Finite asymptotic dimension and the coarse assembly map

TL;DR

This paper demonstrates that for certain coarse spaces with finite asymptotic dimension, the coarse assembly map is a phantom equivalence, simplifying understanding of coarse homology theories.

Contribution

It provides a straightforward proof that the coarse assembly map is a phantom equivalence under conditions of weak transfers and finite asymptotic dimension.

Findings

Coarse assembly map is a phantom equivalence for spaces with finite asymptotic dimension.

Simplifies the proof of the coarse assembly map's properties in this setting.

Applicable to strong coarse homology theories with weak transfers.

Abstract

In this note we give a simple argument for the fact that the coarse assembly map for a strong coarse homology theory with weak transfers and a bornological coarse space of weakly finite homotopical asymptotic dimension is a phantom equivalence.