On the Hilbert uniformization of moduli spaces of flat <em>G</em>-bundles over Riemann surfaces

Abstract

In this thesis, we study the moduli spaces M_(g,q)^m (G)of flat pointed principal G-bundles over Riemann surfaces X. The genus of X is g 0 and G is a fixed Lie group. Further, we are given m 0 permutable marked points in X and a directed base point, that is, a base point Q in X with a tangent vector χ ≠0 in Q. The canonical projection onto the moduli space of Riemann surfaces defines a fiber bundle whose fiber is the representation variety in G. Connected components of M_(g,q)^m (G) are described for several Lie groups G. Homology groups are computed for G=SU(2) and U(1). Some homotopy groups are determined for G=SO(3), SU(2) and U(2). In particular, we analyze moduli spaces of coverings of Riemann surfaces. For ramified and unramified coverings, we combinatorially describe the connected components.<br /> In the second part of this thesis, we construct a cell decomposition for the moduli space of flat G-bundles as an application of a generalized Hilbert uniformization. To this end, we consider the moduli spaces H_(g,q)^m (G) of flat pointed principal G-bundles over Riemann surfaces X of genus g 0 with m 0 permutable punctures (in contrast to marked points) and a directed base point. As a consequence, homology groups can be computed for some examples. Moreover, a stratum of filtered bar complexes of certain finite wreath products of groups can be identified with a disjoint union of moduli spaces. Finally, we investigate stabilization effects of the moduli spaces. First, we consider stabilization maps for g ≫0. Then we compute stable homotopy groups for G=Sp(k), SU(k) and Spin(k) as k ≫ 0.