20 research outputs found
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 as
a free algebra over a colored operad denoted (for "Swiss-Cheese
for links"). This result extends works of Burke and Koytcheff about the
quotient of 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 -dimensional smooth framed manifold
may be compactified so as to admit a right action over the framed little
-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
Lâobjectif de ce travail est lâĂ©tude de lâopĂ©rade Swiss-Cheese SCd qui est une version relative delâopĂ©rade des petits cubes Cd. On montre que les thĂ©orĂšmes classiques dans le cadre des opĂ©rades non colorĂ©es admettent des analogues dans le cas relatif. Il est ainsi possible dâextraire dâune opĂ©rade pointĂ©e O (i.e. un opĂ©rade colorĂ©e sous Ïâ(SCâ) ) un couple dâespaces semi-cosimpliciaux (Oc ; Oâ) dont les semitotalisations sont faiblement Ă©quivalentes Ă une SCâ-algĂšbre explicite. En particulier, on prouve que le couple (â1 ; n ; âm ; n), composĂ© de lâespace des longs nĆuds et de lâespace des longs entrelacs Ă m brins, est faiblement Ă©quivalent Ă une SCâ-algĂšbre explicite. Dans un second temps, on sâintĂ©resse aux couples dâhomologies singuliĂšres et dâhomologies de Hochschild associĂ©s Ă une paire dâespaces semi-cosimpliciaux provenant dâune opĂ©rade pointĂ©e. Dans ce contexte, les couples (Hâ (sTot(Oc)) ; Hâ (sTot(Oâ))) et (HHâ(Oc) ; HHâ(Oâ)) possĂšdent tous deux une structure de Hâ(SCâ)-algĂšbre explicite. On montre alors que le morphisme de Bousfield entre ces deux couples prĂ©serve les structures de Hâ(SCâ)-algĂšbres. Cela nous permet de mieux apprĂ©hender le couple de suites spectrales de Bousfield calculant (Hâ(sTot(Oc)) ; Hâ(sTot(Oâ))). En particulier, on Ă©nonce un critĂšre permettant de faire le lien entre le couple dâhomologies singuliĂšres issu dâune opĂ©rade symĂ©trique multiplicative topologique et la page EÂČ des suites spectrales de Bousfield. La derniĂšre Ă©tape de notre Ă©tude consiste Ă gĂ©nĂ©raliser les prĂ©cĂ©dents rĂ©sultats. Pour cela, on se base sur une conjecture de Dwyer et Hess qui vise Ă identifier une Cdââ-algĂšbre Ă partir dâun morphisme dâopĂ©rades Cd â O. En admettant ce rĂ©sultat, on introduit une opĂ©rade colorĂ©e CCd telle que lâon peut extraire une SCdââ-algĂšbre Ă partir dâun morphisme dâopĂ©rades colorĂ©es CCdâ O. On montre ainsi que le couple dâespaces (âá”â ; n ; TâImm(Ꮇ))(Rá” ; Râż), composĂ© de lâespace des longs nĆuds en dimension d et de lâapproximation polynomiale des (k)-immersions, est faiblement Ă©quivalent Ă une SCdââ-algĂšbre 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 Ï0(SCâ) ), webuild two semi-cosimplicial spaces (Oc ; Oo) such that the pair of semi-totalizations is weakly equivalentto an explicit SCâ-algebra. In particular, we prove that the pair (ââ ; n ; âm; n), composed of the space oflong knots and the space of long links, is weakly equivalent to an explicit SCâ-algebra.We study two homology theories, namely singular and Hochschild homology, of a pair of semicosimplicialspaces arising from a pointed operad. In this context, (Hâ(sTot(Oc)) ; Hâ(sTot(Oo))) and (HHâ(Oc) ; HHâ(Oo)) are equipped with an explicit Hâ(SCâ)-algebra structure. We show that the mapintroduced by Bousfield between these two pairs is a morphism of Hâ(SCâ)-algebras. This result helps us to understand the pair of spectral sequences computing (Hâ(sTot(Oc)) ; Hâ(sTot(Oo))). In particular wegive some conditions on a multiplicative symmetric operad so that the EÂČ pages of the Bousfield spectral sequences are weakly equivalent to Hâ(sTot(Oc)) and Hâ(sTot(Oo)) as Hâ(SCâ)-algebras. Finally we generalize our previous results, relying on a conjecture by Dwyer and Hess. We define acolored operad CCd and obtain an SCdââ-algebra from an operad morphism CCd â O. As a consequence, we prove that the couple of topological spaces (âá”â ; n ; TâImm(Ꮇ))(Rá” ; Râż)), where Ldâ;n is the space of long knots from Rd to Râż and where TâImm(k)(Rá” ; Râż) is the polynomial approximation of the (k)-immersions,is weakly equivalent to an explicit SCd+â-algebra