University of Chicago
Department of Mathematics

Let X be the moduli of SL(n,C), SU(n), GL(n,C), or U(n) valued representations of a rank r free group. We compute the fundamental group of X and show that these four moduli otherwise have identical higher homotopy groups. We then classify the singular stratification of X. This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Lastly, we show that the moduli X are generally not topological manifolds, except for a few examples we explicitly describe.