Cup-one algebras and 1-minimal models

Abstract

In previous work we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod $\cup_1$-products and compatible binomial operations. In this paper we show that binomial cup-one algebras capture homotopy 1-type. In particular, given such an $R$-dga, $(A,d_A)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{F}_p$ (for $p$ a prime), with $H^0(A)=R$ and with $H^1(A)$ a finitely generated, free $R$-module, we show that $A$ admits a functorially defined 1-minimal model, $\rho\colon (\mathcal{M}(A),d)\to (A,d_A)$, which is unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, whose continuous cohomology is isomorphic to that of $\mathcal{M}(A)$. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic torsion-free integral cohomology rings. Moreover, we show that there is an equivalence of categories between isomorphism classes of finitely-generated, torsion-free-nilpotent groups and isomorphism classes of finitely generated 1-minimal models over the integers.