The Omega spectrum for mod 2 $\mathit{KO}$-theory

Abstract

The 8-periodic theory that comes from the $\mathit{KO}$-theory of the mod 2 Moore space is the same as the real first Morava $K$-theory obtained from the homotopy fixed points of the $\mathbb{Z}∕\left(2\right)$ action on the first Morava $K$-theory. The first Morava $K$-theory, $K\left(1\right)$, is just mod 2 $\mathit{KU}$-theory. We compute the homology Hopf algebras for the spaces in this Omega spectrum.