Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices
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| Format: | Preprint |
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2024
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| _version_ | 1866909967904145408 |
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| author | Choi, Suyoung Jang, Hyeontae Vallée, Mathieu |
| author_facet | Choi, Suyoung Jang, Hyeontae Vallée, Mathieu |
| contents | Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on the real moment-angle complex $\mathbb{R}\mathcal{Z}_K$, where $\mathbb{Z}_2$ is the additive group with two elements. We prove that any subgroup of $\mathbb{Z}_2^m$ acting freely on $\mathbb{R}\mathcal{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $\mathcal{Z}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_15600 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices Choi, Suyoung Jang, Hyeontae Vallée, Mathieu Algebraic Topology Combinatorics 57S12 Let $K$ be an $(n-1)$-dimensional piecewise linear sphere on $[m]$, where $m\leq n+4$. There are a canonical action of $m$-dimensional torus $T^m$ on the moment-angle complex $\mathcal{Z}_K$, and a canonical action of $\mathbb{Z}_2^m$ on the real moment-angle complex $\mathbb{R}\mathcal{Z}_K$, where $\mathbb{Z}_2$ is the additive group with two elements. We prove that any subgroup of $\mathbb{Z}_2^m$ acting freely on $\mathbb{R}\mathcal{Z}_K$ is induced by a subtorus of $T^m$ acting freely on $\mathcal{Z}_K$. The proof primarily utilizes a suitably modified method of toric wedge induction and the combinatorial structure of a specific binary matroid of rank $4$. |
| title | Toric wedge induction and toric lifting property for piecewise linear spheres with a few vertices |
| topic | Algebraic Topology Combinatorics 57S12 |
| url | https://arxiv.org/abs/2404.15600 |