Schur-type invariants of branched G-covers of surfaces

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.

15 pages, 7 figures. The proof of the main theorem has been streamlined by replacing Lemma 3.6 of the previous version with the Hopf-Whitney classification. Accepted for publication in Proceedings of the AMS Special Session on Topological Phases of Matter and Quantum Computation

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