Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds
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arXiv
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| Format: | Preprint |
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2024
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| author | Erokhovets, Nikolai |
| author_facet | Erokhovets, Nikolai |
| contents | We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on the real moment-angle manifold $\mathbb R\mathcal{Z}_P$ corresponding to a simple convex $n$ polytope $P$ with $m$ facets. The criterion when the orbit space $\mathbb R\mathcal{Z}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb R\mathcal{Z}_P/H$ homeomorphic to $S^n$ and series of manifolds $M^n=\mathbb R\mathcal{Z}_P/H$ admitting a hyperelliptic involution $τ\in\mathbb Z_2^m/H$, that is an involution $τ$ such that $M^n/\langleτ\rangle$ is homeomorphic to $S^n$. For any simple $3$-polytope $P$ we classify all subgroups $H\subset\mathbb Z_2^m$ such that $\mathbb R\mathcal{Z}_P/H$ is homeomorphic to $S^3$. For any simple $3$-polytope $P$ and any subgroup $H\subset\mathbb Z_2^m$ we classify all hyperelliptic involutions $τ\in\mathbb Z_2^m/H$ acting on $\mathbb R\mathcal{Z}_P/H$. As a corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in $\mathbb Z_2^3$ if and only if it is a rational homology $3$-sphere and if and only if it correspond to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00492 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds Erokhovets, Nikolai Algebraic Topology Combinatorics Geometric Topology 57S12, 57S17, 57S25, 52B05, 52B10, 52B70, 57R19, 57R18, 57R91 We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on the real moment-angle manifold $\mathbb R\mathcal{Z}_P$ corresponding to a simple convex $n$ polytope $P$ with $m$ facets. The criterion when the orbit space $\mathbb R\mathcal{Z}_P/H$ is a topological manifold (perhaps with a boundary) can be extracted from results by M.A. Mikhailova and C. Lange. For any dimension $n$ we construct series of manifolds $\mathbb R\mathcal{Z}_P/H$ homeomorphic to $S^n$ and series of manifolds $M^n=\mathbb R\mathcal{Z}_P/H$ admitting a hyperelliptic involution $τ\in\mathbb Z_2^m/H$, that is an involution $τ$ such that $M^n/\langleτ\rangle$ is homeomorphic to $S^n$. For any simple $3$-polytope $P$ we classify all subgroups $H\subset\mathbb Z_2^m$ such that $\mathbb R\mathcal{Z}_P/H$ is homeomorphic to $S^3$. For any simple $3$-polytope $P$ and any subgroup $H\subset\mathbb Z_2^m$ we classify all hyperelliptic involutions $τ\in\mathbb Z_2^m/H$ acting on $\mathbb R\mathcal{Z}_P/H$. As a corollary we obtain that a $3$-dimensional small cover has $3$ hyperelliptic involutions in $\mathbb Z_2^3$ if and only if it is a rational homology $3$-sphere and if and only if it correspond to a triple of Hamiltonian cycles such that each edge of the polytope belongs to exactly two of them. |
| title | Manifolds realized as orbit spaces of non-free $\mathbb Z_2^k$-actions on real moment-angle manifolds |
| topic | Algebraic Topology Combinatorics Geometric Topology 57S12, 57S17, 57S25, 52B05, 52B10, 52B70, 57R19, 57R18, 57R91 |
| url | https://arxiv.org/abs/2403.00492 |