Publications and Preprints
ON OPERADIC ACTIONS ON SPACES OF KNOTS AND 2-LINKS​
with Etienne Batelier
2020, arXiv:2005.06036
Abstract: In the present work, we realize the space of string 2-links L as a free algebra over a colored operad denoted SCL (for "Swiss-Cheese for links"). This result extends works of Burke and Koytcheff about the quotient of 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.
Projective and Reedy model category structures for (infinitesimal) bimodules over an operad
with Benoit Fresse
and Victor Turchin
2021 (v2), arXiv:1911.03890
Abstract: 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.​
Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules
with Ricardo Campos
and Najib Idrissi
2021 (v2), arXiv:1911.09474
Abstract: In this paper we develop the combinatorics of leveled trees in order to construct explicit resolutions of (co)operads and (co)operadic (co)bimodules. We construct explicit cofibrant resolutions of operads and operadic bimodules in spectra analogous to the ordinary Boardman--Vogt resolutions and we express them as a cobar construction of indecomposable elements. Dually, in the context of CDGAs, we perform similar constructions to obtain fibrant resolutions of Hopf cooperads and Hopf cooperadic cobimodules and we express them as a bar construction of primitive elements.​
On the delooping of (framed) embedding spaces
with Victor Turchin
and Thomas Willwacher
2021 (v3), arXiv:1811.12816
Abstract: 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.
Delooping of high-dimensional spaces of string links
2018, arXiv:1809.00682
Abstract: We study a connection between a multivariable version of the Goodwillie-Weiss' calculus of functors and derived mapping spaces of k-fold bimodules over a family of operads. As our main application, under the assumption di+3≤n for all i∈{1,…,k}, we produce explicit deloopings of high-dimensional spaces of string links modulo immersions from ⊔iRdi to Rn and their polynomial approximations.
A model for framed configuration spaces of points
with Ricardo Campos
and Najib Idrissi
and Thomas Willwacher
2018, arXiv:1807.08319
Abstract: We study configuration spaces of framed points on compact manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of manifolds and in factorization homology. We construct real combinatorial models for these operadic modules, for compact smooth manifolds without boundary.
Delooping the functor calculus tower
with Victor Turchin
2019 (v4), arXiv:1708.02203
Abstract: We study a connection between mapping spaces of bimodules and of infinitesimal bimodules over an operad. As main application and motivation of our work, we produce an explicit delooping of the manifold calculus tower associated to the space of smooth maps Dm→Dn of discs, n≥m, avoiding any given multisingularity and coinciding with the standard inclusion near ∂Dm. In particular, we give a new proof of the delooping of the space of disc embeddings in terms of little discs operads maps with the advantage that it can be applied to more general mapping spaces.
Delooping derived mapping spaces of bimodules over an operad
Journal of Homotopy and Related Structure, 2018.
Abstract: From a map of operads η:O→O′, we introduce a cofibrant replacement of the operad O in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules BimodhO(O;O′) is an algebra over the one dimensional little cubes operad C1. In the present work, we also build an explicit weak equivalence of C1-algebras from the loop space ΩOperadh(O;O′) to BimodhO(O;O′).
From maps between coloured operads to Swiss-Cheese algebras
Annales de l’Institut Fourier Vol 68, 2018.
Abstract: 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.
Swiss-cheese action on the totalization of operads under the monoid actions actions operad
Algebraic & Geometric Topology Vol 16, 2016.
Abstract: We prove that if a pair of semi-cosimplicial spaces (X,Y) arise from a coloured operad then the semi-totalization sTot(Y) has the homotopy type of a relative double loop space and the pair (sTot(X),sTot(Y)) is weakly equivalent to an explicit algebra over the two dimensional Swiss-cheese operad.