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A prismatic decomposition of the Barratt-Eccles operad. (Une décomposition prismatique de l’opérade de Barratt-Eccles.) (French. Abridged English version) Zbl 1016.18005

The article discusses the relationship between two \(E_\infty\)-operads, the Barratt-Eccles operad and the surjection operad. The Barratt-Eccles operad \(\mathcal{W}\) is a simplicial operad formed from the homogeneous bar construction on the symmetric groups. The associated differential graded operad is denoted \(\mathcal{E}\). The surjection operad \(\chi\) was introduced by J. E. McClure and J. H. Smith in their work on the Deligne conjecture [J. Am. Math. Soc. 16, No. 3, 681-704 (2003; Zbl 1014.18005)].
A morphism of operads \(\mathcal{E}\to \chi\) was constructed in previous work of the authors [C. Berger and B. Fresse, “Combinatorial operad actions on cochains”, preprint,
http:arXiv.org/abs/math.AT/0109158]. The main result proved here is the existence of a section \(\chi \to \mathcal{E}\). This comes from comparing the \(E_\infty\)-cellular structures of \(\mathcal{W}\) and of \(\mathcal{E}\), described by C. Berger [Ann. Inst. Fourier 46, 1133-1166 (1996; Zbl 0853.55007)] and J. McClure and J. H. Smith (loc. cit.), respectively. There results a decomposition of the Barratt-Eccles operad into unions of prisms indexed by surjections.

MSC:

18D50 Operads (MSC2010)
55P48 Loop space machines and operads in algebraic topology
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References:

[1] Berger, C., Opérades cellulaires et espaces de lacets itérés, Ann. Inst. Fourier, 46, 1125-1157 (1996) · Zbl 0853.55007
[2] C. Berger, B. Fresse, Combinatorial operad actions on cochains, Prépublication, arXiv:math.AT/0109158; C. Berger, B. Fresse, Combinatorial operad actions on cochains, Prépublication, arXiv:math.AT/0109158 · Zbl 1056.55006
[3] Gabriel, P.; Zisman, M., Calculus of Fractions and Homotopy Theory. Calculus of Fractions and Homotopy Theory, Ergeb. Math. Grenzgeb., 35 (1967), Springer-Verlag · Zbl 0186.56802
[4] Kontsevich, M.; Soibelman, Y., Deformations of algebras over operads and Deligne’s conjecture, (Conférence Moshé Flato 1999, Vol. I. Conférence Moshé Flato 1999, Vol. I, Math. Phys. Stud., 21 (2000), Kluwer Academic), 255-307 · Zbl 0972.18005
[5] J. McClure, J.H. Smith, A solution of Deligne’s conjecture, Prépublication, arXiv:math.QA/9910126; J. McClure, J.H. Smith, A solution of Deligne’s conjecture, Prépublication, arXiv:math.QA/9910126
[6] J. McClure, J.H. Smith, Multivariable cochain operations and little \(n\) math.QA/0106024; J. McClure, J.H. Smith, Multivariable cochain operations and little \(n\) math.QA/0106024 · Zbl 1014.18005
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