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

Let Mg 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 ofMg: these are the points ofMg 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 inM2, 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.