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 \(\mathbb {E}_k\)-ring spectrum with an \(\mathbb {E}_k\)-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 \(\mathbb {E}_1-X(n)\)-algebra with homotopy groups in degree \(2n-1\) killed by an \(\mathbb {E}_1\)-cell. Lastly, we prove analogous theorems for a sequence of \(\mathbb {E}_k\)-ring Thom spectra, for each odd k, which are formally similar to Ravenel’s X(n) spectra and whose colimit is also MU.