Calculating the Euler Characteristic of the Moduli Space of Curves

Abstract

The orbifold Euler characteristic of the moduli space \({{\cal M}_{g;1}}\) of genus g smooth curves with one marked point (g ≥ 1) was calculated by Harer and Zagier: \(\chi ({{\cal M}_{g;1}}) = \zeta (1 - 2g) = - {B_{2g}}/(2g)\), where ζ is the Riemann zeta function and B2g is the (2g)th Bernoulli number. We give a shorter proof of this result using only formal power series and classical combinatorics.