A Cochain Level Proof of Adem Relations in the Mod 2 Steenrod Algebra

TL;DR

This paper provides an explicit cochain-level proof of the Adem relations in the mod 2 Steenrod algebra, using Steenrod's original cochain definitions of Steenrod Squares.

Contribution

It offers a novel cochain-level proof of Adem relations, connecting explicit cochain formulae with fundamental algebraic relations in cohomology operations.

Findings

Explicit cochain formulae for Adem relations

Coboundary expressions for Adem relations

Connection to Steenrod's original cochain definitions

Abstract

In 1947, N.E. Steenrod defined the Steenrod Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup-i products of cocycles. He later recast the construction in more general homological terms, using group homology and acyclic model methods, rather than explicit cochain formulae, to define mod p operations for all primes p. Steenrod's student J. Adem applied the homological point of view to prove fundamental relations, known as the Adem relations, in the algebra of cohomology operations generated by the Steenrod operations. In this paper we give a proof of the mod 2 Adem relations at the cochain level. Specifically, given a mod 2 cocycle, we produce explicit cochain formulae whose coboundaries are the Adem relations among compositions of Steenrod Squares applied to the cocycle, using Steenrod's original cochain definition of the Square operations.