Schur-type invariants of branched G-covers of surfaces

Fix a finite group $G$ and a conjugacy invariant subset $C\subseteq G$. Let $Σ$ be an oriented surface, possibly with punctures. We consider the question of when two homomorphisms $π_1(Σ) \to G$ taking punctures into $C$ are equivalent up to an orientation preserving diffeomorphism of $Σ$. We provide an answer to this question in a stable range, meaning that $Σ$ has enough genus and enough punctures of every conjugacy type in $C$. If $C$ generates $G$, then we can assume $Σ$ 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.

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