Coarse cohomology of the complement

Abstract

In this paper, we define the coarse (co)homology of the complement of a subspace in a metric space, generalizing the coarse (co)homology of Roe. We give a model space which encodes coarse geometric structure of the complement. We also introduce a new approach to coarse Poincaré duality spaces. We prove a version of coarse Alexander duality for these spaces and give a homological criterion for a space to be a coarse $\mathrm{PD}(n)$ space. Our approach is inspired by the work of Kapovich and Kleiner but is somewhat different, and we believe, simpler.