An Orientation Map for Height p-1 Real E Theory

Abstract

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

References in corpus (1)

• Finite subgroups of extended Morava stabilizer groups

Cited by in corpus (1)

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