Configuration Models For Moduli Spaces of Riemann surfaces with boundary

Abstract

In this article we consider Riemann surfacesF of genus g ≥ 0 with n ≥ 1 incoming and m ≥ 1 outgoing boundary circles, where on each incoming circle a point is marked. For the moduli space mg*(m, n) of all suchF of genusg ≥ 0 a configuration space model Radh(m, n) is described: it consists of configurations of h = 2g-2+m+n pairs of radial slits distributed over n annuli; certain combinatorial conditions must be satisfied to guarantee the genusg and exactly m outgoing circles. Our main result is a homeomorphism between Radh(m, n) and Mg*(m,n). The space Radh(m, n) is a non-compact manifold, and the complement of a subcomplex in a finite cell complex. This can be used for homological calculations. Furthermore, the family of spaces Radh(m, n ) form an operad, acting on various spaces connected to conformai field theories.