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The diagonal of the associahedra
[La diagonale de l’associaèdre]
Naruki Masuda1 ; Hugh Thomas2 ; Andy Tonks3 ; Bruno Vallette4
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
Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 121-146.

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.

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.

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DOI : 10.5802/jep.142
Classification : 52B11, 18M75, 18M70, 06A07
Keywords: Associahedra, approximation of the diagonal, operads, fiber polytopes, $\mathrm{A}_\infty $-algebras
Mots-clés : Associaèdres, approximation de la diagonale, opérades, polytopes fibrés, $A_\infty $-algèbres

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
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Naruki Masuda; Hugh Thomas; Andy Tonks; Bruno Vallette. The diagonal of the associahedra. Journal de l’École polytechnique — Mathématiques, Tome 8 (2021), pp. 121-146. doi : 10.5802/jep.142. https://jep.centre-mersenne.org/articles/10.5802/jep.142/

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