From higher Bruhat orders to Steenrod cup-i coproducts

We show that the higher Bruhat orders of Manin and Schechtman provide a useful conceptual framework for understanding Steenrod’s cup-i coproducts, which are used to define the cohomology operations known as Steenrod squares. Indeed, we show that the elements of the (i + 1)-dimensional higher Bruhat order are in bijection with all possible cup-i coproducts on the chain complex of the simplex which give a homotopy between cup-(i − 1) and its opposite. The Steenrod cup-i coproduct and its opposite are then given by the maximal and minimal elements of the higher Bruhat order. This correspondence uses the geometric realisation of the higher Bruhat orders in terms of tilings of cyclic zonotopes, and enables us to give conceptual proofs of the fundamental properties of the cup-i coproducts.