On the Lego-Teichmuller game for finite $G$ cover

TL;DR

This paper introduces a combinatorial framework for constructing and analyzing G-covers of surfaces, establishing a connected and simply-connected complex structure to facilitate the development of G-equivariant Modular Functors.

Contribution

It defines moves and relations on G-covers of surfaces, creating a foundational structure for future G-equivariant Modular Functor construction.

Findings

Defined moves and relations for G-covers

Proved the complex is connected and simply-connected

Lays groundwork for G-equivariant Modular Functors

Abstract

Given a smooth, oriented, closed surface $\Sigma$ of genus zero, possibly with boundary, let $\tilde{\Sigma} \longrightarrow \Sigma$ be a given $G$-cover of $\Sigma$, where $G$ is a given finite group. Let $S_{n}$ denote the standard sphere with $n$ holes. There are many ways of gluing together several $G$-cover of $S_{n}$ to construct the $G$-cover $\ts \longrightarrow \Sigma$, of $\Sigma$. We let $M(\tilde{\Sigma} ,\Sigma)$ be the set of all ways to construct the given $G$-cover, $\tilde{\Sigma} \longrightarrow \Sigma$, of $\Sigma$ from gluing of several $G$-covers of $S_{n}$, here $n$ may vary. In this paper, we define some simple moves and relation which will turn $M(\tilde{\Sigma} ,\Sigma)$ into a connected and simply-connected complex. This will be used in the future paper to construct $G$-equivariant Modular Functor. This $G$-equivariant Modular Functor will be an extension of the usual Modular Functor.