The T-algebra spectral sequence: Comparisons and applications

TL;DR

This paper compares various spectral sequences related to homotopy groups of structured objects, demonstrating their equivalences and applying these results to topological problems like lifting Hirzebruch genera.

Contribution

It establishes conditions under which the T-algebra, Goerss-Hopkins, and unstable Adams spectral sequences agree, and applies these to topological and algebraic problems.

Findings

Spectral sequences are equivalent under certain assumptions.

Hirzebruch genera can be lifted to E_-ring maps.

The forgetful functor from E_-algebras to H_-algebras is neither full nor faithful.

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 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.