Homotopy and Homology for Simplicial Abelian Hopf Algebras

Let A be a simplicial bicommutative Hopf algebra over the field F2 with the property that π0A ∼= F2. We show that π∗A is a functor of the André-Quillen homology of A, where A is regarded as an F2 algebra. Then we give a method for calculating that André-Quillen homology independent of knowledge of π∗A. Let G be an abelian group. Since the work of Serre [19] and Cartan [6], we have know that the mod p homology of an Eilenberg-MacLane space K(G, n), n ≥ 1, depends only on Tors(Z/p,G), s = 0, 1. More is true: the structure of H∗K(G, n) = H∗(K(G, n),Fp) as an unstable coalgebra over the Steenrod algebra depends only on there Tor groups and the Bockstein β : Tor1(Z/p,G)→ Tor0(Z/p,G) = Z/p⊗G which is the connecting homomorphism of the six term exact sequence obtained by tensoring G with the short exact sequence 0→ Z/p→ Z/p → Z/p→ 0. The purpose of this paper to expand on this observation; indeed, our principal result will be that this is an algebraic fact, not a topological one, and an instance of a phenomenon that arises naturally in the study of simplicial bicommutative Hopf algebras. With an appropriate model for an Eilenberg-MacLane space – for example, the simplicial abelian group model – the mod p homology groups of the Eilenberg-MacLane space K(G, n) are the homotopy groups of the simplicial bicommutative Hopf algebra FpK(G, n); that is, H∗K(G, n) = H∗(K(G, n),Fp) ∼= π∗FpK(G, n). Since K(G, n) is connected (as n > 0), we have that π0FpK(G, n) ∼= Fp; we will say that the simplicial Hopf algebra FpK(G, n) is homotopy connected. We will prove, at least when The first author was partially supported by the NSF.