal covers, we will construct wrong way maps in homology of groups. As applications we discuss obstructions against positive scalar curvature metrics and against inessentialness.