• Corpus ID: 119668637

Topological Hochschild homology of X(n)

@article{Beardsley2017TopologicalHH,
  title={Topological Hochschild homology of X(n)},
  author={Jonathan Beardsley},
  journal={arXiv: Algebraic Topology},
  year={2017}
}
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… 
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