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Coronas and Callias type operators in coarse geometry

Ulrich Bunke, Matthias Ludewig

Abstract

We interpret the coarse symbol and index class of a Callias type Dirac operator $D+Ψ$ on a manifold $M$ as a pairing between the coarse symbol and index classes associated to $D$ and K-theory classes of the coarse corona of $M$ or $M$ itself determined by $Ψ$. Local positivity of $D$ and local invertibility of $Ψ$ are incorporated in terms of support conditions on the $K$-theoretic level.

Coronas and Callias type operators in coarse geometry

Abstract

We interpret the coarse symbol and index class of a Callias type Dirac operator on a manifold as a pairing between the coarse symbol and index classes associated to and K-theory classes of the coarse corona of or itself determined by . Local positivity of and local invertibility of are incorporated in terms of support conditions on the -theoretic level.

Paper Structure

This paper contains 25 sections, 47 theorems, 389 equations.

Table of Contents

  1. Introduction
  2. Coarse index theory with supports.
  3. Callias type operators.
  4. Structure of the paper.
  5. Coarse cohomology theories
  6. Uniform bornological coarse spaces and cones
  7. Coarse homology theories
  8. Local and locally finite homology theories
  9. $C^{*}$-algebras and categories and their $K$-theory
  10. $X$-controlled Hilbert spaces and coarse $K$-homology
  11. Coarse coronas
  12. Coarse coronas and a commutator estimate
  13. Uniform coronas and the comparison map
  14. Analytic locally finite $K$-homology and Paschke duality
  15. The coarse corona pairing
  16. ...and 10 more sections

Key Result

Lemma 2.38

We have a functorial fibre sequence

Theorems & Definitions (201)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Example 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8
  • Example 2.9
  • ...and 191 more