Alexandru Suciu
Northeastern University, Boston, USA

In previous work, we introduced the notion of binomial cup-one algebras, which are differential graded algebras endowed with Steenrod cup-one products and compatible binomial operations. Given such an $R$-dga, $(A,d)$, defined over the ring $R=\mathbb{Z}$ or $\mathbb{Z}_p$ (for $p$ a prime) and with $H^1(A)$ a finitely generated, free $R$-module, we show that $A$ admits a functorially defined $1$-minimal model, unique up to isomorphism. Furthermore, we associate to this model a pronilpotent group, $G(A)$, which only depends on the $1$-quasi-isomorphism type of $A$. These constructions, which refine classical notions from rational homotopy theory, allow us to distinguish spaces with isomorphic (torsion-free) cohomology that share the same rational $1$-minimal model, yet whose integral $1$-minimal models are not isomorphic. This is joint work with Richard Porter.