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This is a revision of a paper first posted June 4, 2001. It will appear inthe Journal of the AMS.  In this paper we construct a small $E_\infty$ chain operad $\S$ which actsnaturally on the normalized cochains $S^*X$ of a topological space. We alsoconstruct, for each $n$, a suboperad $\S_n$ which is quasi-isomorphic to thenormalized singular chains of the little $n$-cubes operad. The case $n=2$ leadsto a substantial simplification of our earlier proof of Deligne's Hochschildcohomology conjecture.

Multivariable cochain operations and little 𝑛-cubes