Homology of the Kac-Moody groups. II (Q1814219)

Let \(G\) be a compact, connected, simply connected, simple Lie group and \(\mathfrak g\) its Lie algebra. Let \(X\langle n\rangle\) be the \(n\)-connected cover of the space \(X\). Since \(\pi_ 3(G)\cong\mathbb{Z}\) is the first non- trivial homotopy, there is an \(S^ 1\)-fibration \(S^ 1\rightarrow\Omega G\langle 2\rangle\rightarrow\Omega G\). The homotopy type of the Kac-Moody group \({\mathfrak K}({\mathfrak g}^{(1)})\) is \(\Omega G\langle 2\rangle\times G\). Since the homology of \(G\) is known and \(H_ *(\Omega G\langle 2\rangle;\mathbb{Z})\) is finitely generated, we have only to determine \(H_ *(\Omega G\langle 2\rangle;\mathbb{Z}_{(p)})\) for all prime \(p\) to determine \(H_ *({\mathfrak K}({\mathfrak g}^{(1)});\mathbb{Z})\). The homology of \(G\) has non trivial \(p\)-torsions if and only if \((G,p)\) is one of the following: \[ \begin{gathered}(\text{Spin}(n),2)\;n\geq 7,\;(E_ 6,2),\;(E_ 6,3),\;(E_ 7,2),\;(E_ 7,3),\\ (E_ 8,2),\;(E_ 8,3),\;(E_ 8,5),\;(F_ 4,2),\;(F_ 4,3)\hbox{ and }(G_ 2,2).\end{gathered} \] In Part I [ibid. 29, 449-453 (1989; Zbl 0705.57023)] we computed \(H_ *(\Omega G\langle 2\rangle;\mathbb{Z}_{(p)})\) for such \((G,p)\) except \((\text{Spin}(n),2)\) and \((E_ 6,2)\). The purpose of this paper is to determine it for the groups whose homology has no \(p\)-torsion.