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

Campos, Ricardo; Ducoulombier, Julien; Idrissi, Najib

High. Struct. 5, No. 1, 310-383 (2021).

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.

Reviewer: Hirokazu Nishimura (Tsukuba)

MSC:

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

Keywords:

operads; bimodules over operads; bar construction; model category; homotopy theory; cofibrant and fibrant resolutions

Citations:

Zbl 0927.57028; Zbl 1017.18001; Zbl 0244.55009; Zbl 0165.26204; Zbl 1390.57018

References:

[1]	Greg Arone and Michael Ching. “Operads and chain rules for the calculus of functors”. In: Astérisque 338 (2011), pp. vi+158. issn: 0303-1179. · Zbl 1239.55004
[2]	Gregory Arone and Victor Turchin. “On the rational homology of high dimensional ana-logues of spaces of long knots”. In: Geom. Topol. 18.3 (2014), pp. 1261-1322. issn: 1465-3060. doi: 10.2140/gt.2014.18.1261. arXiv: 1105.1576. · Zbl 1312.57034 · doi:10.2140/gt.2014.18.1261
[3]	Michael Batanin and Martin Markl. “Operadic categories and Duoidal Deligne”s conjec-ture”. In: Adv. Math. 285 (2015), pp. 1630-1687. doi: 10 . 1016 / j . aim . 2015 . 07 . 008. arXiv: 1404.3886. · Zbl 1360.18009 · doi:10.1016/j.aim.2015.07.008
[4]	Clemens Berger and Ieke Moerdijk. “Axiomatic homotopy theory for operads”. In: Com-ment. Math. Helv. 78.4 (2003), pp. 805-831. issn: 0010-2571. doi: 10.1007/s00014-003-0772-y. · Zbl 1041.18011 · doi:10.1007/s00014-003-0772-y
[5]	Clemens Berger and Ieke Moerdijk. “The Boardman-Vogt resolution of operads in monoidal model categories”. In: Topology 45.5 (2006), pp. 807-849. issn: 0040-9383. doi: 10.1016/ j.top.2006.05.001. · Zbl 1105.18007 · doi:10.1016/j.top.2006.05.001
[6]	J. Michael Boardman and Rainer M. Vogt. Homotopy invariant algebraic structures on topological spaces. Lecture Notes in Mathematics 347. Berlin Heidelberg: Springer, 1973, pp. x+257. isbn: 978-3-540-06479-4. doi: 10.1007/BFb0068547. · Zbl 0285.55012 · doi:10.1007/BFb0068547
[7]	J. Michael Boardman and Rainer M. Vogt. “Homotopy-everything H-spaces”. In: Bull. Amer. Math. Soc. 74 (1968), pp. 1117-1122. issn: 0002-9904. doi: 10.1090/S0002-9904-1968-12070-1. · Zbl 0165.26204 · doi:10.1090/S0002-9904-1968-12070-1
[8]	Pedro Boavida de Brito and Michael S. Weiss. “Spaces of smooth embeddings and config-uration categories”. In: J. Topol. 11.1 (2018), pp. 65-143. issn: 1753-8416. doi: 10.1112/ topo.12048. arXiv: 1502.01640. Pre-published. · Zbl 1390.57018 · doi:10.1112/topo.12048
[9]	Michael Ching. “Bar constructions for topological operads and the Goodwillie derivatives of the identity”. In: Geom. Topol. 9 (2005), 833-933 (electronic). issn: 1465-3060. doi: 10.2140/gt.2005.9.833. arXiv: math/0501429. · Zbl 1153.55006 · doi:10.2140/gt.2005.9.833
[10]	Michael Ching. “Bar-cobar duality for operads in stable homotopy theory”. In: J. Topol. 5.1 (2012), pp. 39-80. issn: 1753-8416. doi: 10.1112/jtopol/jtr027. url: https://doi. org/10.1112/jtopol/jtr027. · Zbl 1319.55003 · doi:10.1112/jtopol/jtr027
[11]	Julien Ducoulombier. Delooping of high-dimensional spaces of string links. 2018. arXiv: 1809.00682. Pre-published. · Zbl 1420.55021
[12]	Julien Ducoulombier. “From maps between coloured operads to Swiss-Cheese algebras”. In: Ann. Inst. Fourier 68.2 (2018), pp. 661-724. issn: 0373-0956. doi: 10.5802/aif.3175. arXiv: 1603.07162. · Zbl 1405.18014 · doi:10.5802/aif.3175
[13]	Julien Ducoulombier. “From maps between coloured operads to Swiss-Cheese algebras”. In: Ann. Inst. Fourier 68.2 (2018), pp. 661-724. issn: 0373-0956. arXiv: 1603.07162. url: http://aif.cedram.org/item?id=AIF_2018__68_2_661_0. · Zbl 1405.18014
[14]	Julien Ducoulombier, Benoit Fresse, and Victor Turchin. Projective and Reedy model cat-egory structures for (infinitesimal) bimodules over an operad. 2019. arXiv: 1911.03890. Pre-published.
[15]	Julien Ducoulombier and Victor Turchin. Delooping the functor calculus tower. 2017. arXiv: 1708.02203. Pre-published.
[16]	A. D. Elmendorf et al. Rings, modules, and algebras in stable homotopy theory. Vol. 47. Mathematical Surveys and Monographs. With an appendix by M. Cole. Providence, RI: American Mathematical Society, 1997, pp. xii+249. isbn: 0-8218-0638-6. · Zbl 0894.55001
[17]	Benoit Fresse. “Koszul duality of operads and homology of partition posets”. In: Homo-topy theory: relations with algebraic geometry, group cohomology, and algebraic K-theory. · Zbl 1077.18007
[18]	Vol. 346. Contemp. Math. Providence, RI: Amer. Math. Soc., 2004, pp. 115-215. doi: 10.1090/conm/346/06287. · doi:10.1090/conm/346/06287
[19]	Benoit Fresse. Homotopy of Operads and Grothendieck-Teichmüller Groups. Vol. 1: The Algebraic Theory and its Topological Background. Mathematical Surveys and Monographs 217. Providence, RI: Americal Mathematical Society, 2017. 532 pp. isbn: 978-1-4704-3481-6. · Zbl 1373.55014
[20]	Benoit Fresse. Homotopy of Operads and Grothendieck-Teichmüller Groups. Vol. 2: The Applications of (Rational) Homotopy Theory Methods. Mathematical Surveys and Mono-graphs 217. Providence, RI: Americal Mathematical Society, 2017. 704 pp. isbn: 978-1-4704-3482-3. · Zbl 1375.55007
[21]	Benoit Fresse, Victor Turchin, and Thomas Willwacher. The rational homotopy of mapping spaces of E n operads. 2017. arXiv: 1703.06123. Pre-published. · Zbl 1405.18015
[22]	Benoit Fresse and Thomas Willwacher. Rational homotopy theory of operad modules. 2019. In preparation. · Zbl 1445.18014
[23]	Thomas G. Goodwillie and Michael Weiss. “Embeddings from the point of view of immer-sion theory : Part II”. In: Geom. Topol. 3 (1999), 103-118 (electronic). doi: 10.2140/gt. 1999.3.103. · Zbl 0927.57028 · doi:10.2140/gt.1999.3.103
[24]	Muriel Livernet. “Koszul duality of the category of trees and bar constructions for operads”. In: Operads and universal algebra. Vol. 9. Nankai Ser. Pure Appl. Math. Theoret. Phys. Hackensack, NJ: World Sci. Publ., 2012, pp. 107-138. doi: 10.1142/9789814365123_0006. arXiv: 1102.3622. · Zbl 1308.18011 · doi:10.1142/9789814365123_0006
[25]	Jean-Louis Loday and Bruno Vallette. Algebraic operads. Grundlehren der mathematischen Wissenschaften 346. Berlin-Heidelberg: Springer, p. 634. isbn: 978-3-642-30361-6. doi: 10. 1007/978-3-642-30362-3. · Zbl 1260.18001 · doi:10.1007/978-3-642-30362-3
[26]	Martin Markl, Steve Shnider, and Jim Stasheff. Operads in algebra, topology and physics. Mathematical Surveys and Monographs 96. Providence, RI: Amer. Math. Soc., p. 349. isbn: 0-8218-2134-2.
[27]	J. Peter May. The geometry of iterated loop spaces. Lectures Notes in Mathematics 271. Berlin: Springer-Verlag, 1972, pp. viii+175. doi: 10.1007/BFb0067491. · Zbl 0244.55009 · doi:10.1007/BFb0067491
[28]	Paolo Salvatore. “Configuration operads, minimal models and rational curves.” PhD thesis. University of Oxford, 1998.