Delooping the functor calculus tower

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 D m → D n of discs, n ≥ m, avoiding any given multisingularity and coinciding with the standard inclusion near the boundary ∂D m . 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.

Little discs operads, graph complexes and Grothendieck-Teichmüller groups

“…But subsequent works on the subject have given the expression of the spaces T k Emb c (R m , R n ) in terms of m + 1-fold iterated loop spaces of operadic mapping spaces stated in the above formula. These finer results have been obtained by Boavida-Weiss [9], for all m ≥ 1, by an improvement of the methods used in the study of the Goodwillie-Weiss calculus of embedding spaces, while other authors have obtained general results on mapping spaces of (truncated) operadic bimodules which permit to recover this homotopy identity between the spaces T k Emb c (R m , R n ) and iterated loop spaces of operadic mapping spaces from the form of the results obtained by Sinha and Arone-Turchin in their works (see the articles of Dwyer-Hess [21] and Turchin [57] for the case m = 1, and the article of Ducoulombier-Turchin [20] for the case of general m ≥ 1).…”

“…The paper [20] establishes the general case of the present theorem. This paper also establishes a delooping result…”

“…To be precise, we have the following general statement: 1.6. Theorem (W. Dwyer and K. Hess [21], V. Turchin [58], J. Ducoulombier and V. Turchin [20]). For any m ≥ 1, and for any operad in topological spaces Q such that Q(0) = Q(1) = * , we have a weak-equivalence:…”

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

“…The paper presents a homotopy theory for the categories of bimodules and infinitesimal bimodules over operads. This theory finds important applications in the manifold functor calculus, specifically in the problems of delooping the functor calculus towers [DT,Duc2,DTW]. It has been well known that the arity zero elements essentially complicate the homotopy theory of such objects.…”

“…One of the advantages of the Reedy model structures is that the cofibrant resolutions are smaller as they do not take into account the arity zero component. This makes the constructions of delooping in [DT,Duc2,DTW] more elegant. Reedy model structures enjoy better homotopy behavior.…”