LAUSR

We show that the homotopy groups of a connective $$\mathbb {E}_k$$Ek-ring spectrum with an $$\mathbb {E}_k$$Ek-cell attached along a class $$\alpha $$α in degree n are isomorphic to the homotopy groups of the cofiber of the self-map associated to $$\alpha $$α through degree 2n. Using this, we prove that the $$2n-1$$2n-1st homotopy groups of Ravenel’s X(n) spectra are cyclic for all n. This further implies that, after localizing at a prime, $$X(n+1)$$X(n+1) is homotopically unique as the $$\mathbb {E}_1-X(n)$$E1-X(n)-algebra with homotopy groups in degree $$2n-1$$2n-1 killed by an $$\mathbb {E}_1$$E1-cell. Lastly, we prove analogous theorems for a sequence of $$\mathbb {E}_k$$Ek-ring Thom spectra, for each odd k, which are formally similar to Ravenel’s X(n) spectra and whose colimit is also MU.