Intersection of curves on surfaces and their applications to mapping class groups
Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2711-2762.

We introduce an operation which measures the self intersections of paths on an oriented surface. As applications, we give a criterion of the realizability of a generalized Dehn twist, and derive a geometric constraint on the image of the Johnson homomorphisms.

Nous introduisons une opération qui mesure l’auto-intersection des chemins sur une surface orientée. Comme applications, nous donnons un critère de la réalisabilité d’un twist de Dehn généralisé, et nous obtenons une contrainte géométrique sur l’image des homomorphismes de Johnson.

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Accepted:
Published online:
DOI: 10.5802/aif.3001
Classification: 57N05,  20F34,  32G15
Keywords: Goldman bracket, Turaev cobracket, Lie bialgebra, mapping class group, Dehn twist, Johnson homomorphisms
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Kawazumi, Nariya; Kuno, Yusuke. Intersection of curves on surfaces and their applications to mapping class groups. Annales de l'Institut Fourier, Volume 65 (2015) no. 6, pp. 2711-2762. doi : 10.5802/aif.3001. https://aif.centre-mersenne.org/articles/10.5802/aif.3001/

[1] Cahn, Patricia A generalization of the Turaev cobracket and the minimal self-intersection number of a curve on a surface, New York J. Math., Tome 19 (2013), pp. 253-283 http://nyjm.albany.edu:8000/j/2013/19_253.html | MR: 3084705

[2] Cahn, Patricia; Chernov, Vladimir Intersections of loops and the Andersen-Mattes-Reshetikhin algebra, J. Lond. Math. Soc. (2), Tome 87 (2013) no. 3, pp. 785-801 | Article | MR: 3073676 | Zbl: 1303.57019

[3] Chas, Moira Combinatorial Lie bialgebras of curves on surfaces, Topology, Tome 43 (2004) no. 3, pp. 543-568 | Article | MR: 2041630 | Zbl: 1050.57014

[4] Chas, Moira Minimal intersection of curves on surfaces, Geom. Dedicata, Tome 144 (2010), pp. 25-60 | Article | MR: 2580416 | Zbl: 1186.57015

[5] Chas, Moira; Krongold, Fabiana Algebraic characterization of simple closed curves via Turaev’s cobracket (http://arxiv.org/abs/1009.2620)

[6] Chas, Moira; Krongold, Fabiana An algebraic characterization of simple closed curves on surfaces with boundary, J. Topol. Anal., Tome 2 (2010) no. 3, pp. 395-417 | Article | MR: 2718130 | Zbl: 1245.57021

[7] Chernov, V. Graded Poisson algebras on bordism groups of garlands and their applications (http://arxiv.org/abs/math/0608153)

[8] Church, Thomas Orbits of curves under the Johnson kernel, Amer. J. Math., Tome 136 (2014) no. 4, pp. 943-994 | Article | MR: 3245184

[9] Enomote, E. (private communication)

[10] Enomoto, Naoya; Satoh, Takao New series in the Johnson cokernels of the mapping class groups of surfaces, Algebr. Geom. Topol., Tome 14 (2014) no. 2, pp. 627-669 | Article | MR: 3159965

[11] Goldman, William 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

[12] Hain, Richard Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc., Tome 10 (1997) no. 3, pp. 597-651 | Article | MR: 1431828 | Zbl: 0915.57001

[13] Johnson, Dennis A survey of the Torelli group, Low-dimensional topology (San Francisco, Calif., 1981) (Contemp. Math.) Tome 20, Amer. Math. Soc., Providence, RI, 1983, pp. 165-179 | Article | MR: 718141

[14] Kawazumi, N. Cohomological aspects of Magnus expansions (http://arxiv.org/abs/math/0505497)

[15] Kawazumi, Nariya; Kuno, Yusuke Groupoid-theoretical methods in the mapping class groups of surfaces (http://arxiv.org/abs/1109.6479)

[16] Kawazumi, Nariya; Kuno, Yusuke The logarithms of Dehn twists, Quantum Topol., Tome 5 (2014) no. 3, pp. 347-423 | Article | MR: 3283405

[17] Kontsevich, Maxim 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

[18] Kuno, Yusuke A combinatorial construction of symplectic expansions, Proc. Amer. Math. Soc., Tome 140 (2012) no. 3, pp. 1075-1083 | Article | MR: 2869092 | Zbl: 1241.57025

[19] Kuno, Yusuke The generalized Dehn twist along a figure eight, J. Topol. Anal., Tome 5 (2013) no. 3, pp. 271-295 | Article | MR: 3096306 | Zbl: 1282.57025

[20] Massuyeau, Gwénaël 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

[21] Massuyeau, Gwénaël; Turaev, Vladimir (in preparation)

[22] Massuyeau, Gwénaël; Turaev, Vladimir Fox pairings and generalized Dehn twists, Ann. Inst. Fourier (Grenoble), Tome 63 (2013) no. 6, pp. 2403-2456 http://aif.cedram.org/item?id=AIF_2013__63_6_2403_0 | Numdam | MR: 3237452 | Zbl: 1297.57005

[23] Massuyeau, Gwénaël; Turaev, Vladimir Quasi-Poisson structures on representation spaces of surfaces, Int. Math. Res. Not. IMRN (2014) no. 1, pp. 1-64 | MR: 3158528 | Zbl: 1298.53085

[24] Morita, Shigeyuki On the structure and the homology of the Torelli group, Proc. Japan Acad. Ser. A Math. Sci., Tome 65 (1989) no. 5, pp. 147-150 http://projecteuclid.org/euclid.pja/1195512903 | MR: 1011856 | Zbl: 0684.57006

[25] Morita, Shigeyuki Mapping class groups of surfaces and three-dimensional manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (1991), pp. 665-674 | MR: 1159253 | Zbl: 0743.57007

[26] Morita, Shigeyuki Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J., Tome 70 (1993) no. 3, pp. 699-726 | Article | MR: 1224104 | Zbl: 0801.57011

[27] Morita, Shigeyuki Cohomological structure of the mapping class group and beyond, Problems on mapping class groups and related topics (Proc. Sympos. Pure Math.) Tome 74, Amer. Math. Soc., Providence, RI, 2006, pp. 329-354 | Article | MR: 2264550 | Zbl: 1304.57030

[28] 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., 1975, p. 261-292. Ann. of Math. Studies, No. 84 | MR: 388396 | Zbl: 0325.55002

[29] Putman, Andrew The Johnson homomorphism and its kernel (http://arxiv.org/abs/0904.0467, to appear in J. Reine Angew. Math.)

[30] Putman, Andrew Cutting and pasting in the Torelli group, Geom. Topol., Tome 11 (2007), pp. 829-865 | Article | MR: 2302503 | Zbl: 1157.57010

[31] Schedler, Travis A Hopf algebra quantizing a necklace Lie algebra canonically associated to a quiver, Int. Math. Res. Not. (2005) no. 12, pp. 725-760 | Article | MR: 2146606 | Zbl: 1079.16028

[32] Turaev, V. Intersections of loops in two-dimensional manifolds, Mat. Sb., Tome 106(148) (1978) no. 4, pp. 566-588 | MR: 507817 | Zbl: 0384.57004

[33] Turaev, Vladimir G. Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. École Norm. Sup. (4), Tome 24 (1991) no. 6, pp. 635-704 | Numdam | MR: 1142906 | Zbl: 0758.57011

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