The homology of free spectral Lie algebras and machine computation in algebraic topology

Abstract

In his three seminal papers Thomas Goodwillie constructed the Goodwillie tower, with which one can approximate a homotopy functor on spaces similar to how the Taylor series approximates a smooth function in ordinary calculus. In the case of the identity functor on spaces, Michael Ching showed that the derivatives form an operad in spectra. The algebras for this operad are called spectral Lie algebras. It turns out that the mod 2 homology of a free spectral Lie algebra can be described in terms of the homology of the original spectrum. We will construct a machine computational tool to compute the mod 2 homology of a free spectral Lie algebra as a module over the dual Steenrod algebra.