Homotopy Groups of Free Group Character Varieties

Abstract

Let G be a connected, complex reductive Lie group with maximal compact subgroup K, and let X denote the moduli space of G- or K-valued representations of a rank r free group. In this article, we develop methods for studying the low-dimensional homotopy groups of these spaces and of their subspaces Y of irreducible representations. Our main result is that when G = GL(n,C) or SL(n,C), the second homotopy group of X is trivial. The proof depends on a new general position-type result in a singular setting. This result is proven in the Appendix and may be of independent interest. We also obtain new information regarding the homotopy groups of the subspaces Y. Recent work of Biswas and Lawton determined the fundamental group of X for general G, and we describe the fundamental group of Y. Specializing to the case G = GL(n,C), we explicitly compute the homotopy groups of the smooth locus of X in a large range of dimensions, finding that they exhibit Bott Periodicity. As a further application of our methods (and in particular our general position result) we obtain new results regarding centralizers of subgroups of G and K, motivated by a question of Sikora. Additionally, we use work of Richardson to solve a conjecture of Florentino-Lawton about the singular locus of X, and we give a topological proof that for G= GL(n,C) or SL(n,C), the space X is not a rational Poincaré Duality Space for r>3 and n=2.

43 pages

References in corpus (4)

• The topology of moduli spaces of free group representations
• Fundamental Groups of Character Varieties: Surfaces and Tori
• On the fundamental group of Hom(Z^k,G)
• The Bott cofiber sequence in deformation K-theory and simultaneous similarity in U(n)

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• Flawed groups and the topology of character varieties
• Poisson maps between character varieties: gluing and capping
• Bad Representations and Homotopy of Character Varieties
• Mixed Hodge structures on character varieties of nilpotent groups
• Non-ergodicity on SU(2) and SU(3) character varieties of the once-punctured torus
• Centralizers of irreducible subgroups in the projective special linear group
• Dynamics on the SU(2,1)-character variety of the one-holed torus