Abstract
Let SU(n) be the n-th special unitary group, SO(n) be the n-th special orthogonal group, and SU(n)/SO(n) be the homogenous space. Let SU be the infinite special unitary group, SO be the infinite special orthogonal group, and SU/SO be the homogenous space. By Bott-periodicity, the loop of SU/SO, Ω(SU/SO), is homotopy-equivalent to BO which is the classifying space of the infinite orthogonal group. Hence we have a map h:Ω(SU(n)/SO(n)) → BO, which is induced by looping the natural inclusion map. Furthermore by Lemma 7 in [6] the above natural inclusion map is (n - 2)-equivalence. This suggests us that we can compute the Brown-Peterson homology of Ω(SU(n)/SO(n)), BP * Ω(SU (n)/SO (n)), completely by knowing BP * BO. In [5] the first author gave a complete answer of the Brown-Peterson homology of the classifying space BO, BP * BO. He computed BP * BO by using the 2-primary Adams spectral sequence. With the same techniques, we can also use the Adams spectral sequence to compute BP * Ω(SU (n)/SO(n)).