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arXiv · 2410.15103

Equivariant Poincaré-Hopf theorem

Hongzhi Liu·Hang Wang·Zijing Wang·Shaocong Xiang
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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.

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Hongzhi Liu, Hang Wang, Zijing Wang, Shaocong Xiang. 2024-10-19. Equivariant Poincaré-Hopf theorem. https://arxiv.org/abs/2410.15103

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