## Multivariable cochain operations and little $n$-cubes

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- by James E. McClure and Jeffrey H. Smith
- J. Amer. Math. Soc.
**16**(2003), 681-704 - DOI: https://doi.org/10.1090/S0894-0347-03-00419-3
- Published electronically: January 3, 2003
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## Abstract:

In this paper we construct a small $E_\infty$ chain operad $\mathcal {S}$ which acts naturally on the normalized cochains $S^*X$ of a topological space. We also construct, for each $n$, a suboperad $\mathcal {S}_n$ 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.## References

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## Bibliographic Information

**James E. McClure**- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- Email: mcclure@math.purdue.edu
**Jeffrey H. Smith**- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- Email: jhs@math.purdue.edu
- Received by editor(s): June 25, 2001
- Received by editor(s) in revised form: June 28, 2002
- Published electronically: January 3, 2003
- Additional Notes: The first author was partially supported by NSF grant DMS-9971953. He thanks the Lord for making his work possible

The second author was partially supported by NSF grant DMS-9971953 - © Copyright 2003 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**16**(2003), 681-704 - MSC (2000): Primary 18D50; Secondary 55P48, 16E40
- DOI: https://doi.org/10.1090/S0894-0347-03-00419-3
- MathSciNet review: 1969208