The mod 2 homology of infinite loopspaces

TL;DR

This paper analyzes the mod 2 homology spectral sequence of infinite loopspaces, revealing how algebraic and topological structures interact through operads, Tate constructions, and Dyer-Lashof operations, providing new computational tools.

Contribution

It introduces a universal algebraic spectral sequence for mod 2 homology of infinite loopspaces, linking algebraic and topological data via operad and Tate constructions.

Findings

Spectral sequence converges to H_*(X_0) for 0-connected spectra.

Explicit differentials determined by operad and Tate construction interactions.

Algebraic spectral sequence matches topological one for many spectra, providing bounds.

Abstract

We study the spectral sequence that one obtains by applying mod 2 homology to the Goodwillie tower which sends a spectrum X to the suspension spectrum of its 0th space X_0. This converges strongly to H_*(X_0) when X is 0-connected. The E^1 term is the homology of the extended powers of X, and thus is a well known functor of H_*(X), including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra A, and a left module over the Dyer-Lashof operations. Hopf algebra considerations show that all pages of the spectral sequence are primitively generated, with primitives equal to a subquotient of the primitives in E^1. We use an operad structure on the tower and the Z/2 Tate construction to show how Dyer-Lashof operations and differentials interact. These then determine differentials that hold for any spectrum X. These universal differentials then lead us to construct, for every A-module M, an algebraic spectral sequence depending functorially on M. The algebraic spectral sequence for H_*(X) agrees with the topological spectral sequence for X for many spectra, including suspension spectra and almost all generalized Eilenberg-MacLane spectra, and appears to give an upper bound in general. The E^infty term of the algebraic spectral sequence has form and structure similar to E^1, but now the right A-module structure is unstable. Our explicit formula involves the derived functors of destabilization as studied in the 1980's by W. Singer, J. Lannes and S. Zarati, and P. Goerss.