Fox pairings and generalized Dehn twists
Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2403-2456.

We introduce a notion of a Fox pairing in a group algebra and use Fox pairings to define automorphisms of the Malcev completions of groups. These automorphisms generalize to the algebraic setting the action of the Dehn twists in the group algebras of the fundamental groups of surfaces. This work is inspired by the Kawazumi–Kuno generalization of the Dehn twists to non-simple closed curves on surfaces.

Nous introduisons la notion de “forme de Fox” sur une algèbre de groupe et nous utilisons les formes de Fox pour définir des automorphismes des complétés de Malcev de groupes. Ces automorphismes étendent au cadre algébrique l’action des twists de Dehn sur les algèbres de groupes fondamentaux de surfaces. Ce travail s’inspire de la généralisation des twists de Dehn par Kawazumi–Kuno aux courbes fermées non-simples sur les surfaces.

DOI: 10.5802/aif.2834
Classification: 57M05, 57N05, 20F28, 20F34, 20F38
Keywords: surface, mapping class group, Dehn twist, group, Malcev completion, Fox derivative
Mot clés : surface, groupe de difféotopie, twist de Dehn, groupe, complété de Malcev, dérivation de Fox
Massuyeau, Gwénaël 1; Turaev, Vladimir 2

1 IRMA, Université de Strasbourg & CNRS 7 rue René Descartes 67084 Strasbourg, France
2 Department of Mathematics Indiana University Bloomington IN47405, USA
     author = {Massuyeau, Gw\'ena\"el and Turaev, Vladimir},
     title = {Fox pairings and generalized {Dehn} twists},
     journal = {Annales de l'Institut Fourier},
     pages = {2403--2456},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     doi = {10.5802/aif.2834},
     zbl = {1297.57005},
     mrnumber = {3237452},
     language = {en},
     url = {}
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Massuyeau, Gwénaël; Turaev, Vladimir. Fox pairings and generalized Dehn twists. Annales de l'Institut Fourier, Volume 63 (2013) no. 6, pp. 2403-2456. doi : 10.5802/aif.2834.

[1] Epstein, D. B. A. Curves on 2-manifolds and isotopies, Acta Math., Volume 115 (1966), pp. 83-107 | DOI | MR | Zbl

[2] Garoufalidis, S.; Levine, J. Tree-level invariants of three-manifolds, Massey products and the Johnson homomorphism, Graphs and patterns in mathematics and theoretical physics (Proc. Sympos. Pure Math.), Volume 73, Amer. Math. Soc., Providence, RI, 2005, pp. 173-203 | MR | Zbl

[3] Goldman, W. M. Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., Volume 85 (1986) no. 2, pp. 263-302 | DOI | MR | Zbl

[4] Habegger, N. Milnor, Johnson and the tree-level perturbative invariants Preprint (2000), University of Nantes

[5] Jennings, S. A. The group ring of a class of infinite nilpotent groups, Canad. J. Math., Volume 7 (1955), pp. 169-187 | DOI | MR | Zbl

[6] Kawazumi, N. Cohomological aspects of Magnus expansions preprint (2005) arXiv:math/0505497

[7] Kawazumi, N.; Kuno, Y. Groupoid-theoretical methods in the mapping class groups of surfaces preprint (2011) arXiv:1109.6479

[8] Kawazumi, N.; Kuno, Y. The logarithms of Dehn twists preprint (2010) arXiv:1008.5017

[9] Kontsevich, M. Formal (non)commutative symplectic geometry, The Gel’fand Mathematical Seminars, 1990–1992, Birkhäuser Boston, Boston, MA, 1993, pp. 173-187 | MR | Zbl

[10] Kuno, Y. The generalized Dehn twist along a figure eight preprint (2011) arXiv:1104.2107

[11] Magnus, W.; Karrass, A.; Solitar, D. Combinatorial group theory. Presentations of groups in terms of generators and relations, Dover Publications, Inc., New York, 1976 | MR | Zbl

[12] Massuyeau, G. Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France, Volume 140 (2012) no. 1, pp. 101-161 | Numdam | MR | Zbl

[13] Morita, S. Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces, Groups of diffeomorphisms (Adv. Stud. Pure Math.), Volume 52, Math. Soc. Japan, Tokyo, 2008, pp. 443-468 | MR | Zbl

[14] Papakyriakopoulos, C. D. Planar regular coverings of orientable closed surfaces, Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox) (Ann. of Math. Studies), Princeton Univ. Press, Princeton, N.J., 1975 no. 84, pp. 261-292 | MR | Zbl

[15] Perron, B. A homotopic intersection theory on surfaces: applications to mapping class group and braids, Enseign. Math. (2), Volume 52 (2006) no. 1-2, pp. 159-186 | MR | Zbl

[16] Quillen, D. Rational homotopy theory, Ann. of Math. (2), Volume 90 (1969), pp. 205-295 | DOI | MR | Zbl

[17] Turaev, V. G. Intersections of loops in two-dimensional manifolds, (Russian) Mat. Sb, Volume 106(148) (1978), pp. 566-588 English translation: Math. USSR, Sb. 35 (1979), 229–250 | MR | Zbl

[18] Turaev, V. G. Multiplace generalizations of the Seifert form of a classical knot, (Russian) Mat. Sb, Volume 116(158) (1981), pp. 370-397 English translation: Math. USSR, Sb. 44 (1983), 335–361 | MR | Zbl

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