Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action by an operad that is equivalent to the little 2-cubes operad.

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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.

In this paper we show that if a cosimplicial space has a certain kind of combinatorial structure then its total space has an action of an operad weakly equivalent to the little n-cubes operad. Our results are also valid for cosimplicial spectra.

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