Delooping derived mapping spaces of bimodules over an operad

Abstract

From a map of operads η : O → O , we introduce a cofibrant replacement of O in the category of bimodules over itself such that the corresponding model of the derived mapping space of bimodules Bimod h O (O ; O )is an algebra over the one dimensional little cubes operad C 1 . In the present work, we also build an explicit weak equivalence of C 1 -algebras from the loop spaceIn particular, if we apply the identifications (3) to the map η : C d → C n , then we get an explicit description of the iterated loop space associated to the space of long embeddings and their polynomial approximations:Applications to the Swiss-Cheese operad. Theorem 3.1 is also used by the author [10] in order to extend the previous results to the coloured case using the Swiss-Cheese operad SC d which is a relative version of the little cubes operad C d . In that case, a typical example of SC d -algebra is a pair of topological spaces (since the operad has two colours S = {o, ; c}) of the formwhere f : Y → X is a continuous map between pointed spaces. In particular, if (A ; B) is an SC d -algebra, then A is a C d -algebra, B is a C d−1 -algebra and there is a map τ : A → B which is more or less central up to homotopy (i.e. τ preserves the product and τ(a) × b b × τ(a) with a ∈ A and b ∈ B). In [10], we give a relative version of the delooping (2) such that, together with Theorem 3.1, we are able to identify explicit SC d+1 -algebras. In particular, if η 1 : C d → O is a map of operads and η 2 : O → M is a map of bimodules over O, then, under technical conditions, the pair of spacesis proved to be weakly equivalent to a typical SC d+1 -algebra using the identifications

On the delooping of (framed) embedding spaces

“…Our main result Theorem 4.1 is then that the following is a homotopy fiber sequence (1) Bimod h P (P, Q•X) → X → Operad h (P, Q), where the superscript h is used to show that we consider the derived version of the corresponding mapping spaces. This result can be considered as a generalisation of the delooping result [7,11] (2)…”

“…The operad P may naturally be considered as a reduced bimodule of itself. We will use (a slight variant of) the cofibrant resolution BP of P as a bimodule introduced by Ducoulombier in [7]. The points are equivalence classes [T ; {t v } ; {x v }] where T is a tree, {t v } is a family of real numbers in the interval [0 , 1] indexing the vertices and {x v } is a family of points in WP labelling the vertices.…”

“…In the previous subsection we introduced a cofibrant replacement BP of an operad P in the category of bimodules over WP. In [7], the author uses this resolution in order to equip the corresponding model of the derived mapping space of bimodules with a structure of D 1 -algebra. Then, he shows the following statement: Theorem 3.4.…”

“…Proof. The first truncation map (22) has been proved to be a Serre fibration by the first author in [7] (using projective model category structures). More precisely, we show that each truncation map of the form…”

“…Application 3: The Goodwillie-Weiss calculus. The deloopings (3), ( 4), (7) were obtained in [3,9] using the Goodwillie-Weiss functor calculus on manifolds. In fact one obtains there the deloopings of the Taylor towers…”