Torsion in the homology of the double loop spaces of compact simple Lie groups (Q2785263)

Using Morse theory, one proves that the integral homology of the based loop space on a compact simple Lie group is concentrated in even degrees. A consequence is that \(H_*(\Omega G)\) is torsion free. It does not follow from this, however, that \(H_*(\Omega^2 G)\) is torsion free. The aim of this paper is the determination of the \(p\)-torsion in \(H_*(\Omega^2 G)\) for \(G\) a compact simple Lie group for all primes \(p\). The tools of choice are the Bockstein spectral sequence and the Araki-Kudo-Dyer-Lashof operations that exist on the homology of a three-fold loop space (\(\Omega^2 G \simeq \Omega^3 BG\)). Because the last nontrivial differential in the Bockstein spectral sequence gives an upper bound on the torsion in the homology of a space, the torsion can be bounded from the differentials which in turn depends on the behavior of the homology operations and known homology data for \(H_*(\Omega G)\).