J. DIFFERENTIAL GEOMETRY 27 (1988) 35 53 PERTURBATIVE SEMES AND THE MODULI SPACE OF RIEMANN SURFACES R. C. PENNER Introduction In this paper, we use some techniques from quantum field theory to compute quantities related to symmetry groups of pairs (F,G), where F is a surface and G is a spine of F. As a consequence of our computation, we derive a certain cohomological invariant of the mapping class group MC* of homotopy classes of orientation preserving homeomorphisms of the genus g surface with s points removed. This invariant, the virtual Euler characteristic χ^MC^ of MC*, is defined as follows. If A is a group which contains a torsion free subgroup B of finite index and every such subgroup has finitely generated integral homology, then we define Xv A=χB/[A:B], where xB denotes the usual Euler characteristic of B. (It is an exercise to check that χ υ A is independent of the choice of B.) We will find that *!(2g)! for s ^ 1, g > 0, and 2g 2 +s > 0, where B 2g denotes the 2gth Bernoulli number. This result is proved by Harer and Zagier [5] using computational techniques (rediscovered in [5]) which are related to the perturbative series of quantum field theory (see [2] and §2 below). Herein, we apply a variant of the full perturbative series machine to capture the equivariant combinatorics of certain cell decompositions of foliated fiber bundles over various Teichmuller spaces. In contrast, [5] begins with cell decompositions of the Teichmuller spaces themselves, and only part of the perturbative series machinery is applicable in Received January 31, 1986 and, in revised form, December 15, 1986. The author was partially supported by the National Science Foundation.