The asphericity of locally finite infinite configuration spaces and Weierstrass entire coverings
Fuente:
arXiv
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866911338718035968 |
|---|---|
| author | Teh, Jyh-Haur |
| author_facet | Teh, Jyh-Haur |
| contents | Let $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ denote the locally finite infinite ordered and unordered configuration spaces of the complex plane. We prove that both $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ are aspherical. We further obtain a locally finite analogue of the braid exact sequence, \[ 1\longrightarrow H^{lf}(\infty)\longrightarrow B^{lf}(\infty)\longrightarrow \Aut(\N)\longrightarrow 1, \] where $H^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C))$ and $B^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C)//\Aut(\N))$, the fundamental group of the homotopy quotient of $Conf^{lf}_{\infty}(\C)$ by $\Aut(\N)$. Building on this, we classify connected countably infinite--sheeted covering spaces and give a criterion for when such a covering can be realized from the zero set of a family of entire functions $F:X\times\C\to\C$. In particular, if $π_1(X)$ is free and $H^2(X;\Z)=0$, then every countably infinite--sheeted covering space over $X$ is realizable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21498 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The asphericity of locally finite infinite configuration spaces and Weierstrass entire coverings Teh, Jyh-Haur Algebraic Topology Complex Variables Group Theory Geometric Topology Let $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ denote the locally finite infinite ordered and unordered configuration spaces of the complex plane. We prove that both $Conf^{lf}_{\infty}(\C)$ and $C^{lf}_{\infty}(\C)$ are aspherical. We further obtain a locally finite analogue of the braid exact sequence, \[ 1\longrightarrow H^{lf}(\infty)\longrightarrow B^{lf}(\infty)\longrightarrow \Aut(\N)\longrightarrow 1, \] where $H^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C))$ and $B^{lf}(\infty)=π_1(Conf^{lf}_{\infty}(\C)//\Aut(\N))$, the fundamental group of the homotopy quotient of $Conf^{lf}_{\infty}(\C)$ by $\Aut(\N)$. Building on this, we classify connected countably infinite--sheeted covering spaces and give a criterion for when such a covering can be realized from the zero set of a family of entire functions $F:X\times\C\to\C$. In particular, if $π_1(X)$ is free and $H^2(X;\Z)=0$, then every countably infinite--sheeted covering space over $X$ is realizable. |
| title | The asphericity of locally finite infinite configuration spaces and Weierstrass entire coverings |
| topic | Algebraic Topology Complex Variables Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2512.21498 |