The Euler characteristic of the moduli space of curves
Let \(\Gamma^ 1_ g\), \(g\geq 1\), be the mapping class group consisting of all isotopy classes of base-point and orientation preserving homeomorphisms of a closed, oriented surface of genus \(g.\) Main theorem: \(\chi(\Gamma^ 1_ g)=\zeta (1-2g),\) where \(\zeta(s)\) is the Riemann zeta function, and \(\chi(\Gamma^ 1_ g)=[\Gamma^ 1_ g:\Gamma]^{- 1}\chi (E/\Gamma)\) for any torsion free subgroup \(\Gamma\) of finite index in \(\Gamma^ 1_ g\). E is a contractible space on which \(\Gamma\) acts freely and properly discontinuously. \(\chi(E/\Gamma)\) is the usual Euler characteristic. For every positive integer n, consider a fixed 2n-gon with sides \(S_ 1,...,S_{2n}\). Then denote by \(\epsilon_ g(n)\) the number of ways of grouping \(S_ 1,...,S_{2n}\) into n pairs making a surface of genus \(g\) under suitable identification of sides and denote by \(\lambda_ g(n)\) the number of such groupings which do not contain the special two types of configurations. The authors prove a formula for \(\chi(\Gamma^ 1_ g)\) expressed by \(\lambda_ g(n)\) in theorem 1 and a formula for \(\epsilon_ g(n)\) as (essentially) the coefficient of \(x^{2g}\) in \((x/\tanh (x))^{n+1}\) in theorem 2 and combine them to deduce the main theorem aforementioned. The authors write that the proof of theorem 2 is rather indirect, but the idea of their 'indirect' proof is very interesting: the main point is the integral formula for \(C(n,k)=\sum_{0\leq g\leq n}\epsilon_ g(n)k^{n+1-2g}\) given in {\S} 4. - In {\S} 6, the Euler characteristic of \(\Gamma^ 1_ g\) is given (theorems 4 and 4').
scientific article; zbMATH DE number 3997972