New formulas for cup-$i$ products and fast computation of Steenrod squares

TL;DR

This paper introduces new formulas for the cup-$i$ construction, enabling a faster algorithm for computing Steenrod squares in cohomology, with implications for topology and related fields.

Contribution

It presents novel formulas for the cup-$i$ construction and a fast algorithm for Steenrod square computation, unifying previous approaches under an isomorphic framework.

Findings

New formulas define a cup-$i$ construction

Fast algorithm for Steenrod squares computation

All existing formulas are isomorphic to the new construction

Abstract

Operations on the cohomology of spaces are important tools enhancing the descriptive power of this computable invariant. For cohomology with mod 2 coefficients, Steenrod squares are the most significant of these operations. Their effective computation relies on formulas defining a cup-$i$ construction, a structure on (co)chains which is important in its own right, having connections to lattice field theory, convex geometry and higher category theory among others. In this article we present new formulas defining a cup-$i$ construction, and use them to introduce a fast algorithm for the computation of Steenrod squares on the cohomology of finite simplicial complexes. In forthcoming work we use these formulas to axiomatically characterize the cup-$i$ construction they define, showing additionally that all other formulas in the literature define the same cup-$i$ construction up to isomorphism.