Mednykh's Formula via Lattice Topological Quantum Field Theories

2007/03/05 by Noah Snyder, Snyder, Noah
Computer Science · Mathematics · #20C05 #57R56 #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.math/0703073

openalex publication_date 2007/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Mednykh proved that for any finite group G and any orientable surface S, there is a formula for #Hom(pi₁(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 π₁. 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.

Citations

• A generating function of the number of homomorphisms from a surface group into a finite group

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