Algebraic Goodwillie Spectral Sequence

Let sL be the infinity-category of simplicial restricted Lie algebras over F, the algebraic closure of a finite field F_p. By the work of A. K. Bousfield et al. on the unstable Adams spectral sequence, the category sL can be viewed as an algebraic approximation of the infinity-category of pointed p-complete spaces. We study the functor calculus in the category sL. More specifically, we consider the Taylor tower for the functor L^r of a free simplicial restricted Lie algebra together with the associated Goodwillie spectral sequence. We show that this spectral sequence evaluated at Sigma^l F, l>=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.