Equivariant Poincar\'e-Hopf theorem

TL;DR

This paper develops an algebraic framework using localization algebras to compute equivariant K-homology classes of the Euler characteristic operator, leading to an equivariant Poincaré-Hopf theorem that extends classical topological results.

Contribution

It introduces a novel approach combining localization algebras and Witten deformation techniques to establish an equivariant Poincaré-Hopf theorem in K-homology.

Findings

Computed equivariant K-homology class of Euler characteristic operator

Extended classical Poincaré-Hopf theorem to the equivariant setting

Provided explicit formulas for index-theoretic invariants

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 Poincar\'e-Hopf theorem, extending classical topological insights to the equivariant case, inspired by the results of L\"uck-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.