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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.00665 |
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| _version_ | 1866915001829163008 |
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| author | Bosgraaf, Austin Escher, Christine Searle, Catherine |
| author_facet | Bosgraaf, Austin Escher, Christine Searle, Catherine |
| contents | In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to $\mathbb{Z}_2$-tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the $\Z_2$-torus are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the $\mathbb{Z}_2$-torus to approximately $n/6$ and $n/8$ and are able to classify either the integral cohomology ring or the $\mathbb{Z}_2$-cohomology ring, respectively, of the fixed point set of the $\mathbb{Z}_2$-torus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_00665 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Fixed-Point Sets of $\Z_2$-Tori in Positive Curvature Bosgraaf, Austin Escher, Christine Searle, Catherine Differential Geometry Algebraic Topology 53C20, 57S25 In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to $\mathbb{Z}_2$-tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the $\Z_2$-torus are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the $\mathbb{Z}_2$-torus to approximately $n/6$ and $n/8$ and are able to classify either the integral cohomology ring or the $\mathbb{Z}_2$-cohomology ring, respectively, of the fixed point set of the $\mathbb{Z}_2$-torus. |
| title | On Fixed-Point Sets of $\Z_2$-Tori in Positive Curvature |
| topic | Differential Geometry Algebraic Topology 53C20, 57S25 |
| url | https://arxiv.org/abs/2411.00665 |