Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωn+1G→Ωn+1(G/H)→ΩnH for n=1,2, and the Eilenberg–Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H∗(Ωn(G/H);F2). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.