Let $p$ be an odd prime and let $\mathit{ EO} = E_{ p-1}^{ hC_p}$ be the $C_p$ fixed points of height $p-1$ Morava $E$ theory. We say that a spectrum $X$ has algebraic $\mathit{ EO}$ theory if the splitting of $K_*(X)$ as an $K_*[C_p]$-module lifts to a topological splitting of $\mathit{ EO} \wedge X$. We develop criteria to show that a spectrum has algebraic $\mathit{ EO}$ theory, in particular showing that any connective spectrum with mod $p$ homology concentrated in degrees $2k(p - 1)$ has algebraic $\mathit{ EO}$ theory. As an application, we answer a question posed by Hovey and Ravenel by producing a unital orientation $\mathit{ MY}_{ 4p-4}\to \mathit{ EO}$ analogous to the $\mathit{ MSU}$ orientation of $\mathit{ KO}$ at $p=2$.
Categories: math.AT
