Fundamental Group of Moduli Spaces of Representations

TL;DR

This paper computes the fundamental group of moduli spaces of surface group representations into various Lie groups, expanding understanding of their topological structure in different cases.

Contribution

It provides explicit calculations of the fundamental group of X(G) for multiple classes of Lie groups and surface configurations, filling gaps in the topological understanding.

Findings

Fundamental group of X(G) for n>0 and G reductive or compact.

Fundamental group of X(G) for n=0 and G=GL(m,C), SL(m,C), U(m), SU(m).

Topological properties of moduli spaces of representations.

Abstract

Let S be a surface of genus g with n points removed, G a connected Lie group, and X(G) the moduli space of representations of the fundamental group of S into G. We compute the fundamental group of X(G) when n>0 and G is a real or complex reductive algebraic group, or a compact Lie group; and when n=0 and G=GL(m,C), SL(m,C), U(m), or SU(m).