Wilson Spaces, Snaith Constructions, and Elliptic Orientations

Abstract

We construct a canonical family of even periodic ${\mathbb{E}}_{\infty }$-ring spectra, with exactly one member of the family for every prime $p$ and chromatic height $n$. At height $1$ our construction is due to Snaith, who built complex $K$-theory from ${\mathbb{CP}}^{\infty }$. At height $2$ we replace ${\mathbb{CP}}^{\infty }$ with a $p$-local retract of $\mathrm{BU}⟨6⟩$, producing a new theory that orients elliptic, but not generic, height $2$ Morava $E$-theories. In general our construction exhibits a kind of redshift, whereby $\mathrm{BP}⟨n-1⟩$ is used to produce a height $n$ theory. A familiar sequence of Bocksteins, studied by Tamanoi, Ravenel, Wilson, and Yagita, relates the $K(n)$-localization of our height $n$ ring to work of Peterson and Westerland building $E_n^{hS{\mathbb{G}}^{\pm }}$ from $\mathrm{K}(\mathbb{Z},n+1)$.

41 pages, accepted version. Comments welcome!