On the EO $\mathrm{EO}$ -orientability of vector bundles

We study the orientability of vector bundles with respect to a family of cohomology theories called $\mathrm{EO}$-theories. The $\mathrm{EO}$-theories are higher height analogues of real $K$-theory $\mathrm{KO}$. For each $\mathrm{EO}$-theory, we prove that the direct sum of $i$ copies of any vector bundle is $\mathrm{EO}$-orientable for some specific integer $i$. Using a splitting principal, we reduce to the case of the canonical line bundle over ${\mathrm{CP}}^{\infty }$. Our method involves understanding the action of an order $p$ subgroup of the Morava stabilizer group on the Morava $E$-theory of ${\mathrm{CP}}^{\infty }$. Our calculations have another application: We determine the homotopy type of the $S^1$-Tate spectrum associated to the trivial action of $S^1$ on all $\mathrm{EO}$-theories.