The Morava $K$–theory of $BO(q)$ and $MO(q)$

Abstract

We give an easy proof that the Morava $K$–theories for $BO\left(q\right)$ and $MO\left(q\right)$ are in even degrees. Although this is a known result, it had followed from a difficult proof that ${BP}^{\ast }\left(BO\left(q\right)\right)$ was Landweber flat. Landweber flatness follows from the even Morava $K$–theory. We go further and compute an explicit description of $K{\left(n\right)}_{\ast }\left(BO\left(q\right)\right)$ and $K{\left(n\right)}_{\ast }\left(MO\left(q\right)\right)$ and reconcile it with the purely algebraic construct from Landweber flatness.