The mod 2 homology of free spectral Lie algebras

TL;DR

This paper introduces mod 2 homology operations for spectral Lie algebras derived from Goodwillie calculus and demonstrates they fully describe the mod 2 homology of free algebras on suspension spectra.

Contribution

It adapts Behrens' definition to spectral Lie algebras and proves these operations capture all mod 2 homology for free algebras on simply-connected spaces.

Findings

Mod 2 homology operations are introduced for spectral Lie algebras.

These operations account for all mod 2 homology of free algebras on suspension spectra.

The approach extends understanding of homology in spectral algebra contexts.

Abstract

The Goodwillie derivatives of the identity functor on pointed spaces form an operad in spectra. Adapting a definition of Behrens, we introduce mod 2 homology operations for algebras over this operad and prove these operations account for all the mod 2 homology of free algebras on suspension spectra of simply-connected spaces.