Let \(X\) be a closed oriented surface, of genus \(g\geq 2\). Let \(\Gamma\) be the universal central extension of \(\pi_1X\), generated by the same generators, \(A_1,B_1,\ldots,A_g,B_g\), as \(\pi_1X\), together with a central element \(J\), subject to the relation \(\prod_{i=1}^g[A_i,B_i]=J\). Given a reductive Lie group, let \(\mathcal{R}(G)\) be the space of semisimple representations of \(\Gamma\) in \(G\). In this paper, the authors compute the homotopy groups of \(\mathcal{R}(\mathrm{GL}(n,\mathbb{C}))\) and \(\mathcal{R}(\mathrm{U}(p,q))\), under certain conditions.
For this, they use the correspondence of this theory with the theory of \(G\)-Higgs bundles over the associated compact Riemann surface. More precisely, if \(\mathcal{M}(G)\) denotes the moduli space of polystable \(G\)-Higgs bundles over \(X\), then \(\mathcal{M}(G)\) and \(\mathcal{R}(G)\) are homeomorphic [see N. J. Hitchin, Proc. Lond. Math. Soc., III. 55, 59–126 (1987; Zbl 0634.53045); C. T. Simpson, Publ. Math., Inst. Hautes Étud. Sci. 79, 47–129 (1994; Zbl 0891.14005); Publ. Math., Inst. Hautes Étud. Sci. 80, 5–79 (1995; Zbl 0891.14006); S. K. Donaldson, Proc. Lond. Math. Soc., III. 55, 127–131 (1987; Zbl 0634.53046) and K. Corlette, J. Differ. Geom. 28, No. 3, 361–382 (1988; Zbl 0676.58007)]. Then, the \(L^2\)-norm of the Higgs field is used as a perfect Morse-Bott function in \(\mathcal{M}(\mathrm{GL}(n,\mathbb{C}))\) and \(\mathcal{M}(\mathrm{U}(p,q))\) to reduce the problem to the computation of the homotopy groups of its minimum subvarieties. In the case of \(\mathcal{M}(\mathrm{GL}(n,\mathbb{C}))\), the minimum subvarieties are isomorphic to the moduli spaces of polystable vector bundles over \(X\), whose homotopy groups have been calculated by G. D. Daskalopoulos and K. K. Uhlenbeck [Topology 34, No. 1, 203–315 (1995; Zbl 0835.58005)]. When \(g\geq 3\) and \(n\) and the degree of the vector bundle appearing in a \(\mathrm{GL}(n,\mathbb{C})\)-Higgs bundle are coprime, this procedure yields the homotopy groups of \(\mathcal{M}(\mathrm{GL}(n,\mathbb{C}))\), hence of \(\mathcal{R}(\mathrm{GL}(n,\mathbb{C}))\).
For \(\mathcal{M}(\mathrm{U}(p,q))\), the minimum subvarieties are isomorphic to the moduli spaces of polystable triples and the authors compute their homotopy groups, using the results of S. B. Bradlow, O. García-Prada, P. Gothen [Math. Ann. 328, No. 1–2, 299–351 (2004; Zbl 1041.32008)]. When \(g\geq 3\) and \(p+q\) and the sum of the degrees of the two vector bundles appearing in a \(\mathrm{U}(p,q)\)-Higgs bundle are coprime, this procedure yields the homotopy groups of \(\mathcal{M}(\mathrm{U}(p,q))\), hence of \(\mathcal{R}(\mathrm{U}(p,q))\).