On the fundamental group of Hom\(({\mathbb Z}^k,G)\) (Q431284)

For a given compact Lie group \(G\), the set of homomorphisms from \(\mathbb{Z}^k\) to \(G\), \(\Hom (\mathbb{Z}^k,G)\), is a based space with base point the trivial homomorphism \(1\). The authors show that the fundamental group \(\pi_1(\Hom(\mathbb{Z}^k,G))\) is isomorphic to the product \(\pi_1(G)\), that is, \[ \pi_1(\Hom(\mathbb{Z}^k,G))\cong \pi_1(G)^k, \quad k\geq 1. \] Their result generalizes the one by \textit{E. Torres-Giese} and \textit{D. Sjerve} [Bull. Lond. Math. Soc. 40, No. 1, 65--76 (2008; Zbl 1145.55016)], where \(G\) is either \(\mathrm{SO}(3)\), \(\mathrm{SU}(2)\) or \(\mathrm{U}(2)\). In Section~2, they consider \(G\) simply connected and prove that \(\pi_1(\Hom(\mathbb{Z}^k,G))=1\). In Section~3, for the general case, since the natural inclusion \(G_0\hookrightarrow G\) gives rise to a map \(i_*: \Hom(\mathbb{Z}^k,G_0)\to \Hom(\mathbb{Z}^k,G)\) inducing an isomorphism on fundamental groups, they only consider \(G\) a connected compact Lie group. Finally, in Section~4, some examples are shown, such as: ``Even if \(G\) is simply connected, \(\Hom(\mathbb{Z}^k,G)\) may have connected components with non-trivial \(\pi_1\)'' or ``The space \(\Hom(\mathbb{Z}^3,\mathrm{Spin}(7))\) has two path-connected components''.