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
–spaces
and
–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
–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
–ring maps and that the
forgetful functor from
–algebras
in
–modules
to
–algebras
is neither full nor faithful.
Keywords
spectral sequence, orientations, structured ring spectra,
power operations, rational homotopy theory, unstable
homotopy theory
