Title:  Homotopy groups of character varieties

Abstract:  Given a discrete group \Gamma and a (complex reductive or compact) Lie group G, the character variety X_r (G) is the quotient for the conjugation action of G on Hom(\Gamma, G). When G is complex reductive, this quotient should be interpreted in the sense of Geometric Invariant Theory. When G = GL(n) or SL(n), the subspace of irreducible representation coincides with the smooth locus of X_r (G). The rational homology of these spaces has been studied in various cases by a number of authors, and when G = U(n) or SU(n), the homotopy type of the stable moduli spaces X_r (U) and X_r (SU) are explicitly known. In this talk I'll discuss recent progress on understanding low-dimensional homotopy (and integral homology) of character varieties and of their subspaces of irreducible representations. This is joint work with Indranil Biswas, Carlos Florentino, and Sean Lawton.