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

Abstract

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 $\mathrm{K}$-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 ${\mathbb{CP}}^{\infty }$. Our method involves understanding the action of an order $p$ subgroup of the Morava stabilizer group on the Morava $\mathrm{E}$-theory of ${\mathbb{CP}}^{\infty }$. Our calculations have another application: We determine the homotopy type of the ${\mathrm{S}}^1$-Tate spectrum associated to the trivial action of ${\mathrm{S}}^1$ on all $\mathrm{EO}$-theories.

References in corpus (1)

• An Orientation Map for Height p-1 Real E Theory

Cited by in corpus (1)

• Odd primary analogs of Real orientations