Let \(G\) be a connected, complex reductive Lie group, \(K\subset G\) a maximal compact subgroup and \(F_r\) the free group on \(r\) generators. Let \(\mathfrak{X}_r(G):=\mathrm{Hom}(F_r,G)^*/G\) be the \(G\)-character variety, with \(G\) acting on \(\mathrm{Hom}(F_r,G)\) by conjugation, and \(\mathrm{Hom}(F_r,G)^*\) denoting the polystable locus, i.e. the subspace corresponding to representations whose \(G\)-orbit is closed. \(\mathfrak{X}_r\) is homeomorphic to the GIT quotient \(\mathrm{Hom}(F_r,G)//G\). Let \(\mathfrak{X}_r^{irr}(G)\) be the subspace corresponding to irreducible representations. If \(G\) equals \(\mathsf{GL}_n(\mathbb{C})\), \(\mathsf{SL}_n(\mathbb{C})\), \(\mathsf{U}_n\) or \(\mathsf{SU}_n\), then (if \((r-1)(n-1)\geq 2\)) the irreducible locus coincides with the smooth locus \(\mathfrak{X}_r^{sm}(G)\) of \(\mathfrak{X}_r(G)\).
The main result of this paper is the proof that for the given groups above, namely \(\mathsf{GL}_n(\mathbb{C})\), \(\mathsf{SL}_n(\mathbb{C})\), \(\mathsf{U}_n\) or \(\mathsf{SU}_n\), we have that \(\pi_2(\mathfrak{X}_r(G))=0\).
For such groups, the authors explicitly compute the homotopy groups of \(\mathfrak{X}_r^{sm}(G)\) in a large range of dimensions, and their result exhibits the corresponding Bott periodicity.
If \(X\) is a singular algebraic variety, with smooth locus \(X^{sm}\), the authors also prove a general-position result establishing local conditions guaranteeing that the inclusion \(X^{sm}\subset X\) is \(2\)-connected. This result is of course of independent interest. Then they prove that these conditions are satisfied for certain \(G\)-character varieties.
The above mentioned main theorem, about the second homotopy group, is then proved using the results of the two previous paragraphs.
In addition, the authors also obtain (again using their general-position result) new results on the centralizers of \(G\) and \(K\), and solve a conjecture of Florentino-Lawton about the singular locus of \(\mathfrak{X}_r(G)\) and give a topological proof that for \(\mathsf{GL}_n(\mathbb{C})\) or \(\mathsf{SL}_n(\mathbb{C})\), the space \(\mathfrak{X}_r(G)\) is not a rational Poincaré Duality Space for \(r\geq 4\) and \(n=2\).