The mod 2 homology of infinite loopspaces

Abstract

Applying mod 2 homology to the Goodwillie tower of the functor sending a spectrum $X$ to the suspension spectrum of its $0^{th}$ space, leads to a spectral sequence for computing $H_{\ast }\left({\Omega }^{\infty }X;\mathbb{Z}∕2\right)$, which converges strongly 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 $H_{\ast }\left(X;\mathbb{Z}∕2\right)$, including structure as a bigraded Hopf algebra, a right module over the mod 2 Steenrod algebra $\mathcal{A}$, and a left module over the Dyer–Lashof operations. This paper is an investigation of how this structure is transformed through the spectral sequence.

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 action on the tower, and the Tate construction, to determine how Dyer–Lashof operations act on the spectral sequence. In particular, $E^{\infty }$ has Dyer–Lashof operations induced from those on $E^1$.

We use our spectral sequence Dyer–Lashof operations to determine differentials that hold for any spectrum $X$. The formulae for these universal differentials then lead us to construct an algebraic spectral sequence depending functorially on an $\mathcal{A}$–module $M$. The topological spectral sequence for $X$ agrees with the algebraic spectral sequence for $H_{\ast }\left(X;\mathbb{Z}∕2\right)$ for many spectra $X$, including suspension spectra and almost all Eilenberg–Mac Lane spectra. The $E^{\infty }$ term of the algebraic spectral sequence has form and structure similar to $E^1$, but now the right $\mathcal{A}$–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.