Multivariable cochain operations and little n-cubes

James E. McClure; Jeffrey H. Smith

paper

Multivariable cochain operations and little n-cubes

arXiv:math/0106024

Abstract

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 $§$ which acts naturally on the normalized cochains $S^{*} X$ of a topological space. We also construct, for each $n$ , a suboperad $§_{n}$ 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.