Groups associated to 1-minimal models for binomial $\cup_1$-algebras
We give an explicit, cochain-level algebraic model for the pronilpotent completion of a group with finitely generated first cohomology. To each binomial $\cup_1$-dga $(A,d_A)$ over $R=\mathbb{Z}$ or $\mathbb{F}_p$ ($p$ prime) -- a differential graded algebra endowed with a Steenrod $\cup_1$-product and a compatible binomial operation -- we associate a pronilpotent group $G(A)$ that depends only on the 1-quasi-isomorphism type of $A$, provided $H^0(A)=R$ and $H^1(A)$ is a finitely generated free $R$-module. This group arises functorially from the 1-minimal model of $A$, which is unique up to isomorphism. When $A=C^*(X;R)$ is the cochain algebra of a connected CW-complex $X$ with $H^1(X;R)$ finitely generated, the group $G(A)$ recovers the Bousfield--Kan $R$-completion of $π_1(X)$ when $R=\mathbb{F}_p$, and its pro-torsion-free-nilpotent completion when $R=\mathbb{Z}$. Moreover, the group $G(A)$ comes equipped with a natural inverse system $\{G_n(A)\}_{n\ge 1}$ whose structure maps $G_{n+1}(A)\to G_n(A)$ are surjective. If $A=C^*(X;R)$, then $G_n(A)$ is the quotient of $π_1(X)$ by the $(n+1)$th term of the fastest descending central series whose successive quotients are free $R$-modules. We give a purely algebraic necessary and sufficient criterion that, given an isomorphism $G_n(A)\cong G_n(B)$, determines whether $G_{n+1}(A)\cong G_{n+1}(B)$, and we illustrate the use of this criterion with examples distinguishing spaces with isomorphic cohomology rings.