From maps between coloured operads to Swiss-Cheese algebras

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 SC_{1}. Thereafter, we show that the pair formed by the space of long knots and the polynomial approximation of (k)-immerions from R^{d} to R^{n} is an SC_{d+1}-algebra assuming the Dwyer-Hess'conjecture

On Operadic Actions on Spaces of Knots and 2-Links

In the present work, we realize the space of string 2-links $\mathcal{L}$ as a free algebra over a colored operad denoted $\mathcal{SCL}$ (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of $\mathcal{L}$ by its center and is compatible with Budney's freeness theorem for long knots. From an algebraic point of view, our main result refines Blaire, Burke and Koytcheff's theorem on the monoid of isotopy classes of string links. Topologically, it expresses the homotopy type of the isotopy class of a string 2-link in terms of the homotopy types of the classes of its prime factors.Comment: Comments are welcom

On the delooping of (framed) embedding spaces

It is known that the bimodule derived mapping spaces between two operads have a delooping in terms of the operadic mapping space. We show a relative version of that statement. The result has applications to the spaces of disc embeddings fixed near the boundary and framed disc embeddings.Comment: arXiv admin note: text overlap with arXiv:1704.0706

A model for configuration spaces of points

The configuration space of points on a $D$-dimensional smooth framed manifold may be compactified so as to admit a right action over the framed little $D$-disks operad. We construct a real combinatorial model for these modules, for compact smooth manifolds without boundary

Projective and Reedy model category structures for (infinitesimal) bimodules over an operad

We construct and study projective and Reedy model category structures for bimodules and infinitesimal bimodules over topological operads. Both model structures produce the same homotopy categories. For the model categories in question, we build explicit cofibrant and fibrant replacements. We show that these categories are right proper and under some conditions left proper. We also study the extension/restriction adjunctions.Comment: All comments on this work are welcom

The Swiss-Cheese operad and applications to the space of long knots

L’objectif de ce travail est l’étude de l’opérade Swiss-Cheese SCd qui est une version relative del’opérade des petits cubes Cd. On montre que les théorèmes classiques dans le cadre des opérades non colorées admettent des analogues dans le cas relatif. Il est ainsi possible d’extraire d’une opérade pointée O (i.e. un opérade colorée sous π₀(SC₁) ) un couple d’espaces semi-cosimpliciaux (Oc ; O₀) dont les semitotalisations sont faiblement équivalentes à une SC₂-algèbre explicite. En particulier, on prouve que le couple (ℒ1 ; n ; ℒm ; n), composé de l’espace des longs nœuds et de l’espace des longs entrelacs à m brins, est faiblement équivalent à une SC₂-algèbre explicite. Dans un second temps, on s’intéresse aux couples d’homologies singulières et d’homologies de Hochschild associés à une paire d’espaces semi-cosimpliciaux provenant d’une opérade pointée. Dans ce contexte, les couples (H∗ (sTot(Oc)) ; H∗ (sTot(O₀))) et (HH∗(Oc) ; HH∗(O₀)) possèdent tous deux une structure de H∗(SC₂)-algèbre explicite. On montre alors que le morphisme de Bousfield entre ces deux couples préserve les structures de H∗(SC₂)-algèbres. Cela nous permet de mieux appréhender le couple de suites spectrales de Bousfield calculant (H∗(sTot(Oc)) ; H∗(sTot(O₀))). En particulier, on énonce un critère permettant de faire le lien entre le couple d’homologies singulières issu d’une opérade symétrique multiplicative topologique et la page E² des suites spectrales de Bousfield. La dernière étape de notre étude consiste à généraliser les précédents résultats. Pour cela, on se base sur une conjecture de Dwyer et Hess qui vise à identifier une Cd₊₁-algèbre à partir d’un morphisme d’opérades Cd → O. En admettant ce résultat, on introduit une opérade colorée CCd telle que l’on peut extraire une SCd₊₁-algèbre à partir d’un morphisme d’opérades colorées CCd→ O. On montre ainsi que le couple d’espaces (ℒᵈ₁ ; n ; T∞Imm(ᴷ))(Rᵈ ; Rⁿ), composé de l’espace des longs nœuds en dimension d et de l’approximation polynomiale des (k)-immersions, est faiblement équivalent à une SCd₊₁-algèbre explicite.The aim of this work is to study the Swiss-Cheese operad, denoted by SCd, which is a relative version of the little cubes operad Cd.We show that the classical theorems in the context of uncolored operads can begeneralized to the relative case. From a pointed operad O (i.e. a two colored operad under π0(SC₁) ), webuild two semi-cosimplicial spaces (Oc ; Oo) such that the pair of semi-totalizations is weakly equivalentto an explicit SC₂-algebra. In particular, we prove that the pair (ℒ₁ ; n ; ℒm; n), composed of the space oflong knots and the space of long links, is weakly equivalent to an explicit SC₂-algebra.We study two homology theories, namely singular and Hochschild homology, of a pair of semicosimplicialspaces arising from a pointed operad. In this context, (H∗(sTot(Oc)) ; H∗(sTot(Oo))) and (HH∗(Oc) ; HH∗(Oo)) are equipped with an explicit H∗(SC₂)-algebra structure. We show that the mapintroduced by Bousfield between these two pairs is a morphism of H∗(SC₂)-algebras. This result helps us to understand the pair of spectral sequences computing (H∗(sTot(Oc)) ; H∗(sTot(Oo))). In particular wegive some conditions on a multiplicative symmetric operad so that the E² pages of the Bousfield spectral sequences are weakly equivalent to H∗(sTot(Oc)) and H∗(sTot(Oo)) as H∗(SC₂)-algebras. Finally we generalize our previous results, relying on a conjecture by Dwyer and Hess. We define acolored operad CCd and obtain an SCd₊₁-algebra from an operad morphism CCd → O. As a consequence, we prove that the couple of topological spaces (ℒᵈ₁ ; n ; T∞Imm(ᴷ))(Rᵈ ; Rⁿ)), where Ld₁;n is the space of long knots from Rd to Rⁿ and where T∞Imm(k)(Rᵈ ; Rⁿ) is the polynomial approximation of the (k)-immersions,is weakly equivalent to an explicit SCd+₁-algebra