Cochain level May-Steenrod operations

TL;DR

This paper introduces explicit, computable definitions of Steenrod operations at all primes using operadic methods, generalizing classical cup-i products for cochains.

Contribution

It provides a new operadic framework for defining Steenrod operations at all primes, enhancing their computability and conceptual understanding.

Findings

Defines multioperations generalizing cup-i products

Provides explicit cochain formulas for all primes

Facilitates effective computation of Steenrod operations

Abstract

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, the need to have an effectively computable definition of Steenrod operations has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-$i$ products on the simplicial and cubical cochains of spaces.