Branched coarse coverings and transfer maps

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.