Configuration models for moduli spaces of riemann surfaces with boundary

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Abh. Math. Sem. Univ. Hamburg 76 (2006), 191-233 Configuration Models for Moduli Spaces of Riemann Surfaces with Boundary B y C . - F . BODIGHEIMER Abstract. In this article we consider Riemann surfaces F 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 93t~(m, n) of all such F of genus g > 0 a configuration space model 9~aOh(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 genus g and exactly m outgoing circles. Our main result is a homeomorphism between fftnOh(m, n) and ~ ( m , n). The space 9~a~h(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 ~aOh(m, n) form an operad, acting on various spaces connected to conformal field theories. Contents 1. I n t r o d u c t i o n 1.1. Surfaces a n d their M o d u l i 1.2. Results 1.3. O v e r v i e w o f t h e S e c t i o n s 2. T h e M o d u l i S p a c e s 9 ~ (m, n ) 2.1. R i e m a n n Surfaces 2.2. M o d u l i S p a c e s 2.3. Teichmtiller S p a c e s 2.4. M a p p i n g Class G r o u p s 3. Radial Slit C o n f i g u r a t i o n s 3.1. Slit A n n u l i 3.2. E x a m p l e s o f C o n f i g u r a t i o n s 4. T h e Surface A s s o c i a t e d to a C o n f i g u r a t i o n 4.1. ...
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