Equivariant Localization of $K$-homological Euler Class for almost connected Lie Groups
Operator Algebras
2025-08-22 v2 Functional Analysis
K-Theory and Homology
Abstract
Using the Witten deformation and localization algebra techniques, we compute the -equivariant -homology class of the de Rham operator on a proper cocompact -spin manifold, where is an almost connected Lie group. By applying a -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.
Keywords
Cite
@article{arxiv.2507.21415,
title = {Equivariant Localization of $K$-homological Euler Class for almost connected Lie Groups},
author = {Hongzhi Liu and Hang Wang and Zijing Wang and Shaocong Xiang},
journal= {arXiv preprint arXiv:2507.21415},
year = {2025}
}