Schur-type invariants of branched G-covers of surfaces

TL;DR

This paper characterizes when two branched G-covers of surfaces are equivalent using homological invariants called branched Schur invariants, with implications for topological quantum field theories and topological phases.

Contribution

It introduces branched Schur invariants as a new tool to classify branched G-covers of surfaces in a stable range, extending previous invariants to include branching data.

Findings

Provides a classification criterion for branched G-covers using homology classes.

Defines branched Schur invariants taking values in a torsor over a quotient of the Schur multiplier.

Discusses applications to G-equivariant TQFT and topological phases.

Abstract

Fix a finite group $G$ and a conjugacy invariant subset $C\subseteq G$. Let $\Sigma$ be an oriented surface, possibly with punctures. We consider the question of when two homomorphisms $\pi_1(\Sigma) \to G$ taking punctures into $C$ are equivalent up to an orientation preserving diffeomorphism of $\Sigma$. We provide an answer to this question in a stable range, meaning that $\Sigma$ has enough genus and enough punctures of every conjugacy type in $C$. If $C$ generates $G$, then we can assume $\Sigma$ has genus 0 (or any other constant). The main tool is a classifying space for (framed) $C$-branched $G$-covers, and related homology classes we call branched Schur invariants, since they take values in a torsor over a quotient of the Schur multiplier $H_2(G)$. We conclude with a brief discussion of applications to $(2+1)$-dimensional $G$-equivariant TQFT and symmetry-enriched topological phases.