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The relative index theorem and a characterization of Fredholm operators

Magnus Fries

TL;DR

The work generalizes the relative index theorem to a broad class of hypoelliptic differential operators on non-compact spaces, showing that index changes under local operator perturbations are local computations. It establishes a complete geometric characterization of Fredholmness via invertibility (and coercivity) at infinity and connects this to unbounded KK-theory through truly unbounded Kasparov modules, providing a flexible framework for index theory beyond operators with compact resolvent. The paper also develops a robust abstract perturbation theory, demonstrates multiple computational approaches to bounded KK-cycles from truly unbounded data, and situates the results within KK-theory via a two-point compactification construction, thereby extending index theory to non-compact and filtered settings with both complex and real $K$-theoretic aspects.

Abstract

We extend the relative index theorem on non-compact manifolds to encompass a wide variety of hypoelliptic differential operators of arbitrary order, demonstrating that the change in index when changing a differential operator locally can be calculated locally. We also show that the notion of invertibility at infinity (and coercive at infinity) is not only sufficient condition for an operator to be Fredholm but also necessary, resulting in a general geometric characterization of Fredholmness. This characterization connects to a model for unbounded \(KK\)-theory which assumes the operator to be Fredholm instead of having (locally) compact resolvent, and thus provides a convenient tool for index theory on non-compact spaces.

The relative index theorem and a characterization of Fredholm operators

TL;DR

The work generalizes the relative index theorem to a broad class of hypoelliptic differential operators on non-compact spaces, showing that index changes under local operator perturbations are local computations. It establishes a complete geometric characterization of Fredholmness via invertibility (and coercivity) at infinity and connects this to unbounded KK-theory through truly unbounded Kasparov modules, providing a flexible framework for index theory beyond operators with compact resolvent. The paper also develops a robust abstract perturbation theory, demonstrates multiple computational approaches to bounded KK-cycles from truly unbounded data, and situates the results within KK-theory via a two-point compactification construction, thereby extending index theory to non-compact and filtered settings with both complex and real -theoretic aspects.

Abstract

We extend the relative index theorem on non-compact manifolds to encompass a wide variety of hypoelliptic differential operators of arbitrary order, demonstrating that the change in index when changing a differential operator locally can be calculated locally. We also show that the notion of invertibility at infinity (and coercive at infinity) is not only sufficient condition for an operator to be Fredholm but also necessary, resulting in a general geometric characterization of Fredholmness. This characterization connects to a model for unbounded -theory which assumes the operator to be Fredholm instead of having (locally) compact resolvent, and thus provides a convenient tool for index theory on non-compact spaces.

Paper Structure

This paper contains 17 sections, 27 theorems, 103 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Examples
  3. The classical relative index theorem
  4. The splitting theorem
  5. Extending index theorems to non-compact manifolds
  6. Isolating non-elliptic points
  7. Riemann surfaces and the Riemann-Roch theorem
  8. The relative index theorem
  9. Preliminaries for Hilbert modules and Fredholm operators
  10. Relatively compact perturbations
  11. Bunke's unitary
  12. A characterization of Fredholmness
  13. Persson's lemma
  14. Unbounded Kasparov modules without locally compact resolvent
  15. Truly unbounded Kasparov modules
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $D_{11}, D_{22}, D_{12}, D_{21}$ be regular closed realizations of differential operators with coefficients in a unital $C^*$-algebra $B$ as in the previous paragraph. For $i,j=1,2$ assume that Then $D_{11}$ and $D_{22}$ are $B$-Fredholm if and only if $D_{12}$ and $D_{21}$ are, and if so in $K_0(B)$.

Figures (2)

  • Figure 1: A schematic illustration of the manifolds $M_{11}$ and $M_{22}$ that coincide on an open set $U$.
  • Figure 2: A schematic illustration of the manifolds $M_{11}, M_{22}, M_{12}$ and $M_{21}$ in the geometric setup for the relative index theorem.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • ...and 51 more