Mod 2 cohomology of 2-compact groups of low rank

Abstract

We determine the mod 2 cohomology algebra over the Steenrod algebra $\mathcal{A}_{2}$ of the classifying space of loop groups $LG$ where $G=Spin(7)$, $Spin(8)$, $Spin(9)$, $F_{4}$ and $DI(4)$. Then we show they are isomorphic as algebras over $\mathcal{A}_{2}$ to the mod 2 cohomology of the 2-compact groups of type $G$.