Differential graded algebras, Steenrod cup-one products, binomial operations, and Massey products

TL;DR

This paper develops a new algebraic framework for differential graded algebras incorporating binomial and Massey products, enhancing the understanding of homotopy invariants in algebraic topology.

Contribution

It introduces binomial cup-one differential graded algebras over integers and prime fields, and defines restricted Massey products with reduced indeterminacy.

Findings

Defined free binomial cup-one differential graded algebras

Established properties of these algebras

Introduced restricted triple Massey products with stronger invariants

Abstract

Motivated by the construction of Steenrod cup-$i$ products in the singular cochain algebra of a space and in the algebra of non-commutative differential forms, we define a category of binomial cup-one differential graded algebras over the integers and over prime fields of positive characteristic. The Steenrod and Hirsch identities bind the cup-product, the cup-one product, and the differential in a package that we further enhance with a binomial ring structure arising from a ring of integer-valued rational polynomials. This structure allows us to define the free binomial cup-one differential graded algebra generated by a set and derive its basic properties. It also provides the context for defining restricted triple Massey products, which have a smaller indeterminacy than the classical ones, and hence, give stronger homotopy type invariants.