Equivariant Localization of $K$-homological Euler Class for almost connected Lie Groups

TL;DR

This paper develops a method to compute the equivariant $K$-homology class of the de Rham operator on certain Lie group manifolds, using localization techniques to facilitate index calculations.

Contribution

It introduces a localization approach for the equivariant $K$-homology class via Morse-Bott perturbation, providing explicit formulas and tools for index theory in this setting.

Findings

Explicit computation of the $K$-homology class using localization.

Derivation of an equivariant Poincaré-Hopf formula.

Development of tools for equivariant index calculations.

Abstract

Using the Witten deformation and localization algebra techniques, we compute the $G$-equivariant $K$-homology class of the de Rham operator on a proper cocompact $G$-spin manifold, where $G$ is an almost connected Lie group. By applying a $G$-invariant Morse-Bott perturbation, this class is localized near the zero set of the perturbation and can be identified explicitly with an element in the representation rings associated to some isotropy subgroups. The result yields an equivariant Poincar\'e-Hopf formula and supplies concise tools for equivariant index computations.