Steenrod operations via higher Bruhat orders

TL;DR

This paper links higher Bruhat orders to Steenrod squares in cohomology, providing a geometric interpretation and a unified framework for understanding their coproducts.

Contribution

It establishes a novel correspondence between higher Bruhat orders and Steenrod coproducts, offering geometric insights and a comprehensive construction.

Findings

Steenrod's original coproducts are recovered from extremal elements.

All reasonable coproducts can be derived from the proposed construction.

The geometric interpretation involves zonotopal tilings.

Abstract

The purpose of this paper is to establish a correspondence between the higher Bruhat orders of Yu. I. Manin and V. Schechtman, and the cup-$i$ coproducts defining Steenrod squares in cohomology. To any element of the higher Bruhat orders we associate a coproduct, recovering Steenrod's original ones from extremal elements in these orders. Defining this correspondence involves interpreting the coproducts geometrically in terms of zonotopal tilings, which allows us to give conceptual proofs of their properties and show that all reasonable coproducts arise from our construction.