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

TL;DR

This paper establishes a relationship between homotopy groups of certain ring spectra with attached cells and cofibers of associated self-maps, applying this to analyze Ravenel's X(n) spectra and related Thom spectra.

Contribution

It introduces a new method to compute homotopy groups of $E_k$-ring spectra with attached cells and applies this to prove properties of Ravenel's X(n) spectra and related Thom spectra.

Findings

Homotopy groups of connective $E_k$-ring spectra with attached cells are isomorphic to cofibers of associated self-maps.

The $(2n-1)$-th homotopy groups of Ravenel's $X(n)$ spectra are cyclic.

$X(n+1)$ is homotopically unique after localization, with specific homotopy group properties.

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 $MU$.