Models for the moduli spaces of Riemann surfaces

We consider Riemann surfaces F the boundary of which is partitioned into m outgoing and n incoming boundary circles; we assume that mn > 1. For the moduli space Mg(mn) of all such F of genus g ≥ 0 a configuration space model Radg(mn) is developed: it consists of configurations of 2g + m pairs of radial slits on n annuli satisfying a certain combinatorial condition to guarantee exactly m outgoing circles. The space Radg(mn) is homotopy-equivalent to Mg(mn).

The basic tool is to describe the conformal structure of F by the gradient flow of a uniquely determined harmonic function with constant boundary values. The critical flow lines dissect F into annuli with slits.

The space Radg(mn) is a non-compact manifold. Its closure provides a finite cell complex, which can be used for homological calculations. Furthermore, the family of spaces Radg(mn) form an operad, and we discuss various other spaces connected to this structure.

Carl-Friedrich Bödigheimer