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

Abstract

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