Cochain level May–Steenrod operations

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