From maps between coloured operads to Swiss-Cheese algebras

Ducoulombier, Julien

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

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 $d+1$.

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.

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

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