Homology of the double and triple loop space of \(SO(n)\) (Q1906886)

We compute the mod 2 homology of the double loop space of a Lie group \(\text{SO} (n)\) exploiting the Serre spectral sequence and the Eilenberg-Moore spectral sequence and determine some of the Steenrod actions. The Dyer-Lashof operations on the homology of the iterated loop space are widely used. Using the Bockstein spectral sequence we get the 2-torsion information of the homology with integer coefficients. We also compute the triple loop space case. The structures of the homology of the double and triple loop space of \(\text{SO} (n)\) depend on the congruence of \(n\bmod 8\). Here we study \(\text{Spin} (n)\) instead of \(\text{SO} (n)\).