Euler Characteristic of the Moduli of Riemann Surfaces

Abstract

In this thesis we study the topology of the Deligne-Mumford compactified moduli space Mg;n of n pointed genus g stable Riemann surfaces. We decode the combinatorial information of the space through dual graphs of decorated Riemann surfaces in it. Finally, we extend the work of Harer and Zagier in [HZ86] to obtain the generating series of (and hence calculate) the Euler characteristic of the space in terms of Euler characteristics of Mg;n: