The General Notion of Descent in Coarse Geometry

TL;DR

This paper introduces a new coarse geometric notion called coarsely excisive functors, explores their properties, and relates them to existing conjectures like Farrell-Jones and Baum-Connes, proposing a conjecture about their isomorphism.

Contribution

It defines coarsely excisive functors in coarse geometry and connects them to topological excisiveness, assembly maps, and major conjectures in the field.

Findings

Coarsely excisive functors yield topologically excisive functors via cones.

Proposes a conjecture that the coarse assembly map is an isomorphism for certain spaces.

Shows that the coarse isomorphism conjecture implies injectivity of an equivariant assembly map.

Abstract

In this article, we introduce the notion of a functor on coarse spaces being coarsely excisive- a coarse analogue of the notion of a functor on topological spaces being excisive. Further, taking cones, a coarsely excisive functor yields a topologically excisive functor, and for coarse topological spaces there is an associated coarse assembly map from the topologically exicisive functor to the coarsely excisive functor. We conjecture that this coarse assembly map is an isomorphism for uniformly contractible spaces with bounded geometry, and show that the coarse isomorphism conjecture, along with some mild technical conditions, implies that a correspoding equivariant assembly map is injective. Particular instances of this equivariant assembly map are the maps in the Farrell-Jones conjecture, and in the Baum-Connes conjecture.