On the homology of the Kac-Moody groups and the cohomology of the 3-connective covers of Lie groups

Abstract

Let $G$ be a compact, 1-connected, simple Lie group of exceptional type, $g$ its Lie algebra, and $p$ an odd prime. In this paper, the mod $p$ homology of the Kac-Moody group $K(g^{(1)})$ and the mod $p$ cohomology of the 3-connective cover over $G$ are determined as Hopf algebras over the Steenrod algebra for every case that the integral homology of $G$ has $p$-torsion.