A Theorem on Multiplicative Cell Attachments with an Application to Ravenel's X(n) Spectra

Abstract

We show that the homotopy groups of a connective $E_k$-ring spectrum with an $E_k$-cell attached along a class $\alpha$ in degree $n$ are isomorphic to the homotopy groups of the cofiber of the self-map associated to $\alpha$ through degree $2n$. Using this, we prove that the $2n-1^{st}$ homotopy groups of Ravenel's $X(n)$ spectra are cyclic for all $n$. This further implies that, after localizing at a prime, $X(n+1)$ is homotopically unique as the $E_1$-$X(n)$-algebra with homotopy groups in degree $2n-1$ killed by an $E_1$-cell. Lastly, we prove analogous theorems for a sequence of $E_k$-ring Thom spectra, for each odd $k$, which are formally similar to Ravenel's $X(n)$ spectra and whose colimit is also $MU$.

Final version, accepted for publication at J. Homotopy Rel. Struct

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• An $\infty$-categorical approach to $R$-line bundles, $R$-module Thom spectra, and twisted $R$-homology
• Relative Thom Spectra Via Operadic Kan Extensions
• Kan extensions and the calculus of modules for $\infty$-categories

Cited by in corpus (3)

• M/F-Theory as Mf-Theory
• Left Bousfield localization without left properness
• Higher chromatic Thom spectra via unstable homotopy theory