Equivariant Poincaré-Hopf theorem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909356561268736 |
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| author | Liu, Hongzhi Wang, Hang Wang, Zijing Xiang, Shaocong |
| author_facet | Liu, Hongzhi Wang, Hang Wang, Zijing Xiang, Shaocong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_15103 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Poincaré-Hopf theorem Liu, Hongzhi Wang, Hang Wang, Zijing Xiang, Shaocong Algebraic Topology Differential Geometry Operator Algebras 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. |
| title | Equivariant Poincaré-Hopf theorem |
| topic | Algebraic Topology Differential Geometry Operator Algebras |
| url | https://arxiv.org/abs/2410.15103 |