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.
Highlights
The primary goal of this paper is to produce explicit coboundary formulae that yield the Adem relations between compositions of Steenrod Squares
The second ingredient is the construction of certain very specific chain homotopies between pairs of chain maps from chains on BV4 to chains on BΣ4, where V4 ⊂ Σ4 is the normal subgroup of order 4 in the symmetric group
He did not have the structured method using operads to generate specific coboundary formulae, but just the fact that a cohomology operation is determined by a homology class in the symmetric group surely corresponded to some theoretically possible behind the scenes explicit coboundary computations
Summary
Sr , starting from the left, and first permuting the elements of the ith block among themselves by applying the conjugation of the element of Σsi by the unique order preserving map from the ith block to {1, . This produces an automorphism of each block The blocks with their new internal orderings are permuted among themselves, according to the element of Σr .16. It is easy to see from the block description of the operad structure map that the triple (T 1 , (T 2 , T 3 )) ∈ Σ2 × (Σ2 × Σ2) maps to the permutation c 3 b 2 a 1 ∈ D8 ⊂ Σ4. End(V )n = Hom(V ⊗n, V ), with the obvious operad structure and action of the symmetric groups. To give V the structure of an algebra over a symmetric operad P is to give a map of symmetric operads P → End(V )
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