Topological Hochschild homology of X(n)

TL;DR

This paper establishes that Ravenel's spectrum X(2) is the universal E_1-S-algebra of characteristic η, enabling new insights into complex orientations and topological Hochschild homology of related spectra.

Contribution

It proves that X(2) is the versal E_1-S-algebra of characteristic η and describes the THH of certain E_2-ring Thom spectra mapping from X(2).

Findings

X(2) is the universal E_1-S-algebra of characteristic η.

Every E_1-S-algebra R of characteristic η admits an A_∞ complex orientation.

The topological Hochschild homology of certain spectra has a simple description.

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.