Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, p. 1-62

We continue our investigation of spaces of long embeddings (long embeddings are high-dimensional analogues of long knots). In previous work we showed that when the dimensions are in the stable range, the rational homology groups of these spaces can be calculated as the homology of a direct sum of certain finite graph-complexes, which we described explicitly. In this paper, we establish a similar result for the rational homotopy groups of these spaces. We also put emphasis on the different ways the calculations can be done. In particular we describe three different graph-complexes computing these rational homotopy groups. We also compute the generating functions of the Euler characteristics of the summands in the homological splitting.

On continue notre étude des espaces de plongements longs (les plongements longs sont des analogues en dimension supérieure des nœuds longs). Dans notre travail précédent, on a montré que dans le cas où les dimensions sont dans le rang stable l’homologie rationnelle de ces espaces peut être calculée comme l’homologie d’un certain complexe de graphes que l’on a décrit explicitement. Dans ce travail, on établit un résultat similaire pour les groupes d’homotopie rationnelle de ces espaces. On met aussi un accent sur les différentes façons d’effectuer ces calculs. En particulier, on décrit trois complexes de graphes différents calculant les groupes d’homotopie en question. On calcule également les fonctions génératrices des caractéristiques eulériennes des termes d’une décomposition en somme directe des complexes calculant les groupes d’homologie.

DOI : https://doi.org/10.5802/aif.2924
Classification:  57R40,  57R42,  55P48,  55P62,  18D50
Keywords: Spaces of embeddings, little discs operad, rational homotopy, graph-complexes
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     author = {Arone, Gregory and Turchin, Victor},
     title = {Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     pages = {1-62},
     doi = {10.5802/aif.2924},
     zbl = {1329.57035},
     zbl = {06496533},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2015__65_1_1_0}
}
Arone, Gregory; Turchin, Victor. Graph-complexes computing the rational homotopy of high dimensional analogues of spaces of long knots. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 1-62. doi : 10.5802/aif.2924. http://www.numdam.org/item/AIF_2015__65_1_1_0/

[1] ArnolʼD, V. I. The cohomology ring of the group of dyed braids, Mat. Zametki, Tome 5 (1969), pp. 227-231 | MR 242196 | Zbl 0277.55002

[2] Arone, Greg; Lambrechts, Pascal; Turchin, Victor; Volić, Ismar Coformality and rational homotopy groups of spaces of long knots, Math. Res. Lett., Tome 15 (2008) no. 1, pp. 1-14 | Article | MR 2367169 | Zbl 1148.57033

[3] Arone, Gregory; Lambrechts, Pascal; Volić, Ismar Calculus of functors, operad formality, and rational homology of embedding spaces, Acta Math., Tome 199 (2007) no. 2, pp. 153-198 | Article | MR 2358051 | Zbl 1154.57026

[4] Arone, Gregory; Turchin, Victor On the rational homology of high-dimensional analogues of spaces of long knots, Geom. Topol., Tome 18 (2014) no. 3, pp. 1261-1322 | Article | MR 3228453

[5] Bar-Natan, Dror On the Vassiliev knot invariants, Topology, Tome 34 (1995) no. 2, pp. 423-472 | Article | MR 1318886 | Zbl 0898.57001

[6] Budney, Ryan Little cubes and long knots, Topology, Tome 46 (2007) no. 1, pp. 1-27 | Article | MR 2288724 | Zbl 1114.57003

[7] Budney, Ryan A family of embedding spaces, Groups, homotopy and configuration spaces, Geom. Topol. Publ., Coventry (Geom. Topol. Monogr.) Tome 13 (2008), pp. 41-83 | Article | MR 2508201 | Zbl 1158.57035

[8] Cattaneo, Alberto S.; Cotta-Ramusino, Paolo; Longoni, Riccardo Configuration spaces and Vassiliev classes in any dimension, Algebr. Geom. Topol., Tome 2 (2002), p. 949-1000 (electronic) | Article | MR 1936977 | Zbl 1029.57009

[9] Cattaneo, Alberto S.; Rossi, Carlo A. Wilson surfaces and higher dimensional knot invariants, Comm. Math. Phys., Tome 256 (2005) no. 3, pp. 513-537 | Article | MR 2161270 | Zbl 1101.57012

[10] Cohen, F. R.; Taylor, L. R. On the representation theory associated to the cohomology of configuration spaces, Algebraic topology (Oaxtepec, 1991), Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 146 (1993), pp. 91-109 | Article | MR 1224909 | Zbl 0806.57012

[11] Cohen, Frederick R.; Lada, Thomas J.; May, J. Peter The homology of iterated loop spaces, Springer-Verlag, Berlin-New York, Lecture Notes in Mathematics, Vol. 533 (1976), pp. vii+490 | MR 436146 | Zbl 0334.55009

[12] Conant, James; Gerlits, Ferenc; Vogtmann, Karen Cut vertices in commutative graphs, Q. J. Math., Tome 56 (2005) no. 3, pp. 321-336 | Article | MR 2161246 | Zbl 1187.05029

[13] Dasbach, Oliver T. On the combinatorial structure of primitive Vassiliev invariants. II, J. Combin. Theory Ser. A, Tome 81 (1998) no. 2, pp. 127-139 | Article | MR 1603869 | Zbl 0888.57009

[14] Fresse, Benoit Koszul duality of operads and homology of partition posets, Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic K -theory, Amer. Math. Soc., Providence, RI (Contemp. Math.) Tome 346 (2004), pp. 115-215 | Article | MR 2066499 | Zbl 1077.18007

[15] Gerstenhaber, Murray; Schack, S. D. A Hodge-type decomposition for commutative algebra cohomology, J. Pure Appl. Algebra, Tome 48 (1987) no. 3, pp. 229-247 | Article | MR 917209 | Zbl 0671.13007

[16] Getzler, E.; Jones, J. D. S. Operads, homotopy algebra and iterated integrals for double loop spaces (arXiv:hep-th/9403055)

[17] Hirsch, Morris W. Immersions of manifolds, Trans. Amer. Math. Soc., Tome 93 (1959), pp. 242-276 | Article | MR 119214 | Zbl 0113.17202

[18] Klyachko, A. A. Lie elements in the tensor algebra, Siberian Math. J., Tome 15 (1974), pp. 914-920 | Article | Zbl 0325.15018

[19] Kontsevich, Maxim; Soibelman, Yan Deformations of algebras over operads and the Deligne conjecture, Conférence Moshé Flato 1999, Vol. I (Dijon), Kluwer Acad. Publ., Dordrecht (Math. Phys. Stud.) Tome 21 (2000), pp. 255-307 | MR 1805894 | Zbl 0972.18005

[20] Lambrechts, Pascal; Turchin, Victor Homotopy graph-complex for configuration and knot spaces, Trans. Amer. Math. Soc., Tome 361 (2009) no. 1, pp. 207-222 | Article | MR 2439404 | Zbl 1158.57030

[21] Lambrechts, Pascal; Turchin, Victor; Volić, Ismar The rational homology of spaces of long knots in codimension >2, Geom. Topol., Tome 14 (2010) no. 4, pp. 2151-2187 | Article | MR 2740644 | Zbl 1222.57020

[22] Lambrechts, Pascal; Volić, Ismar Formality of the little N-disks operad, To appear in Memoirs of the AMS (Preprint arXiv:0808.0457)

[23] Lehrer, G. I. Equivariant cohomology of configurations in R d , Algebr. Represent. Theory, Tome 3 (2000) no. 4, pp. 377-384 (Special issue dedicated to Klaus Roggenkamp on the occasion of his 60th birthday) | Article | MR 1808133 | Zbl 1161.57304

[24] Lehrer, G. I.; Solomon, Louis On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, J. Algebra, Tome 104 (1986) no. 2, pp. 410-424 | Article | MR 866785 | Zbl 0608.20010

[25] Loday, Jean-Louis Opérations sur l’homologie cyclique des algèbres commutatives, Invent. Math., Tome 96 (1989) no. 1, pp. 205-230 | Article | MR 981743 | Zbl 0686.18006

[26] Loday, Jean-Louis; Vallette, Bruno Algebraic operads, Springer, Heidelberg, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Tome 346 (2012), pp. xxiv+634 | Article | MR 2954392 | Zbl 1260.18001

[27] Merkulov, Sergei; Vallette, Bruno Deformation theory of representations of prop(erad)s. I, J. Reine Angew. Math., Tome 634 (2009), pp. 51-106 | Article | MR 2560406 | Zbl 1187.18006

[28] Moskovich, Daniel; Ohtsuki, Tomotada Vanishing of 3-loop Jacobi diagrams of odd degree, J. Combin. Theory Ser. A, Tome 114 (2007) no. 5, pp. 919-930 | Article | MR 2333141 | Zbl 1118.57015

[29] Pirashvili, Teimuraz Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. (4), Tome 33 (2000) no. 2, pp. 151-179 | Article | Numdam | MR 1755114 | Zbl 0957.18004

[30] Robinson, Alan; Whitehouse, Sarah The tree representation of Σ n+1 , J. Pure Appl. Algebra, Tome 111 (1996) no. 1-3, pp. 245-253 | Article | MR 1394355 | Zbl 0865.55010

[31] Sakai, Keiichi Configuration space integrals for embedding spaces and the Haefliger invariant, J. Knot Theory Ramifications, Tome 19 (2010) no. 12, pp. 1597-1644 | Article | MR 2755492 | Zbl 1223.57024

[32] Sakai, Keiichi; Watanabe, Tadayuki 1-loop graphs and configuration space integral for embedding spaces, Math. Proc. Cambridge Philos. Soc., Tome 152 (2012) no. 3, pp. 497-533 | Article | MR 2911142 | Zbl 1243.57022

[33] Salvatore, Paolo Knots, operads, and double loop spaces, Int. Math. Res. Not. (2006), pp. Art. ID 13628, 22 | Article | MR 2276349 | Zbl 1131.55004

[34] Ševera, Pavol; Willwacher, Thomas Equivalence of formalities of the little discs operad, Duke Math. J., Tome 160 (2011) no. 1, pp. 175-206 | Article | MR 2838354 | Zbl 1241.18008

[35] Sinha, Dev P. A pairing between graphs and trees (arXiv:math/0502547)

[36] Sinha, Dev P. Operads and knot spaces, J. Amer. Math. Soc., Tome 19 (2006) no. 2, p. 461-486 (electronic) | Article | MR 2188133 | Zbl 1112.57004

[37] Sinha, Dev P. The homology of the little discs operad, Séminaire et Congrès de Société Mathématique de France, Tome 26 (2011), pp. 255-281

[38] Tourtchine, V. On the homology of the spaces of long knots, Advances in topological quantum field theory, Kluwer Acad. Publ., Dordrecht (NATO Sci. Ser. II Math. Phys. Chem.) Tome 179 (2004), pp. 23-52 | Article | MR 2147415 | Zbl 1117.57023

[39] Tourtchine, V. On the other side of the bialgebra of chord diagrams, J. Knot Theory Ramifications, Tome 16 (2007) no. 5, pp. 575-629 | Article | MR 2333307 | Zbl 1151.57029

[40] Turchin, Victor Hodge-type decomposition in the homology of long knots, J. Topol., Tome 3 (2010) no. 3, pp. 487-534 | Article | MR 2684511 | Zbl 1205.57023

[41] Vassiliev, V. A. Complements of discriminants of smooth maps: topology and applications, American Mathematical Society, Providence, RI, Translations of Mathematical Monographs, Tome 98 (1992), pp. vi+208 (Translated from the Russian by B. Goldfarb) | MR 1168473

[42] Watanabe, Tadayuki Configuration space integral for long n-knots and the Alexander polynomial, Algebr. Geom. Topol., Tome 7 (2007), pp. 47-92 | Article | MR 2289804 | Zbl 1133.57016

[43] Weibel, Charles A. An introduction to homological algebra, Cambridge University Press, Cambridge, Cambridge Studies in Advanced Mathematics, Tome 38 (1994), pp. xiv+450 | Article | MR 1269324 | Zbl 0797.18001

[44] Weiss, Michael S. Homology of spaces of smooth embeddings, Q. J. Math., Tome 55 (2004) no. 4, pp. 499-504 | Article | MR 2104688 | Zbl 1065.57030

[45] Whitehouse, S. Gamma Homology of Commutative Algebras and Some Related Representations of the Symmetric Group, Warwick University (1994) (Ph. D. Thesis)

[46] Willwacher, T. M. Kontsevich’s graph complex and the Grothendieck-Teichmueller Lie algebra (To appear in Invent. Math. Preprint arXiv:1009.1654) | MR 3348138