Towards the algebra structure of the Morava \(K\)-theory of the orthogonal groups (Q1382653)

It is well known that \(BP_*= \mathbb{Z}_{(2)} [v_1,v_2, \dots]\) where the degree of \(v_i\) is \(2(2^i-1)\). Using the Sullivan-Baas technique, Jack Morava kills \(\{v_i \mid i<\ell\}\), where \(\ell>0\), to get \(P(\ell)\) a \(BP\)-module theory with coefficient ring \(P(\ell)_* =\mathbb{Z}/2 [v_\ell, v_{\ell+1}, \dots]\). In this note, the author gives the generators for \(K(\ell)\)-homology of Lie groups \(\text{SO} (2^{\ell+1}-1)\) and \(\text{Spin} (2^{\ell +1}-1)\) -- see Proposition 2.4. and section 3. In section 5, he also determines a complete set of algebra relations for the generators -- i.e. the algebra structure -- of \(\text{Pl}_* \text{SO} (2^{\ell +1}-1)\) and \(\text{Pl}_* \text{Spin} (2^{\ell +1}_1-1)\) respectively. The formulas look too complicated to be reproduced here. The results of this paper can also be used to give another derivation of the Hopf algebra structure of \(H^* (\text{Spin} (n); \mathbb{Z}_2)\) -- see Corollary 5.6. combined with Proposition 2.1.