Algebraic Goodwillie spectral sequence

Abstract

Let $\mathit{s}\mathcal{L}$ be the $\infty$-category of simplicial restricted Lie algebras over $\mathbf{F} = \overline{\mathbf{F}}_p$, the algebraic closure of a finite field $\mathbf{F}_p$. By the work of A. K. Bousfield et al. on the unstable Adams spectral sequence, the category $\mathit{s}\mathcal{L}$ can be viewed as an algebraic approximation of the $\infty$-category of pointed $p$-complete spaces. We study the functor calculus in the category $\mathit{s}\mathcal{L}$. More specifically, we consider the Taylor tower for the functor $L^r\colon \mathcal{M}\mathrm{od}^{\geq 0}_{\mathbf{F}} \to \mathit{s}\mathcal{L}$ of a free simplicial restricted Lie algebra together with the associated Goodwillie spectral sequence. We show that this spectral sequence evaluated at $\Sigma^l \mathbf{F}$, $l\geq 0$ degenerates on the third page after a suitable re-indexing, which proves an algebraic version of the Whitehead conjecture. In our proof we compute explicitly the differentials of the Goodwillie spectral sequence in terms of the $\Lambda$-algebra of A. K. Bousfield et al. and the Dyer-Lashof-Lie power operations, which naturally act on the homology groups of a spectral Lie algebra. As an essential ingredient of our calculations, we establish a general Leibniz rule in functor calculus associated to the composition of mapping spaces, which conceptualizes certain formulas of W. H. Lin. Also, as a byproduct, we identify previously unknown Adem relations for the Dyer-Lashof-Lie operations in the odd-primary case.