Homotopy Groups of Free Group Character Varieties

TL;DR

This paper investigates the low-dimensional homotopy groups of moduli spaces of representations of free groups into complex reductive Lie groups, revealing trivial second homotopy groups for certain cases and exhibiting Bott periodicity in specific subspaces.

Contribution

Develops new methods for studying homotopy groups of free group character varieties, including a general position result in singular settings and explicit computations for G=GL(n,C).

Findings

Second homotopy group of X is trivial for G=GL(n,C) or SL(n,C).

Homotopy groups of the smooth locus of X exhibit Bott Periodicity.

Fundamental group of the irreducible representation subspace Y is described.

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 Poincar\'e Duality Space for r>3 and n=2.