Branched coarse coverings and transfer maps

TL;DR

This paper introduces branched coarse coverings and transfer maps in coarse homology, demonstrating their structure in coarse K-homology and applying these concepts to Atiyah's L^2-index theorem and Higson's counterexample to the coarse Baum-Connes conjecture.

Contribution

It develops the theory of branched coarse coverings and transfers, extending coarse K-homology with new structures and applying them to important index theorems and conjectures.

Findings

Coarse K-homology theories admit transfer structures.

Established versions of Atiyah's L^2-index theorem in coarse homotopy theory.

Provided a new argument related to Higson's counterexample to the coarse Baum-Connes conjecture.

Abstract

We introduce the concepts of branched coarse coverings and transfers between coarse homology theories along them. We show that various versions of coarse $K$-homology theories admit the additional structure of transfers. We show versions of Atiyah's $L^{2}$-index theorem in coarse homotopy theory and apply them to give a new argument for the corresponding step in Higson's counterexample to the coarse Baum-Connes conjecture.