An orientation map for height $p−1$ real $E$ theory

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

algebraic topology

Audience: researchers in the topic

MIT topology seminar

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