Shaocong Xiang | Searcharxiv

arXiv2025

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

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.

math.OA↗

arXiv2024

Equivariant Poincar\'e-Hopf theorem

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.

math.AT↗