Geometric Constructions of Mod $p$ Cohomology Operations

Abstract

The Brown Representability Theorem implies that cohomology operations can be represented by continuous maps between Eilenberg-Maclane spaces. These Eilenberg-Maclane spaces have explicit geometric models as spaces of cycles on round spheres and spaces of relative cycles on unit disks, due to the Almgren Isomorphism Theorem. A. Nabutovsky asked what maps between spaces of cycles represent the Steenrod squares. In this work we answer this question by constructing maps with explicit formulas from spaces of cycles on spheres to spaces of relative cycles on disks that represent all Steenrod squares, as well as all Steenrod powers and Bockstein homomorphisms on mod $p$ cohomology, for all primes $p$.