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Transgressions and Chern characters in coarse homotopy theory

Ulrich Bunke

TL;DR

Transgressions and Chern characters are developed in a broad, functorial framework for equivariant coarse geometry, connecting coarse and BM-type theories through the Higson corona. The authors introduce analytic and topological transgressions, a coarse algebraic Chern character, and various (periodic) cyclic and Borel–Moore constructions, establishing a large commutative diagram that intertwines coarse, topological, and analytic K-homology with algebraic and cyclic invariants. They prove commutativity under finite-group and CW-complex hypotheses, and extend the framework via motivic transgression to broader settings, including a homotopical description of analytic K-homology as BM for certain cases. The work also develops transfer-enabled Borelification and pairing with cohomology classes, laying groundwork for index-theoretic applications and potential Novikov-type consequences in coarse geometry.

Abstract

This paper investigates a variety of coarse homology theories and natural transformations between them. We in particular study the commutativity of a square relating analytical and topological transgressions with algebraic and homotopy theoretic Chern characters. Here a transgression is a natural transformation from a coarse homology theory to a functor which factorizes over the Higson corona functor, and a Chern character is a transformation from a $K$-theory like coarse or Borel-Moore type homology theory to an ordinary version.

Transgressions and Chern characters in coarse homotopy theory

TL;DR

Transgressions and Chern characters are developed in a broad, functorial framework for equivariant coarse geometry, connecting coarse and BM-type theories through the Higson corona. The authors introduce analytic and topological transgressions, a coarse algebraic Chern character, and various (periodic) cyclic and Borel–Moore constructions, establishing a large commutative diagram that intertwines coarse, topological, and analytic K-homology with algebraic and cyclic invariants. They prove commutativity under finite-group and CW-complex hypotheses, and extend the framework via motivic transgression to broader settings, including a homotopical description of analytic K-homology as BM for certain cases. The work also develops transfer-enabled Borelification and pairing with cohomology classes, laying groundwork for index-theoretic applications and potential Novikov-type consequences in coarse geometry.

Abstract

This paper investigates a variety of coarse homology theories and natural transformations between them. We in particular study the commutativity of a square relating analytical and topological transgressions with algebraic and homotopy theoretic Chern characters. Here a transgression is a natural transformation from a coarse homology theory to a functor which factorizes over the Higson corona functor, and a Chern character is a transformation from a -theory like coarse or Borel-Moore type homology theory to an ordinary version.

Paper Structure

This paper contains 32 sections, 88 theorems, 491 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. The functors and transformations
  4. Commutativity
  5. Functors and natural transformations
  6. Bornological coarse spaces and the Higson corona functor
  7. Equivariant coarse homology theories
  8. Uniform bornological coarse spaces, topological bornological spaces, and locally compact Hausdorff spaces
  9. Equivariant topological coarse $K$-homology
  10. Equivariant analytic $K$-homology
  11. The analytic transgression
  12. Equivariant algebraic coarse $K$-homology
  13. Equivariant coarse periodic cyclic homology
  14. The algebraic coarse Chern character
  15. Equivariant periodic coarse homology
  16. ...and 17 more sections

Key Result

Proposition 1.1

We have an equivalence of spectrum-valued functors

Theorems & Definitions (225)

  • Proposition 1.1
  • Remark 1.2
  • Proposition 1.3: \ref{['kophertgertgterg']}
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Example 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 215 more