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On the contact mapping class group of the contactization of the \(A_m\)-Milnor fiber. (English. French summary) Zbl 1388.53085

The aim of this paper is to construct an embedding of the full braid group on \(m+1\) strands \(B_{m+1}\), \(m\geq1\), into the contact mapping class group of the contactization \(Q\times S^1\) of the \(A_m\)-Milnor fiber \(Q\). The construction uses the embedding of \(B_{m+1}\) into the symplectic mapping class group of \(Q\) due to M. Khovanov and P. Seidel [J. Am. Math. Soc. 15, No. 1, 203–271 (2002; Zbl 1035.53122)], and a natural lifting homomorphism. In order to show that the composed homomorphism is still injective, the authors use a partially linearized variant of the Chekanov-Eliashberg dga for Legendrians which lie above one another in \(Q\times\mathbb{R}\), reducing the proof to Floer homology. As corollaries they obtain a contribution to the contact isotopy problem for \(Q\times S^1\), as well as the fact that in dimension 4, the lifting homomorphism embeds the symplectic mapping class group of \(Q\) into the contact mapping class group of \(Q\times S^1\).
The paper is organized as follows: Section 1 is an introduction to the subject stating the main result. In Section 2, the authors gather the necessary preliminaries. The main technical tool they use is the dg bimodule associated to a two-component Legendrian link. This is a simplification of the Chekanov-Eliashberg dga, adapted to their geometric situation whereby the components of the link lie one above the other. The authors introduce all necessary algebraic definitions, namely the stable tame isomorphism type and linearized homology, analogously to the standard treatment in Legendrian contact homology. Finally, Section 3 contains the proof of the main result.

MSC:

53D05 Symplectic manifolds (general theory)
53D10 Contact manifolds (general theory)
53D12 Lagrangian submanifolds; Maslov index
53D40 Symplectic aspects of Floer homology and cohomology
53D42 Symplectic field theory; contact homology
32S55 Milnor fibration; relations with knot theory

Citations:

Zbl 1035.53122
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References:

[1] Audin, M., Lafontaine, J. (eds.): Holomorphic curves in symplectic geometry. In: Progress in Mathematics, vol 117. Birkhäuser, Basel (1994) · Zbl 0802.53001
[2] Birman, Joan S.: Braids, links, and mapping class groups. Annals of Mathematics Studies, No. 82. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1974) · Zbl 0297.57001
[3] Bourgeois, F., Chantraine, B.: Bilinearized Legendrian contact homology and the augmentation category. J. Symplectic Geom. 12(3), 553-583 (2014) · Zbl 1308.53119 · doi:10.4310/JSG.2014.v12.n3.a5
[4] Casals, R., Spáčil, O.: Chern-Weil theory and the group of strict contactomorphisms. J. Topol. Anal. 8(1), 59-87 (2016) · Zbl 1339.53075 · doi:10.1142/S1793525316500035
[5] Chekanov, Y.: Differential algebra of Legendrian links. Invent. Math. 150(3), 441-483 (2002) · Zbl 1029.57011 · doi:10.1007/s002220200212
[6] Dimitroglou Rizell, G., Golovko, R.: Estimating the number of Reeb chords using a linear representation of the characteristic algebra. Algebra Geom. Topol. 5, 2885-2918 (2015) · Zbl 1330.53107
[7] Ekholm, T., Etnyre, J., Sullivan, M.: The contact homology of Legendrian submanifolds in \[{\mathbb{R}}^{2n+1}\] R2n+1. J. Differ. Geom. 71(2), 177-305 (2005) · Zbl 1103.53048 · doi:10.4310/jdg/1143651770
[8] Ekholm, T., Etnyre, J., Sullivan, M.: Legendrian contact homology in \[P\times{\mathbb{R}}P\]×R. Trans. Am. Math. Soc 359(7), 3301-3335 (2007). (electronic) · Zbl 1119.53051 · doi:10.1090/S0002-9947-07-04337-1
[9] Ekholm, T., Etnyre, J.B., Sabloff, J.M.: A duality exact sequence for Legendrian contact homology. Duke Math. J. 150(1), 1-75 (2009) · Zbl 1193.53179 · doi:10.1215/00127094-2009-046
[10] Eliashberg, Y., Givental, A., Hofer, H.: Introduction to symplectic field theory. Geom. Funct. Anal. Special Volume(Part II), 560-673 (2000) (GAFA 2000 (Tel Aviv, 1999)) · Zbl 0989.81114
[11] Floer, A.: Morse theory for Lagrangian intersections. J. Differ. Geom. 28(3), 513-547 (1988) · Zbl 0674.57027 · doi:10.4310/jdg/1214442477
[12] Floer, A.: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. 41(6), 775-813 (1988) · Zbl 0633.53058 · doi:10.1002/cpa.3160410603
[13] Frauenfelder, U., Schlenk, F.: Volume growth in the component of the Dehn-Seidel twist. Geom. Funct. Anal. 15(4), 809-838 (2005) · Zbl 1086.37029 · doi:10.1007/s00039-005-0526-7
[14] Giroux, E., Massot, P.: On the contact mapping class group of Legendrian circle bundles. Compos. Math. 153(2), 294-312 (2017) · Zbl 1378.57025 · doi:10.1112/S0010437X16007776
[15] Khovanov, M., Seidel, P.: Quivers, Floer cohomology, and braid group actions. J. Am. Math. Soc. 15(1), 203-271 (2002) · Zbl 1035.53122 · doi:10.1090/S0894-0347-01-00374-5
[16] Massot, P., Niederkrüger, K.: Examples of non-trivial contact mapping classes in all dimensions. Int. Math. Res. Not. 15, 4784-4806 (2016) · Zbl 1404.57045 · doi:10.1093/imrn/rnv396
[17] Milnor, J.: Singular points of complex hypersurfaces. Annals of Mathematics Studies, No. 61. Princeton University Press, Princeton; University of Tokyo Press, Tokyo (1968) · Zbl 0184.48405
[18] Seidel, P.: Floer homology and symplectic isotopy problem. PhD thesis, Oxford (1997) · Zbl 1029.57011
[19] Seidel, P.: Lagrangian two-spheres can be symplectically knotted. J. Differ. Geom. 52(1), 145-171 (1999) · Zbl 1032.53068 · doi:10.4310/jdg/1214425219
[20] Seidel, P.: Graded Lagrangian submanifolds. Bull. Soc. Math. France 128(1), 103-149 (2000) · Zbl 0992.53059 · doi:10.24033/bsmf.2365
[21] Seidel, P.: Fukaya categories and Picard-Lefschetz theory. In: Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich (2008) · Zbl 1159.53001
[22] Viterbo, C.: Intersection de sous-variétés lagrangiennes, fonctionnelles d’action et indice des systèmes hamiltoniens. Bull. Soc. Math. France 115(3), 361-390 (1987) · Zbl 0639.58018 · doi:10.24033/bsmf.2082
[23] Weiwei, W.: Exact Lagrangians in \[A_n\] An-surface singularities. Math. Ann. 359(1-2), 153-168 (2014) · Zbl 1315.53099
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