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 -ring spectrum with an -cell attached along a class in degree are isomorphic to the homotopy groups of the cofiber of the self-map associated to through degree 2 . Using this, we prove that the 2 −1 homotopy groups of Ravenel's ( ) spectra are cyclic for all . This further implies that, after localizing at a prime, ( + 1) is homotopically unique as the 1 -( )-algebra with homotopy groups in degree 2 − 1 killed by an 1 -cell. Lastly, we prove analogous theorems for a sequence of -ring Thom spectra, for each odd , which are formally similar to Ravenel's ( ) spectra and whose colimit is also .-modules (̄ ) → ∕ ∕ which induces an isomorphism on homotopy groups in degrees less than 2 .Here, the spectrum ∕ ∕ is the versal -algebra on of characteristic , described for = ∞ in [20] and for all in [3]. It is constructed as by "attaching a -cell" to along in the category of --algebras. Often, versal algebras of this form are conceptually interesting but have difficult to determine homotopy groups, so the above theorem may be of interest beyond its uses in this paper.In Section 3 we apply the above theorem to prove that 2 −1 ( ( )) is cyclic, and that, moreover, it is generated by a certain element which was identified in [5]. This is Corollary 2:Corollary. The element