The Brown-Peterson homology and nilpotence of the infinite special orthogonal group (Q5917746)

We study the relations of the stable homotopy groups of a certain unstable map \(h: \text{BO}\to \text{SO}\), and we prove that in \(h_\sharp: \pi^s-* (\text{BO}_+)\to \pi^s_* (\text{SO}_+)\), the image of \(h_\sharp\) is contained in the ideal generated by the nilpotent elements of \(\pi^s_* (\text{SO}_+)\). Also along the way we compute part of the Brown-Peterson homology of \(\text{SO}\).