Coarse cohomology of the complement

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 Poincar\'e 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 PD($n$) space. Our approach is inspired by the work of Kapovich and Kleiner, but is somewhat different, and we believe, simpler.