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<p>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 <i>K</i>-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 and Rosenberg (2003). This work thus offers a new perspective on the localization techniques in the equivariant <i>K</i>-homology, highlighting their utility in deriving explicit formulas for index-theoretic invariants.</p>

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The equivariant Poincaré-Hopf theorem

  • 期刊论文

  • 发表日期：

    2026年6月23日

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  • Hongzhi Liu,
  • Hang Wang,
  • Zijing Wang,
  • Shaocong Xiang

摘要

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 and Rosenberg (2003). 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.