The Brown-Peterson homology and nilpotence of the infinite special orthogonal group (Q5917663)

Let \(G_{2m,k}\) be the Grassmannian of \(2m\) planes in \(R^{2m + k}\). Let \(\rho : G_{2m,k} \to \text{SO} (2m + k)\) be the map that sends a \(2m\) plane \(\gamma\) to the orthogonal transformation which is the antipodal map on \(\gamma\) and the identity on the orthogonal complement of \(\gamma\). In the limit as \(m\) and \(k\) go to infinity \(\rho\) induces a map \(h : \text{BO} \to \text{SO}\). Let \(h_\# : \pi^S_*(\text{BO}_+) \to \pi^S_* (\text{SO}_+)\) be the induced map in stable homotopy. The main result of this paper is that the image of \(h_\#\) is contained in the ideal generated by the nilpotent elements of \(\pi^S_* (\text{SO}_+)\). This is of interest because it is known that the Kervaire invariant element in \(\pi^S_* (\text{SO}_+)\) is nonnilpotent. The proof relies on technical arguments using the Adams spectral sequence for \(\text{BP}_* \text{SO}\).