The diagonal of the associahedra
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 121-146.

This paper introduces the first general method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps.

Cet article introduit pour la première fois une méthode générale permettant de résoudre le problème de l’approximation de la diagonale de familles de polytopes satisfaisant à une propriété de cohérence par faces. On retrouve les cas classiques des simplexes et des cubes et on résout celui des associaèdres, appelés aussi polytopes de Stasheff. On montre que ce dernier cas vérifie une formule cellulaire facile à énoncer. Pour la première fois, nous munissons une famille de réalisations des associaèdres (celle de Loday) d’une structure d’opérade topologique cellulaire, dont nous montrons qu’elle est compatible avec les diagonales.

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DOI: 10.5802/jep.142
Classification: 52B11,  18M75,  18M70,  06A07
Keywords: Associahedra, approximation of the diagonal, operads, fiber polytopes, A -algebras
Naruki Masuda 1; Hugh Thomas 2; Andy Tonks 3; Bruno Vallette 4

1 Johns Hopkins University, Department of Mathematics 3400 N. Charles Street, Baltimore, MD 21218, USA
2 Département de mathématiques, Université du Québec à Montréal Local PK-5151, 201, Avenue du Président-Kennedy, Montréal, Canada
3 Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom
4 Laboratoire Analyse, Géométrie et Applications, Université Sorbonne Paris Nord, CNRS, UMR 7539 93430 Villetaneuse, France
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Naruki Masuda; Hugh Thomas; Andy Tonks; Bruno Vallette. The diagonal of the associahedra. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 121-146. doi : 10.5802/jep.142. https://jep.centre-mersenne.org/articles/10.5802/jep.142/

[AA17] M. Aguiar & F. Ardila - “Hopf monoids and generalized permutahedra”, 2017 | 1709.07504

[Amo17] L. Amorim - “The Künneth theorem for the Fukaya algebra of a product of Lagrangians”, Internat. J. Math. 28 (2017) no. 4, article ID 1750026, 38 pages | MR: 3639044 | Zbl: 1368.53057

[Bro59] E. H. Brown Jr. - “Twisted tensor products. I”, Ann. of Math. (2) 69 (1959), p. 223-246 | Article | MR: 105687 | Zbl: 0199.58201

[BS92] L. J. Billera & B. Sturmfels - “Fiber polytopes”, Ann. of Math. (2) 135 (1992) no. 3, p. 527-549 | Article | MR: 1166643 | Zbl: 0762.52003

[BV73] J. M. Boardman & R. M. Vogt - Homotopy invariant algebraic structures on topological spaces, Lect. Notes in Math., vol. 347, Springer-Verlag, Berlin, 1973 | MR: 420609 | Zbl: 0285.55012

[CFZ02] F. Chapoton, S. Fomin & A. Zelevinsky - “Polytopal realizations of generalized associahedra”, Canad. J. Math. 45 (2002) no. 4, p. 537-566 | Article | MR: 1941227 | Zbl: 1018.52007

[CSZ15] C. Ceballos, F. Santos & G. M. Ziegler - “Many non-equivalent realizations of the associahedron”, Combinatorica 35 (2015) no. 5, p. 513-551 | Article | MR: 3437894 | Zbl: 1389.52013

[EML54] S. Eilenberg & S. Mac Lane - “On the groups H(Π,n). II. Methods of computation”, Ann. of Math. (2) 60 (1954), p. 49-139 | Article | MR: 65162

[EZ53] S. Eilenberg & J. A. Zilber - “On products of complexes”, Amer. J. Math. 75 (1953), p. 200-204 | Article | MR: 52767 | Zbl: 0050.17301

[For08] S. Forcey - “Convex hull realizations of the multiplihedra”, Topology Appl. 156 (2008) no. 2, p. 326-347 | Article | MR: 2475119 | Zbl: 1158.55012

[GKZ08] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky - Discriminants, resultants and multidimensional determinants, Modern Birkhäuser Classics, Birkhäuser Boston, Inc., Boston, MA, 2008 | Zbl: 1138.14001

[GZ97] M. R. Gaberdiel & B. Zwiebach - “Tensor constructions of open string theories. I. Foundations”, Nuclear Phys. B 505 (1997) no. 3, p. 569-624 | Article | MR: 1490781 | Zbl: 0911.53044

[Lod04] J.-L. Loday - “Realization of the Stasheff polytope”, Arch. Math. (Basel) 83 (2004) no. 3, p. 267-278 | MR: 2108555 | Zbl: 1059.52017

[LV12] J.-L. Loday & B. Vallette - Algebraic operads, Grundlehren Math. Wiss., vol. 346, Springer-Verlag, Berlin, 2012 | MR: 2954392 | Zbl: 1260.18001

[May72] J. May - The geometry of iterated loop spaces, Lect. Notes in Math., vol. 271, Springer-Verlag, Berlin, 1972 | MR: 420610

[MS06] M. Markl & S. Shnider - “Associahedra, cellular W-construction and products of A -algebras”, Trans. Amer. Math. Soc. 358 (2006) no. 6, p. 2353-2372 | Article | MR: 2204035 | Zbl: 1093.18005

[MSS02] M. Markl, S. Shnider & J. D. Stasheff - Operads in algebra, topology and physics, Math. Surveys and Monographs, vol. 96, American Mathematical Society, Providence, RI, 2002 | MR: 1898414 | Zbl: 1017.18001

[Pro11] A. Prouté - “A -structures. Modèles minimaux de Baues-Lemaire et Kadeishvili et homologie des fibrations”, Repr. Theory Appl. Categ. (2011) no. 21, p. 1-99, Reprint of the 1986 original, With a preface to the reprint by Jean-Louis Loday | MR: 2844537 | Zbl: 1245.55007

[Sei08] P. Seidel - Fukaya categories and Picard-Lefschetz theory, Zürich Lectures in Advanced Math., European Mathematical Society, Zürich, 2008 | Article | MR: 2441780 | Zbl: 1159.53001

[Ser51] J.-P. Serre - “Homologie singulière des espaces fibrés. Applications”, Ann. of Math. (2) 54 (1951), p. 425-505 | Article | Zbl: 0045.26003

[SS97] J. D. Stasheff & S. Shnider - “From operads to “physically” inspired theories (Appendix B)”, in Operads: Proceedings of Renaissance Conferences (Hartford, CT/Luminy, 1995), Contemp. Math., vol. 202, American Mathematical Society, Providence, RI, 1997, p. 53-81

[Sta63] J. D. Stasheff - “Homotopy associativity of H-spaces. I & II”, Trans. Amer. Math. Soc. 108 (1963), p. 275-292 & 293–312 | MR: 158400 | Zbl: 0114.39402

[Sta70] J. D. Stasheff - H-spaces from a homotopy point of view, Lect. Notes in Math., vol. 161, Springer-Verlag, Berlin, 1970 | MR: 270372 | Zbl: 0205.27701

[Ste47] N. E. Steenrod - “Products of cocycles and extensions of mappings”, Ann. of Math. (2) 48 (1947), p. 290-320 | Article | MR: 22071 | Zbl: 0030.41602

[SU04] S. Saneblidze & R. Umble - “Diagonals on the permutahedra, multiplihedra and associahedra”, Homology Homotopy Appl. 6 (2004) no. 1, p. 363-411 | Article | MR: 2118493 | Zbl: 1069.55015

[Tam51] D. Tamari - Monoïdes préordonnés et chaînes de Malcev, Thèse de Mathématique, Paris, 1951

[Zie95] G. M. Ziegler - Lectures on polytopes, Graduate Texts in Math., vol. 152, Springer-Verlag, New York, 1995 | MR: 1311028 | Zbl: 0823.52002

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