Moduli spaces of branched coverings of the plane

Abstract

In this thesis I build a strong connection between Hurwitz spaces of branched coverings of the complex plane, and moduli spaces M_{g,n} of Riemann surfaces of genus g ≥ 0 with n ≥ 1 boundary components.<br /> I refine a construction by Bödigheimer, which gives a combinatorial model for M_{g,n} based on slit pictures on the complex plane. For an integer d ≥ n and a splitting d = d_1 +...+ d_n with d_i ≥ 1, I consider a certain moduli space O_{g,n}[d_1 ,... , d_n] of branched coverings of Riemann surfaces Σ_{g,n} to CP_1 of degree d, with some prescribed behaviour near the n marked points of Σ_{g,n}.<br /> The main results of the thesis are the following. <ul> <li>The spaceO_{g,n}[d_1 ,... , d_n] is a complex manifold of complex dimension d + 2g + n − 2 and has a combinatorial cell structure analogue to Bödigheimer’s model.</li> <li>For d ≥ 2g + n − 1 the space O_{g,n}[d_1 ,... , d_n] is homotopy equivalent to M_{g,n}.</li> <li>The space O_{g,n}[d_1 ,... , d_n] has a natural filtration whose strata are the classical Hurwitz spaces of d-fold coverings of C branched over k points, where k depends on the stratum. Hence the construction of O_{g,n}[d_1 ,... , d_n] creates a bridge between the theory of configuration spaces and braids on one side, and the theory of moduli spaces of Riemann surfaces on the other side.</li> <li>The cellular chain complex of O_{g,n}[d_1 ,... , d_n] can be simplified using a technique due to Balázs Visy, thus obtaining a direct summand of the reduced cobar complex of V(d). The latter is a certain bialgebra in the category of S_d -Yetter-Drinfeld modules: this roughly means that V(d) is a graded and S_d -graded abelian group with an action of S_d , a unit, a counit, a multiplication and a comultiplication, and all these structures are interrelated. Therefore also the homology of O_{g,n}[d_1 ,... , d_n] is a direct summand of the cohomology of the reduced cobar complex of V(d). </ul>