From maps between coloured operads to Swiss-Cheese algebras

[Construction d’une algèbre Swiss-Cheese à partir d’un morphisme d’opérades colorées]

Ducoulombier, Julien¹

¹ ETH Zurich Ramistrasse 101 809 Zurich (Switzerland)

Annales de l'Institut Fourier, Tome 68 (2018) no. 2, pp. 661-724.

Abstract (VO)

In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad ${\mathrm{𝒮𝒞}}_1$. Thereafter, we show that the pair formed by the space of long embeddings and the manifold calculus limit of $\left(l\right)$-immersions from ${ℝ}^d$ to ${ℝ}^n$ is an ${\mathrm{𝒮𝒞}}_{d+1}$-algebra.

Résumé

A partir d’un morphisme d’opérades colorées, on introduit un couple d’espaces topologiques que l’on identifie explicitement à une algèbre sous l’opérade Swiss-Cheese de dimension $1$. Nous sommes alors en mesure d’identifier le couple formé des plongements longs et de l’approximation polynomiale des $\left(l\right)$-immersions de ${ℝ}^d$ vers ${ℝ}^n$ à une algèbre sous l’opérade Swiss-Cheese de dimension $d+1$.

Détail

Reçu le : 2016-06-07
Révisé le : 2017-04-25
Accepté le : 2017-07-13
Publié le : 2018-04-18

DOI : 10.5802/aif.3175

Classification : 18D50, 55P35, 57Q45
Keywords: coloured operads, loop spaces, space of knots, model category
Mots-clés : opérades colorées, espaces de lacets, espaces de plongements, catégorie modèle

Licence : CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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