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#### On the Hilbert uniformization of moduli spaces of flat G-bundles over Riemann surfaces

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https://hdl.handle.net/20.500.11811/6011

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##### Citation

Stein, L. (2013). On the Hilbert uniformization of moduli spaces of flat G-bundles over Riemann surfaces. PhD Thesis, Rheinischen Friedrich-Wilhelms-Universität Bonn, Bonn.

Cite as: https://hdl.handle.net/21.11116/0000-0004-1CA1-7

##### 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.

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.

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.