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Localization for coarse homology theories

Ulrich Bunke, Luigi Caputi

Abstract

We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version. We discuss the special case of algebraic and topological equivariant coarse $K$-homology and obtain the coarse analog of Segal's localization theorem.

Localization for coarse homology theories

Abstract

We introduce the notion of a Bredon-style equivariant coarse homology theory. We show that such a Bredon-style equivariant coarse homology theory satisfies localization theorems and that a general equivariant coarse homology theory can be approximated by a Bredon-style version. We discuss the special case of algebraic and topological equivariant coarse -homology and obtain the coarse analog of Segal's localization theorem.

Paper Structure

This paper contains 16 sections, 45 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Localization theorems in equivariant topology
  3. The Abstract Localization Theorem for presheaves on the orbit category
  4. The Abstract Localization Theorem for topological $G$-spaces
  5. The Segal localization theorem revisited
  6. Fixed points in bornological coarse spaces
  7. Conventions
  8. Motivic fixed points
  9. The motivic orbit functor
  10. Bredon-style equivariant coarse homology theories
  11. Associated Bredon-style homology theories
  12. Coarse Assembly Maps
  13. Localization for coarse homology theories
  14. The Coarse Abstract Localization Theorems
  15. Localization for equivariant coarse topological $K$-homology
  16. ...and 1 more sections

Key Result

Theorem \oldthetheorem

Assume: Then, the morphism $X^{{\mathcal{F}}}\to X$ induces an equivalence $E^{G}(X^{{\mathcal{F}}})\to E^{G}(X)$.

Theorems & Definitions (126)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Theorem \oldthetheorem: Abstract Localization Theorem I
  • proof
  • Example \oldthetheorem
  • Definition \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 116 more