## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday. Scott Morrison and David Pennys, eds. Proceedings of the Centre for Mathematics and its Applications, v. 46. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 2017), 389 - 398

### Mednykh's Formula via Lattice Topological Quantum Field Theories

#### Abstract

Mednykh [Me78] proved that for any finite group G and any orientable surface S, there is a formula for $# \mathrm{Hom}(\pi_1(S),G)$ 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 [FS06]. Both of these proofs use character theory and an explicit presentation for $\pi_1$. These results have been reproven using quantum field theory ([FQ93], [MY05], and others). 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.

#### Article information

**Dates**

First available in Project Euclid:
21 February 2017

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1487646034

**Mathematical Reviews number (MathSciNet)**

MR3635678

**Zentralblatt MATH identifier**

06990161

#### Citation

Snyder, Noah. Mednykh's Formula via Lattice Topological Quantum Field Theories. Proceedings of the 2014 Maui and 2015 Qinhuangdao Conferences in Honour of Vaughan F.R. Jones’ 60th Birthday, 389--398, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 2017. https://projecteuclid.org/euclid.pcma/1487646034