The augmentation category map induced by exact Lagrangian cobordisms

Pan, Yu¹

¹Mathematics Department, Duke University, Durham, NC, United States

Algebraic and Geometric Topology, Tome 17 (2017) no. 3, pp. 1813-1870

Résumé

To a Legendrian knot, one can associate an A∞ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study this functor and establish a long exact sequence relating the corresponding cohomology of morphisms of the two ends. As applications, we prove that the functor between augmentation categories is injective on the level of equivalence classes of objects and find new obstructions to the existence of exact Lagrangian cobordisms in terms of linearized contact homology and ruling polynomials.

Détail

DOI : 10.2140/agt.2017.17.1813

Classification : 53D42, 57R17, 53D12, 57M50
Keywords: the augmentation category, Lagrangian cobordisms, wrapped Floer homology, Legendrian knots

Bibliographie

[1] J A Baldwin, S Sivek, Invariants of Legendrian and transverse knots in monopole knot homology, preprint (2014)

[2] F Bourgeois, B Chantraine, Bilinearized Legendrian contact homology and the augmentation category, J. Symplectic Geom. 12 (2014) 553 | DOI

[3] F Bourgeois, J M Sabloff, L Traynor, Lagrangian cobordisms via generating families: construction and geography, Algebr. Geom. Topol. 15 (2015) 2439 | DOI

[4] B Chantraine, Lagrangian concordance of Legendrian knots, Algebr. Geom. Topol. 10 (2010) 63 | DOI

[5] B Chantraine, Lagrangian concordance is not a symmetric relation, Quantum Topol. 6 (2015) 451 | DOI

[6] B Chantraine, G Dimitroglou Rizell, P Ghiggini, R Golovko, Floer homology and Lagrangian concordance, from: "Proceedings of the Gökova Geometry–Topology Conference" (editors S Akbulut, D Auroux, T Önder), GGT (2015) 76

[7] Y Chekanov, Differential algebra of Legendrian links, Invent. Math. 150 (2002) 441 | DOI

[8] Y V Chekanov, Invariants of Legendrian knots, from: "Proceedings of the International Congress of Mathematicians, II" (editor T Li), Higher Ed. Press (2002) 385

[9] W Chongchitmate, L Ng, An atlas of Legendrian knots, Exp. Math. 22 (2013) 26 | DOI

[10] G Civan, P Koprowski, J Etnyre, J M Sabloff, A Walker, Product structures for Legendrian contact homology, Math. Proc. Cambridge Philos. Soc. 150 (2011) 291 | DOI

[11] C Cornwell, L Ng, S Sivek, Obstructions to Lagrangian concordance, Algebr. Geom. Topol. 16 (2016) 797 | DOI

[12] G Dimitroglou Rizell, Exact Lagrangian caps and non-uniruled Lagrangian submanifolds, Ark. Mat. 53 (2015) 37 | DOI

[13] G Dimitroglou Rizell, Lifting pseudo-holomorphic polygons to the symplectisation of P × R and applications, Quantum Topol. 7 (2016) 29 | DOI

[14] T Ekholm, Rational symplectic field theory over Z2 for exact Lagrangian cobordisms, J. Eur. Math. Soc. 10 (2008) 641 | DOI

[15] T Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology, from: "Perspectives in analysis, geometry, and topology" (editors I Itenberg, B Jöricke, M Passare), Progr. Math. 296, Springer (2012) 109 | DOI

[16] T Ekholm, J B Etnyre, J M Sabloff, A duality exact sequence for Legendrian contact homology, Duke Math. J. 150 (2009) 1 | DOI

[17] T Ekholm, J Etnyre, M Sullivan, The contact homology of Legendrian submanifolds in R2n+1, J. Differential Geom. 71 (2005) 177 | DOI

[18] T Ekholm, J Etnyre, M Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions, Internat. J. Math. 16 (2005) 453 | DOI

[19] T Ekholm, K Honda, T Kálmán, Legendrian knots and exact Lagrangian cobordisms, J. Eur. Math. Soc. 18 (2016) 2627 | DOI

[20] T Ekholm, T Kálmán, Isotopies of Legendrian 1–knots and Legendrian 2–tori, J. Symplectic Geom. 6 (2008) 407 | DOI

[21] Y Eliashberg, Invariants in contact topology, Doc. Math. (1998) 327

[22] Y Eliashberg, A Givental, H Hofer, Introduction to symplectic field theory, Geom. Funct. Anal. (2000) 560 | DOI

[23] J B Etnyre, L L Ng, J M Sabloff, Invariants of Legendrian knots and coherent orientations, J. Symplectic Geom. 1 (2002) 321

[24] A Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988) 513 | DOI

[25] E Getzler, J D S Jones, A∞–algebras and the cyclic bar complex, Illinois J. Math. 34 (1990) 256

[26] B Keller, Introduction to A–infinity algebras and modules, Homology Homotopy Appl. 3 (2001) 1 | DOI

[27] C Leverson, Augmentations and rulings of Legendrian knots, J. Symplectic Geom. 14 (2016) 1089 | DOI

[28] M Menke, On the augmentation categories of positive braid closures, preprint (2015)

[29] D Nadler, E Zaslow, Constructible sheaves and the Fukaya category, J. Amer. Math. Soc. 22 (2009) 233 | DOI

[30] L L Ng, Computable Legendrian invariants, Topology 42 (2003) 55 | DOI

[31] L Ng, D Rutherford, Satellites of Legendrian knots and representations of the Chekanov–Eliashberg algebra, Algebr. Geom. Topol. 13 (2013) 3047 | DOI

[32] L Ng, D Rutherford, V Shende, S Sivek, The cardinality of the augmentation category of a Legendrian link, preprint (2015)

[33] L Ng, D Rutherford, V Shende, S Sivek, E Zaslow, Augmentations are sheaves, preprint (2015)

[34] J M Sabloff, L Traynor, Obstructions to Lagrangian cobordisms between Legendrians via generating families, Algebr. Geom. Topol. 13 (2013) 2733 | DOI

[35] P Seidel, Fukaya categories and Picard–Lefschetz theory, Eur. Math. Soc. (2008) | DOI

[36] J D Stasheff, Homotopy associativity of H–spaces, I, Trans. Amer. Math. Soc. 108 (1963) 275 | DOI

[37] J D Stasheff, Homotopy associativity of H–spaces, II, Trans. Amer. Math. Soc. 108 (1963) 293