Mednykh's Formula via Lattice Topological Quantum Field Theories

TL;DR

This paper presents a simplified proof of Mednykh's formula relating the number of homomorphisms from surface fundamental groups to finite groups, using elementary topology and lattice topological quantum field theories instead of character theory.

Contribution

It introduces a new elementary approach to Mednykh's formula using lattice topological quantum field theories, avoiding complex character theory and explicit presentations.

Findings

Simplified proof of Mednykh's formula using elementary topology.

Extension of the formula to nonorientable surfaces.

Demonstration of lattice TQFT as a unifying tool.

Abstract

Mednykh proved that for any finite group G and any orientable surface S, there is a formula for #Hom(pi_1(S), G) in terms of the Euler characteristic of S and the dimensions of the irreducible representations of G. A similar formula in the nonorientable case was proved by Frobenius and Schur. Both of these proofs use character theory and an explicit presentation for \pi_1. These results have been reproven using quantum field theory. Here we present a greatly simplified proof of these results which uses only elementary topology and combinatorics. The main tool is an elementary invariant of surfaces attached to a semisimple algebra called a lattice topological quantum field theory.