Cup i-produit sur les formes différentielles non commutatives et carrés de Steenrod

In this paper, we define in the framework of noncommutative differential forms the notion of cup i-product introduced by Steenrod in [N.E. Steenrod, Product of cocycles and extensions of mappings, Ann. Math. 48 (2) (1947)]. This cup i-product permits to define an operator br,s who will be a generalisation of the operator b defined in Hochschild and cyclic homology. With the operator br,s we define in an explicitly way the Steenrod and Thomas–Pontrjagin squares.