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