Note on topologically singular points in the moduli space of Riemann surfaces of genus 2

Abstract

Let \(\mathcal {M}_{g}\) be the moduli space of Riemann surfaces of genus g. Rauch (Bull Am Math Soc 68:390–394, 1962) focused his attention on and determined the so-called topological singular points of \(\mathcal {M}_{g}\): these are the points of \(\mathcal {M}_{g}\) whose neighbourhoods are not homeomorphic to a ball. In a previous paper, the authors produced a topological proof for Rauch’s result for genera \(>\,2\); however, the methods used there do not apply to the genus 2 case. The only known proof for the remaining and important case, i.e., the case of singular points in \(\mathcal {M}_{2}\), is to be found in an article by Igusa (Ann Math 72(3):612–649, 1960) and it lays on methods from algebraic geometry. Here, we present a topological proof for this case too.