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Multivariable cochain operations and little $n$-cubes


Authors: James E. McClure and Jeffrey H. Smith
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
Published electronically: January 3, 2003
MathSciNet review: 1969208
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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.

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Additional 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

DOI: https://doi.org/10.1090/S0894-0347-03-00419-3
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
Article copyright: © Copyright 2003 American Mathematical Society