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Equivariant Poincaré-Hopf theorem

Hongzhi Liu, Hang Wang, Zijing Wang, Shaocong Xiang

Abstract

In this paper, we employ the framework of localization algebras to compute the equivariant K-homology class of the Euler characteristic operator, a central object in studying equivariant index theory on manifolds. This approach provides a powerful algebraic language for analyzing differential operators on equivariant structures and allows for the application of Witten deformation techniques in a K-homological context. Utilizing these results, we establish an equivariant version of the Poincaré-Hopf theorem, extending classical topological insights to the equivariant case, inspired by the results of Lück-Rosenberg. This work thus offers a new perspective on the localization techniques in the equivariant K-homology, highlighting their utility in deriving explicit formulas for index-theoretic invariants.

Equivariant Poincaré-Hopf theorem

Abstract

In this paper, we employ the framework of localization algebras to compute the equivariant K-homology class of the Euler characteristic operator, a central object in studying equivariant index theory on manifolds. This approach provides a powerful algebraic language for analyzing differential operators on equivariant structures and allows for the application of Witten deformation techniques in a K-homological context. Utilizing these results, we establish an equivariant version of the Poincaré-Hopf theorem, extending classical topological insights to the equivariant case, inspired by the results of Lück-Rosenberg. This work thus offers a new perspective on the localization techniques in the equivariant K-homology, highlighting their utility in deriving explicit formulas for index-theoretic invariants.

Paper Structure

This paper contains 10 sections, 9 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Equivariant Roe algebras and Equivariant localized Roe algebras
  4. Local index
  5. Equivariant Euler Operator And Localization Trick
  6. Localization trick
  7. Local information
  8. Higher index map and Poincaré-Hopf Theorem on K-homology level
  9. Non-equivariant case
  10. equivariant case

Key Result

Theorem 1.1

The K-homology class $[d+d^*]$ is in the image of the following maps: We have:

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 10 more