The $T$–algebra spectral sequence: Comparisons and applications

Abstract

In previous work with Niles Johnson the author constructed a spectral sequence for computing homotopy groups of spaces of maps between structured objects such as $G$–spaces and ${\mathcal{ℰ}}_n$–ring spectra. In this paper we study special cases of this spectral sequence in detail. Under certain assumptions, we show that the Goerss–Hopkins spectral sequence and the $T$–algebra spectral sequence agree. Under further assumptions, we can apply a variation of an argument due to Jennifer French and show that these spectral sequences agree with the unstable Adams spectral sequence.

From these equivalences we obtain information about the filtration and differentials. Using these equivalences we construct the homological and cohomological Bockstein spectral sequences topologically. We apply these spectral sequences to show that Hirzebruch genera can be lifted to ${\mathcal{ℰ}}_{\infty }$–ring maps and that the forgetful functor from ${\mathcal{ℰ}}_{\infty }$–algebras in $H{\bar{\mathbb{F}}}_p$–modules to $H_{\infty }$–algebras is neither full nor faithful.