Topological Hochschild homology of X(n)

Abstract

We show that Ravenel's spectrum $X(2)$ is the versal $E_1$-$S$-algebra of characteristic $\eta$. This implies that every $E_1$-$S$-algebra $R$ of characteristic $\eta$ admits an $E_1$-ring map $X(2)\to R$, i.e. an ${\mathbb{A}}_{\infty }$ complex orientation of degree 2. This implies that $R^{\ast }(\mathbb{C}{P}^2)\cong R_{\ast }[x]/x^3$. Additionally, if $R$ is an ${\mathbb{E}}_2$-ring Thom spectrum admitting a map (of homotopy ring spectra) from $X(2)$, e.g. $X(n)$, its topological Hochschild homology has a simple description.

This short note has been on my website since 2015, but I noticed that it was cited recently, so thought I would give it a slightly more permanent home. As always, comments and feedback welcome. 2 pages