Diagrams for primitive cycles in spaces of pure braids and string links
Annales de l'Institut Fourier, Online first, 63 p.

The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich’s CDGA of diagrams and Chen’s iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of string links. We conjecture that graphing is injective on all rational homotopy classes.

L’espace des lacets basés d’un espace de configurations des points dans un espace euclidien peut être vu comme un espace de tresses pures dans un espace euclidien d’une dimension superieure. Nous continuons notre travail sur ces espaces du point de vue de l’ADGC de Kontsevich des diagrammes et des intégrales de Chen. Nous construisons une connexion série puissance qui donne une isomorphisme d’algèbre de Hopf entre l’homologie de l’espace des tresses pures et la construction cobar sur les diagrammes. Il mappe les classes primitives aux arbres trivalents modulo la relation IHX. En conséquence, nous établissons une bijection entre les invariants de Milnor des entrelacs sphériques bruniens et certaines intégrales de Chen. Enfin nous montrons que le graphe induit des injections d’un certain sous-module de l’homotopie des espaces de configurations dans l’homotopie des espaces de longs entrelacs. Nous conjecturons qu’il induit des injections de toute l’homotopie rationelle des espaces de configurations.

Revised after acceptance:
Online First:
DOI: 10.5802/aif.3616
Classification: 55P35, 55R80, 57K45, 20F36, 58D10, 81Q30
Keywords: Spaces of braids, loop spaces, bar and cobar constructions, configuration space integrals, Chen’s iterated integrals, formality, graph complexes, spaces of high-dimensional string links, generalized Milnor invariants.
Mot clés : Espaces de tresses, espaces de lacets, constructions bar et cobar, intégrales d’espaces de configurations, intégrales itérées de Chen, formalité, complexes de graphes, espaces de longs entrelacs de grande dimension, invariants de Milnor généralisés.
Komendarczyk, Rafal 1; Koytcheff, Robin 2; Volić, Ismar 3

1 Department of Mathematics, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118
2 Department of Mathematics, University of Louisiana at Lafayette, PO Box 43568, Lafayette, LA 70504
3 Department of Mathematics, Wellesley College, 106 Central Street, Wellesley, MA 02481
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Komendarczyk, Rafal; Koytcheff, Robin; Volić, Ismar. Diagrams for primitive cycles in spaces of pure braids and string links. Annales de l'Institut Fourier, Online first, 63 p.

[1] Arnold, Vladimir I. The cohomology ring of the group of dyed braids, Mat. Zametki, Volume 5 (1969), pp. 227-231 | MR

[2] Arone, Gregory; Krushkal, Vyacheslav Embedding obstructions in d from the Goodwillie–Weiss calculus and Whitney disks (2021) (https://arxiv.org/abs/2101.10995)

[3] Artin, Emil Theory of braids, Ann. Math., Volume 48 (1947), pp. 101-126 | DOI | MR | Zbl

[4] Bar-Natan, Dror On the Vassiliev knot invariants, Topology, Volume 34 (1995) no. 2, pp. 423-472 | DOI | MR | Zbl

[5] Boardman, J. Michael; Steer, Brian On Hopf invariants, Comment. Math. Helv., Volume 42 (1967), pp. 180-221 | DOI | MR | Zbl

[6] Bott, Raoul; Tu, Loring W. Differential forms in algebraic topology, Graduate Texts in Mathematics, 82, Springer, 1982, xiv+331 pages | DOI | MR | Zbl

[7] Budney, Ryan; Conant, James; Koytcheff, Robin; Sinha, Dev Embedding calculus knot invariants are of finite type, Algebr. Geom. Topol., Volume 17 (2017) no. 3, pp. 1701-1742 | DOI | MR | Zbl

[8] Budney, Ryan; Conant, James; Scannell, Kevin; Sinha, Dev New perspectives on self-linking, Adv. Math., Volume 191 (2005) no. 1, pp. 78-113 | DOI | MR | Zbl

[9] Chen, Kuo-Tsai Iterated integrals of differential forms and loop space homology, Ann. Math., Volume 97 (1973), pp. 217-246 | DOI | MR | Zbl

[10] Chen, Kuo-Tsai Extension of C function algebra by integrals and Malcev completion of π 1 , Adv. Math., Volume 23 (1977) no. 2, pp. 181-210 | DOI | MR | Zbl

[11] Chen, Kuo-Tsai Iterated path integrals, Bull. Am. Math. Soc., Volume 83 (1977) no. 5, pp. 831-879 | DOI | MR | Zbl

[12] Chen, Kuo-Tsai; Fox, Ralph H.; Lyndon, Roger C. Free differential calculus. IV. The quotient groups of the lower central series, Ann. Math., Volume 68 (1958), pp. 81-95 | DOI | MR | Zbl

[13] Chibrikov, E. S. A right normed basis for free Lie (super)algebras, Algebra and model theory 5, Novosibirsk State Technical University, 2005, p. 28-38, 281 | MR | Zbl

[14] Cohen, Frederick R.; Gitler, Samuel On loop spaces of configuration spaces, Trans. Am. Math. Soc., Volume 354 (2002) no. 5, pp. 1705-1748 | DOI | MR | Zbl

[15] Cohen, Frederick R.; Hemmer, David J.; Nakano, Daniel K. The Lie module and its complexity, Bull. Lond. Math. Soc., Volume 48 (2016) no. 1, pp. 109-114 | DOI | MR | Zbl

[16] Cohen, Frederick R.; Lada, Thomas J.; May, J. Peter The homology of iterated loop spaces, Lecture Notes in Mathematics, 533, Springer, 1976, vii+490 pages | DOI | MR | Zbl

[17] Conant, James Homotopy approximations to the space of knots, Feynman diagrams, and a conjecture of Scannell and Sinha, Am. J. Math., Volume 130 (2008) no. 2, pp. 341-357 | DOI | MR | Zbl

[18] Conant, James; Schneiderman, Rob; Teichner, Peter Tree homology and a conjecture of Levine, Geom. Topol., Volume 16 (2012) no. 1, pp. 555-600 | DOI | MR | Zbl

[19] Deligne, Pierre; Griffiths, Phillip; Morgan, John; Sullivan, Dennis Real homotopy theory of Kähler manifolds, Invent. Math., Volume 29 (1975) no. 3, pp. 245-274 | DOI | Zbl

[20] DeTurck, Dennis; Gluck, Herman; Komendarczyk, Rafal; Melvin, Paul; Shonkwiler, Clayton; Vela-Vick, David Shea Generalized Gauss maps and integrals for three-component links: Toward higher helicities for magnetic fields and fluid flows, J. Math. Phys., Volume 54 (2013) no. 1, 013515, 48 pages | DOI | MR | Zbl

[21] Elliott, Robin Iterated Integrals in Quantitative Topology (2020) (https://arxiv.org/abs/2012.08937)

[22] Fadell, Edward R.; Husseini, Sufian Y. Geometry and topology of configuration spaces, Springer Monographs in Mathematics, Springer, 2001, xvi+313 pages | DOI | MR | Zbl

[23] Félix, Yves; Halperin, Stephen; Thomas, Jean-Claude Rational homotopy theory, Graduate Texts in Mathematics, 205, Springer, 2001, xxxiv+535 pages | DOI | MR

[24] Freedman, Michael; Krushkal, Vyacheslav Geometric complexity of embeddings in d , Geom. Funct. Anal., Volume 24 (2014) no. 5, pp. 1406-1430 | DOI | MR | Zbl

[25] Fresse, Benoit; Turchin, Victor; Willwacher, Thomas On the rational homotopy type of embedding spaces of manifolds in n (2020) (https://arxiv.org/abs/2008.08146)

[26] Habegger, Nathan; Masbaum, Gregor The Kontsevich integral and Milnor’s invariants, Topology, Volume 39 (2000) no. 6, pp. 1253-1289 | DOI | MR | Zbl

[27] Habegger, Nathan; Pitsch, Wolfgang Tree level Lie algebra structures of perturbative invariants, J. Knot Theory Ramifications, Volume 12 (2003) no. 3, pp. 333-345 | DOI | MR | Zbl

[28] Haefliger, André Whitehead products and differential forms, Differential topology, foliations and Gelfand-Fuks cohomology (Proc. Sympos., Pontifícia Univ. Católica, Rio de Janeiro, 1976) (Lecture Notes in Mathematics), Volume 652, Springer (1978), pp. 13-24 | DOI | MR | Zbl

[29] Hain, Richard M. Iterated integrals and homotopy periods, Memoirs of the American Mathematical Society, 47, American Mathematical Society, 1984 no. 291, iv+98 pages | DOI | MR | Zbl

[30] Hatcher, Allen Algebraic topology, Cambridge University Press, 2002, xii+544 pages | MR | Zbl

[31] Kohno, Toshitake Série de Poincaré–Koszul associée aux groupes de tresses pures, Invent. Math., Volume 82 (1985) no. 1, pp. 57-75 | DOI | MR | Zbl

[32] Kohno, Toshitake Monodromy representations of braid groups and Yang-Baxter equations, Ann. Inst. Fourier, Volume 37 (1987) no. 4, pp. 139-160 | DOI | Numdam | MR | Zbl

[33] Kohno, Toshitake Vassiliev invariants of braids and iterated integrals, Arrangements – Tokyo 1998 (Advanced Studies in Pure Mathematics), Volume 27, Kinokuniya, Tokyo, 2000, pp. 157-168 | DOI | MR | Zbl

[34] Kohno, Toshitake Loop spaces of configuration spaces and finite type invariants, Invariants of knots and 3-manifolds (Kyoto, 2001) (Geometry and Topology Monographs), Volume 4, Geometry and Topology Publications, 2002, pp. 143-160 | DOI | MR | Zbl

[35] Kohno, Toshitake Bar complex, configuration spaces and finite type invariants for braids, Topology Appl., Volume 157 (2010) no. 1, pp. 2-9 | DOI | MR | Zbl

[36] Komendarczyk, Rafal The third order helicity of magnetic fields via link maps, Commun. Math. Phys., Volume 292 (2009) no. 2, pp. 431-456 | DOI | MR | Zbl

[37] Komendarczyk, Rafal The third order helicity of magnetic fields via link maps. II, J. Math. Phys., Volume 51 (2010) no. 12, 122702, 16 pages | DOI | MR | Zbl

[38] Komendarczyk, Rafal; Koytcheff, Robin; Volić, Ismar Diagram Complexes, Formality, and Configuration Space Integrals for Spaces of Braids, Q. J. Math., Volume 71 (2020) no. 2, pp. 729-779 | DOI | MR | Zbl

[39] Kontsevich, Maxim Operads and motives in deformation quantization, Lett. Math. Phys., Volume 48 (1999) no. 1, pp. 35-72 | DOI | MR | Zbl

[40] Kosanović, Danica Embedding calculus and grope cobordism of knots (2020) (https://arxiv.org/abs/2010.05120)

[41] Koschorke, Ulrich A generalization of Milnor’s μ-invariants to higher-dimensional link maps, Topology, Volume 36 (1997) no. 2, pp. 301-324 | DOI | MR | Zbl

[42] Koytcheff, Robin; Volić, Ismar Milnor invariants of string links, trivalent trees, and configuration space integrals, Topology Appl., Volume 251 (2019), pp. 47-69 | DOI | MR | Zbl

[43] Kuperberg, Greg; Thurston, Dylan P. Perturbative 3-manifold invariants by cut-and-paste topology (1999) (https://arxiv.org/abs/math/9912167)

[44] Lambrechts, Pascal; Turchin, Victor Homotopy graph-complex for configuration and knot spaces, Trans. Am. Math. Soc., Volume 361 (2009) no. 1, pp. 207-222 | DOI | MR | Zbl

[45] Lambrechts, Pascal; Volić, Ismar Formality of the little N-disks operad, Memoirs of the American Mathematical Society, 230, American Mathematical Society, 2014 no. 1079, viii+116 pages | DOI | MR | Zbl

[46] Levine, Jerome Addendum and correction to: “Homology cylinders: an enlargement of the mapping class group”, Algebr. Geom. Topol., Volume 2 (2002), pp. 1197-1204 | DOI | MR | Zbl

[47] Lin, Xiao-Song Power series expansions and invariants of links, Geometric topology (Athens, GA, 1993) (AMS/IP Studies in Advanced Mathematics), Volume 2, American Mathematical Society, 1997, pp. 184-202 | MR | Zbl

[48] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald Combinatorial group theory. Presentations of groups in terms of generators and relations, Dover Publications, 2004, xii+444 pages | MR | Zbl

[49] Milnor, John W.; Moore, John C. On the structure of Hopf algebras, Ann. Math., Volume 81 (1965), pp. 211-264 | DOI | MR | Zbl

[50] Orr, Kent E. Homotopy invariants of links, Invent. Math., Volume 95 (1989) no. 2, pp. 379-394 | DOI | MR | Zbl

[51] Petrogradsky, Victor M. Witt’s formula for restricted Lie algebras, Adv. Appl. Math., Volume 30 (2003) no. 1-2, pp. 219-227 Formal power series and algebraic combinatorics (Scottsdale, AZ, 2001) | DOI | MR | Zbl

[52] Sinha, Dev A pairing between graphs and trees (2005) (https://arxiv.org/abs/math/0502547)

[53] Sinha, Dev The topology of spaces of knots: cosimplicial models, Am. J. Math., Volume 131 (2009) no. 4, pp. 945-980 | DOI | MR | Zbl

[54] Sinha, Dev; Walter, Ben Lie coalgebras and rational homotopy theory II: Hopf invariants, Trans. Am. Math. Soc., Volume 365 (2012) no. 2, pp. 861-883 | DOI | MR | Zbl

[55] Songhafouo Tsopméné, Paul Arnaud; Turchin, Victor Euler characteristics for spaces of string links and the modular envelope of , Homology Homotopy Appl., Volume 20 (2018) no. 2, pp. 115-144 | DOI | MR | Zbl

[56] Songhafouo Tsopméné, Paul Arnaud; Turchin, Victor Rational homology and homotopy of high-dimensional string links, Forum Math., Volume 30 (2018) no. 5, pp. 1209-1235 | DOI | MR | Zbl

[57] Suciu, Alexander I.; Wang, He Formality properties of finitely generated groups and Lie algebras, Forum Math., Volume 31 (2019) no. 4, pp. 867-905 | DOI | MR | Zbl

[58] Suciu, Alexander I.; Wang, He Taylor expansions of groups and filtered-formality, Eur. J. Math., Volume 6 (2020) no. 3, pp. 1073-1096 | DOI | MR | Zbl

[59] Sullivan, Dennis Infinitesimal computations in topology, Publ. Math., Inst. Hautes Étud. Sci. (1977) no. 47, pp. 269-331 | DOI | Numdam | MR | Zbl

[60] Walter, Ben The configuration basis of a Lie algebra and its dual, 2010 (https://arxiv.org/abs/1010.4765)

[61] Willerton, Simon On the Vassiliev invariants for knots and for pure braids, Ph. D. Thesis, University of Edinburgh (1997) (available at http://simonwillerton.staff.shef.ac.uk/ftp/thesis.pdf)

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