Mednykh's Formula via Lattice Topological Quantum Field Theories
arXiv:math/0703073
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 π_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.
11 pages, 6 figures. v3 clarifies the dicussion of the Frobenius-Schur indicators and improves the argument in the quaternionic case
References in corpus (7)
- Chern-Simons Theory with Finite Gauge Group
- Categorical Aspects of Topological Quantum Field Theories
- State sum construction of two-dimensional open-closed Topological Quantum Field Theories
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- Dijkgraaf-Witten invariants of surfaces and projective representations of groups
- Lattice Topological Field Theory on Non-Orientable Surfaces
- A generating function of the number of homomorphisms from a surface group into a finite group