Multivariable cochain operations and little 𝑛-cubes

Abstract

In this paper we construct a small E ∞ E_\infty chain operad S \mathcal {S} which acts naturally on the normalized cochains S ∗ X S^*X of a topological space. We also construct, for each n n , a suboperad S n \mathcal {S}_n which is quasi-isomorphic to the normalized singular chains of the little n n -cubes operad. The case n = 2 n=2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture.