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Boardman-Vogt resolutions and bar/cobar constructions of (co)operadic (co)bimodules. (English) Zbl 1483.18024

Given two smooth manifolds \(M\) and \(N\), \(\mathrm{Emb}(M,N)\) denotes the space of smooth embeddings \(M\rightarrow N\) endowed with the compact-open topology. It is difficult to determine its homotopy type or even its rational homotopy type. It is the principal object in knot theory to understand the connected components of \(\mathrm{Emb}(S^{1} ,S^{3})\). Two key ingredients are involved in the authors’ approach to \(\mathrm{Emb}(M,N)\).
1.
One is Goodwillie-Weise calculus [T. G. Goodwillie and M. Weiss, Geom. Topol. 3, 103–118 (1999; Zbl 0927.57028)], which gives information about \(\mathrm{Emb}(M,N)\) in case of \(\dim N-\dim M\geq3\), allowing of expressing \(\mathrm{Emb}(M,N)\) as the limit of a tower of polynomial approximations \(T_{k}\mathrm{Emb}(M,N)\).
2.
The other is operads [M. Markl et al., Operads in algebra, topology and physics. Providence, RI: American Mathematical Society (AMS) (2002; Zbl 1017.18001)], which are combinatorial objects encoding algebraic structures. The little disks operads \(E_{n}\), introduced initially in the study of iterated loop spaces [J. P. May, The geometry of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009); J. M. Boardman and R. M. Vogt, Bull. Am. Math. Soc. 74, 1117–1122 (1968; Zbl 0165.26204)], play a central role in this theory. An element of \(E_n\) is a configuration of numbered disjoint \(n\)-disks in the unit \(n\)-disk. The operadic structure consists in plugging a configuration of disks inside one of the disks of another configuration. Algebras over these little disks operads are precisely spaces of the homotopy type of iterated loop spaces under some technical conditions.

Ways of computing the rational homotopy type of \(\mathrm{Emb}(M,N)\) with combinatorial methods have been developed using these two ingredients, particularly, for the space of compactly supported embeddings \(\mathrm{Emb}_{c}(\mathbb{R}^{d},\mathbb{R}^{n})\), which was shown to be weakly equivalent to the \((d+1)\)-iterated space of the derived mapping space of operads \(\mathrm{Operad}^{h}(E_{d} ,E_{n})\) [P. B. De Brito and M. Weiss, J. Topol. 11, No. 1, 65–143 (2018; Zbl 1390.57018); J. Ducoulombier and V. Turchin, “Delooping the functor calculus tower”, Preprint, arXiv:1708.02203], which, beholden to Suillivan’s rational homotopy theory and the formality of the little disks operads, is to be expressed in terms of hairy graph complexes [B. Fresse et al., “The rational homotopy of mapping spaces of \(E_n\) operads”, Preprint, arXiv:1703.06123], where the Boardman-Vogt construction of a cooperad was identified with the bar construction of an explicit operad.
The principal objective in this paper is to extend these computations to study the space of compactly supported embeddings \(\mathrm{Emb}_{c}(\mathbb{R}^{d_{1}}\sqcup\cdots\sqcup\mathbb{R}^{d_{k}},\mathbb{R}^{n})\) or the space of string links. Goodwillie-Weise calculus is to be replaced by multivariable Goodwillie-Weise calculus, with which the space of string links should be expressed as a mapping space of operadic bimodules [J. Ducoulombier, “Delooping of high-dimensional spaces of string links”, Preprint, arXiv:1809.00682]. The authors extend the Boardman-Vogt resolution and the bar-cobar resolution to deal with (co)operadic (co)bimodules. In classical Boardman-Vogt construction for (co)operads, free constructions are to be defined using planar trees, while, for bimodules, one has to consider categories of leveled trees, i.e., trees whose leaves are all at the same height.

MSC:

18M70 Algebraic operads, cooperads, and Koszul duality
18M75 Topological and simplicial operads
18N40 Homotopical algebra, Quillen model categories, derivators
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