  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.
