Fox pairings and generalized Dehn twists  [ Formes de Fox et twists de Dehn généralisés ]
Annales de l'Institut Fourier, Tome 63 (2013) no. 6, p. 2403-2456
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.
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.
DOI : https://doi.org/10.5802/aif.2834
Classification:  57M05,  57N05,  20F28,  20F34,  20F38
Mots clés: surface, groupe de difféotopie, twist de Dehn, groupe, complété de Malcev, dérivation de Fox
@article{AIF_2013__63_6_2403_0,
     author = {Massuyeau, Gw\'ena\"el and Turaev, Vladimir},
     title = {Fox pairings and generalized Dehn twists},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {63},
     number = {6},
     year = {2013},
     pages = {2403-2456},
     doi = {10.5802/aif.2834},
     mrnumber = {3237452},
     zbl = {1297.57005},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2013__63_6_2403_0}
}
Massuyeau, Gwénaël; Turaev, Vladimir. Fox pairings and generalized Dehn twists. Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2403-2456. doi : 10.5802/aif.2834. http://www.numdam.org/item/AIF_2013__63_6_2403_0/

[1] Epstein, D. B. A. Curves on 2-manifolds and isotopies, Acta Math., Tome 115 (1966), pp. 83-107 | Article | MR 214087 | Zbl 0136.44605

[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, Amer. Math. Soc., Providence, RI (Proc. Sympos. Pure Math.) Tome 73 (2005), pp. 173-203 | MR 2131016 | Zbl 1086.57013

[3] Goldman, W. M. Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Invent. Math., Tome 85 (1986) no. 2, pp. 263-302 | Article | MR 846929 | Zbl 0619.58021

[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., Tome 7 (1955), pp. 169-187 | Article | MR 68540 | Zbl 0066.01302

[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 1247289 | Zbl 0821.58018

[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 422434 | Zbl 0362.20023

[12] Massuyeau, G. Infinitesimal Morita homomorphisms and the tree-level of the LMO invariant, Bull. Soc. Math. France, Tome 140 (2012) no. 1, pp. 101-161 | Numdam | MR 2903772 | Zbl 1248.57009

[13] Morita, S. Symplectic automorphism groups of nilpotent quotients of fundamental groups of surfaces, Groups of diffeomorphisms, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 52 (2008), pp. 443-468 | MR 2509720 | Zbl 1166.57012

[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), Princeton Univ. Press, Princeton, N.J. (Ann. of Math. Studies) (1975) no. 84, pp. 261-292 | MR 388396 | Zbl 0325.55002

[15] Perron, B. A homotopic intersection theory on surfaces: applications to mapping class group and braids, Enseign. Math. (2), Tome 52 (2006) no. 1-2, pp. 159-186 | MR 2255532 | Zbl 1161.57009

[16] Quillen, D. Rational homotopy theory, Ann. of Math. (2), Tome 90 (1969), pp. 205-295 | Article | MR 258031 | Zbl 0191.53702

[17] Turaev, V. G. Intersections of loops in two-dimensional manifolds, (Russian) Mat. Sb, Tome 106(148) (1978), pp. 566-588 (English translation: Math. USSR, Sb. 35 (1979), 229–250) | MR 507817 | Zbl 0384.57004

[18] Turaev, V. G. Multiplace generalizations of the Seifert form of a classical knot, (Russian) Mat. Sb, Tome 116(158) (1981), pp. 370-397 (English translation: Math. USSR, Sb. 44 (1983), 335–361) | MR 665689 | Zbl 0484.57002