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A Lefschetz fixed-point formula for noncompact fixed-point sets

Peter Hochs

TL;DR

The paper develops a Lefschetz-type equivariant index theorem for isometries on complete manifolds with possibly noncompact fixed-point sets by constructing a localised trace on a new algebra of asymptotically local operator paths. It integrates Roe-style noncompact index techniques with a localisation algebra, yielding a formula that equates a numerically defined localisation index to the Atiyah–Segal–Singer integrand on the fixed-point set, in a setting with compact symmetry groups. The approach unifies prior Roe and fixed-point results, while providing geometric obstructions to positive scalar curvature in Spin manifolds via asymptotic trace limits. The results generalise several known indices and offer a robust framework for equivariant indices in noncompact settings with controlled geometry.

Abstract

We obtain an equivariant index theorem, or Lefschetz fixed-point formula, for isometries from complete Riemannian manifolds to themselves. The fixed-point set of such an isometry may be noncompact. We build on techniques developed by Roe. Key new ingredients are a localised functional on operators with bounded smooth kernels, and an algebra (reminiscent of Yu's localisation algebra) of `asymptotically local' operators, on which this functional has an asymptotic trace property. As consequences, we show that some earlier indices used are special cases of the one we introduce here, and obtain an obstruction to positive scalar curvature.

A Lefschetz fixed-point formula for noncompact fixed-point sets

TL;DR

The paper develops a Lefschetz-type equivariant index theorem for isometries on complete manifolds with possibly noncompact fixed-point sets by constructing a localised trace on a new algebra of asymptotically local operator paths. It integrates Roe-style noncompact index techniques with a localisation algebra, yielding a formula that equates a numerically defined localisation index to the Atiyah–Segal–Singer integrand on the fixed-point set, in a setting with compact symmetry groups. The approach unifies prior Roe and fixed-point results, while providing geometric obstructions to positive scalar curvature in Spin manifolds via asymptotic trace limits. The results generalise several known indices and offer a robust framework for equivariant indices in noncompact settings with controlled geometry.

Abstract

We obtain an equivariant index theorem, or Lefschetz fixed-point formula, for isometries from complete Riemannian manifolds to themselves. The fixed-point set of such an isometry may be noncompact. We build on techniques developed by Roe. Key new ingredients are a localised functional on operators with bounded smooth kernels, and an algebra (reminiscent of Yu's localisation algebra) of `asymptotically local' operators, on which this functional has an asymptotic trace property. As consequences, we show that some earlier indices used are special cases of the one we introduce here, and obtain an obstruction to positive scalar curvature.
Paper Structure (20 sections, 33 theorems, 110 equations)

This paper contains 20 sections, 33 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Dirac operators
  4. Exhaustions
  5. A functional
  6. Asymptotically local operators
  7. Uniform operators
  8. Asymptotically local operators
  9. A subalgebra
  10. A $K$-theoretic index
  11. Functional calculus
  12. Heat operators
  13. The index
  14. An asymptotic trace property
  15. Estimates of partial integrals
  16. ...and 5 more sections

Key Result

Lemma 2.4

There exists a functional associated to $(U_j)_{j=1}^{\infty}$.

Theorems & Definitions (82)

  • Definition 2.1
  • Example 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 3.1
  • Lemma 3.2
  • ...and 72 more