Previously we introduced the notion of binomial cup-one algebras,
which are differential graded algebras endowed with Steenrod
-products and
compatible binomial operations. In this paper we show that binomial cup-one algebras capture homotopy
-type. In particular,
given such an
-dga,
, defined over
the ring
or
(for
a prime), with
and with
a finitely generated, free
-module, we show that
admits a functorially
defined
-minimal
model,
,
which is unique up to isomorphism. Furthermore, we associate to this model a
pronilpotent group, whose continuous cohomology is isomorphic to that of
. 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
-minimal
models over the integers.
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