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 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 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
E_\infty-ring maps and that the forgetful functor from E_\infty-algebras in
H\overline{F}_p-modules to H_\infty-algebras is neither full nor faithful.Comment: Minor revisions and more than a few typo corrections. To appear in
Algebraic and Geometric Topolog
