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The general notion of descent in coarse geometry

Paul D Mitchener

Algebraic & Geometric Topology 10 (2010) 2419–2450

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 excisive 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 corresponding 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.

coarse geometry, descent
Mathematical Subject Classification 2000
Primary: 55N20
Secondary: 20F05
Received: 24 February 2010
Revised: 23 September 2010
Accepted: 2 October 2010
Published: 19 December 2010
Paul D Mitchener
School of Mathematics and Statistics
University of Sheffield
Hicks Building
S3 7RH