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.
Since the first appearance of this work as the author's PhD thesis, an analogous computation of the mod @math homology for odd primes @math has been carried out by Kjaer @cite_0 . In that setting the story is less complete: due to a lack of an analogue of Priddy's trace formula [Lemma 1.4.3] Behrens , the relations between the homology operations have yet to be fully determined.
@cite_0
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The derivatives of the identity functor on spaces in Goodwillie calculus forms an operad in spectra. Antolin-Camarena computed the mod 2 homology of free algebras over this operad for 1-connected spectra. In this present paper we carry out similar computations for mod p homology for odd primes p.