Hyperelliptic four-manifolds defined by vector-colorings of simple polytopes
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866918357637267456 |
|---|---|
| author | Erokhovets, Nikolai |
| author_facet | Erokhovets, Nikolai |
| contents | Toric topology assigns to each simple convex $n$-polytope $P$ with $m$ facets an $n$-dimensional real moment angle manifold $\mathbb RZ_P$ with a canonical action of $\mathbb Z_2^m=(\mathbb Z/2\mathbb Z)^m$. We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on $\mathbb RZ_P$. The orbit space $N(P,H)=\mathbb RZ_P/H$ has an action of $\mathbb Z_2^m/H$. For general $n$ we introduce the notion of a Hamiltonian $C(n,k)$-subcomplex in the boundary of an $n$-polytope $P$ generalizing the notions of a Hamiltonian cycle ($k=2$), Hamiltonian theta-subgraph ($k=3$) and Hamiltonian $K_4$-subgraph ($k=4)$ in the $1$-skeleton of a $3$-polytope. Each $C(n,k)$-subcomplex $C\subset \partial P$ corresponds to a subgroup $H_C\subset\mathbb Z_2^m$ such that $N(P,H_C)\simeq S^n$. We prove that in dimensions $n\leqslant 4$ this correspondence is a bijection. Any subgroup $H\subset \mathbb Z_2^m$ defines a complex $C(P,H)\subset \partial P$. We prove that each Hamiltonian $C(n,k)$-subcomplex $C\subset C(P,H)$ inducing $H$ corresponds to a hyperelliptic involution $τ_C\in\mathbb Z_2^m/H$ on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to $S^n$) and in dimensions $n\leqslant 4$ this correspondence is a bijection. We prove that for the geometries $\mathbb X= \mathbb S^4$, $\mathbb S^3\times\mathbb R$, $\mathbb S^2\times \mathbb S^2$, $\mathbb S^2\times \mathbb R^2$, $\mathbb S^2\times \mathbb L^2$, and $\mathbb L^2\times \mathbb L^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of $H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in $\mathbb Z_2^m/H$, and for $\mathbb X=\mathbb R^4$, $\mathbb L^4$, $\mathbb L^3\times \mathbb R$ and $\mathbb L^2\times \mathbb R^2$ there are no such polytopes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20575 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hyperelliptic four-manifolds defined by vector-colorings of simple polytopes Erokhovets, Nikolai Algebraic Topology Combinatorics Geometric Topology 57S12, 57S17, 57S25, 52B05, 52B70, 57R19, 57R18, 57R91 Toric topology assigns to each simple convex $n$-polytope $P$ with $m$ facets an $n$-dimensional real moment angle manifold $\mathbb RZ_P$ with a canonical action of $\mathbb Z_2^m=(\mathbb Z/2\mathbb Z)^m$. We consider (non-necessarily free) actions of subgroups $H\subset \mathbb Z_2^m$ on $\mathbb RZ_P$. The orbit space $N(P,H)=\mathbb RZ_P/H$ has an action of $\mathbb Z_2^m/H$. For general $n$ we introduce the notion of a Hamiltonian $C(n,k)$-subcomplex in the boundary of an $n$-polytope $P$ generalizing the notions of a Hamiltonian cycle ($k=2$), Hamiltonian theta-subgraph ($k=3$) and Hamiltonian $K_4$-subgraph ($k=4)$ in the $1$-skeleton of a $3$-polytope. Each $C(n,k)$-subcomplex $C\subset \partial P$ corresponds to a subgroup $H_C\subset\mathbb Z_2^m$ such that $N(P,H_C)\simeq S^n$. We prove that in dimensions $n\leqslant 4$ this correspondence is a bijection. Any subgroup $H\subset \mathbb Z_2^m$ defines a complex $C(P,H)\subset \partial P$. We prove that each Hamiltonian $C(n,k)$-subcomplex $C\subset C(P,H)$ inducing $H$ corresponds to a hyperelliptic involution $τ_C\in\mathbb Z_2^m/H$ on the manifold $N(P,H)$ (that is, an involution with the orbit space homeomorphic to $S^n$) and in dimensions $n\leqslant 4$ this correspondence is a bijection. We prove that for the geometries $\mathbb X= \mathbb S^4$, $\mathbb S^3\times\mathbb R$, $\mathbb S^2\times \mathbb S^2$, $\mathbb S^2\times \mathbb R^2$, $\mathbb S^2\times \mathbb L^2$, and $\mathbb L^2\times \mathbb L^2$ there exists a compact right-angled $4$-polytope $P$ with a free action of $H$ such that the geometric manifold $N(P,H)$ has a hyperelliptic involution in $\mathbb Z_2^m/H$, and for $\mathbb X=\mathbb R^4$, $\mathbb L^4$, $\mathbb L^3\times \mathbb R$ and $\mathbb L^2\times \mathbb R^2$ there are no such polytopes. |
| title | Hyperelliptic four-manifolds defined by vector-colorings of simple polytopes |
| topic | Algebraic Topology Combinatorics Geometric Topology 57S12, 57S17, 57S25, 52B05, 52B70, 57R19, 57R18, 57R91 |
| url | https://arxiv.org/abs/2407.20575 |