Homology of the Kač-Moody groups. I (Q917986)

Let G be a compact, connected, simply connected, simple Lie group of exceptional type and \({\mathcal G}\) its Lie algebra. The homotopy type of the Kač-Moody Lie group \({\mathcal K}({\mathcal G}^{(1)})\) is \(\Omega G<2>\times G\) where \(\Omega G<2>\) is the 2-connected cover of the space of loops on G [cf. \textit{V. G. Kač}, Infinite dimensional groups with applications, Publ., Math. Sci. Res. Inst. 4, 167-216 (1985; Zbl 0614.22006)]. The purpose of this pape is to determine \(H_*({\mathcal K}({\mathcal G}^{(1)});{\mathbb{Z}})\). The homology \(H_*(G;{\mathbb{Z}})\) is known and therefore we need only determine \(H_*(\Omega G<2>;{\mathbb{Z}})\). Since \(H_*(\Omega G<2>;{\mathbb{Z}})\) is a finitely generated abelian group for any *, it is sufficient to determine \(H_*(\Omega G<2>;{\mathbb{Z}}_{(p)})\) for all prime p. In this paper \(H_*(\Omega G<2>;{\mathbb{Z}}_{(p)})\) except \((G,p)=(E_ 6,2)\) is determined. The cases \((E_ 6,2)\) and G of classical type are determined in part II.