Coarse cohomology of the complement

TL;DR

This paper introduces a new framework for coarse (co)homology of complements in metric spaces, providing models, duality results, and criteria for coarse Poincaré duality spaces, advancing coarse geometric analysis.

Contribution

It generalizes Roe's coarse (co)homology to complements, introduces a model space for coarse structure, and offers a new, simpler approach to coarse Poincaré duality spaces and Alexander duality.

Findings

Established a coarse Alexander duality theorem.

Provided a homological criterion for coarse PD(n) spaces.

Developed a model space capturing coarse geometric structure.

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 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.