HPT and cocyclic operations

Let C be the category of cochain complexes. A cocyclic operation of degree i is a natural transformation O on C of the form O:C !C +i that preserves cocycles. In the paper under review the authors design an algebraic-combinatorial machinery based on the homological perturbation theory which generates cocyclic operations starting from a given Eilenberg-Zilber contraction. Working with chain complexes of simplicial sets K, for every positive integer p there is a contraction from C (K×p) to C (K) p, obtained by composition of Eilenberg-Zilber contractions C (K ×K) ! C (K) C (K). The explicit formulation of these morphisms in terms of face and degeneracy operators provides explicit formulae for cocyclic operations, as the main result (Theorem 3.3) shows. The paper contains two applications of this method: an alternative proof of Theorem 3.1 of [R. Gonz´alez-D´ıaz and P. Real Jurado, J. Pure Appl. Algebra 139 (1999), no. 1-3, 89–108; MR1700539 (2000i:55024)] about Steenrod cocyclic operations, and an explicit formula for the first Adem cohomology operation 2 at the cocyclic level. The general case q[c] for a q-cocycle c is studied in detail by the authors in the preprint [“Computing Adem secondary cohomology operations”; per bibl.].