An effective proof of the Cartan formula: the even prime

The Cartan formula encodes the relationship between the cup product and the action of the Steenrod algebra in $\mathbb F_p$-cohomology. In this work, we present an effective proof of the Cartan formula at the cochain level when the field is $\mathbb F_2$. More explicitly, for an arbitrary pair of cocycles and any non-negative integer, we construct a natural coboundary that descends to the associated instance of the Cartan formula. Our construction works for general algebras over the Barratt-Eccles operad, in particular, for the singular cochains of spaces

A combinatorial E∞{E}_\infty algebra structure on cubical cochains

Cubical cochains are equipped with an associative product, dual to the Serre diagonal, lifting the graded ring structure in cohomology. In this work we introduce through explicit combinatorial methods an extension of this product to a full $E_\infty$ structure. We also study the Cartan-Serre map relating the cubical and simplicial singular cochains of spaces, showing this classical quasi-isomorphism is a map of $E_\infty$ algebras.Comment: 14 pages, 2 figure

Cochain level May-Steenrod operations

Steenrod defined in 1947 the Steenrod squares on the mod 2 cohomology of spaces using explicit cochain formulae for the cup-$i$ products; a family of coherent homotopies derived from the broken symmetry of Alexander--Whitney's chain approximation to the diagonal. He later defined his homonymous operations for all primes using the homology of symmetric groups. This approach enhanced the conceptual understanding of the operations and allowed for many advances, but lacked the concreteness of their definition at the even prime. In recent years, thanks to the development of new applications of cohomology, the need to have an effectively computable definition of Steenrod operations has become a key issue. Using the operadic viewpoint of May, this article provides such definitions at all primes introducing multioperations that generalize the Steenrod cup-$i$ products on the simplicial and cubical cochains of spaces.Comment: Published in Forum Mathematicum, 202