Lifting pseudo-holomorphic polygons to the symplectisation of \(P \times \mathbb{R}\) and applications. (English) Zbl 1346.53074

Dimitroglou Rizell, Georgios

Quantum Topol. 7, No. 1, 29-105 (2016).

The author studies the lifting problem of pseudoholomorphic polygons on an exact symplectic manifold \(P\) to the symplectization of the contactization, i.e., to \(\mathbb R \times (P \times \mathbb R)\). Denote by \(\Pi(\Lambda)\) the \(P\)-projection of a given closed Legendrian submanifold \(\Lambda\), which defines an exact Lagrangian immersion in \(P\). Let \(\mathbb R \times \Lambda \subset \mathbb R \times (P \times \mathbb R)\) be the corresponding Lagrangian cylinder. Then the author proves that a pseudoholomorphic polygon in \(P\) having boundary on \(\Pi(\Lambda)\) can be lifted to a pseudoholomorphic disc in the \(\mathbb R \times (P \times \mathbb R)\) having boundary on \(\mathbb R \times \Lambda\). As a result it is shown that Legendrian contact homology may be equivalently defined by counting either of these objects. Using the result, the author gives a proof that the linearized Legendrian contact homology induced by an exact Lagrangian filling is isomorphic to the singular homology of the filling. Such an isomorphism was first observed by Seidel and its proof was previously outlined by T. Ekholm [Prog. Math. 296, 109–145 (2012; Zbl 1254.57024)]. The proof is based on some vanishing result of wrapped Floer cohomology of the pair \((L,L')\) of exact Lagrangian fillings in the symplectization \(\mathbb R \times (P \times \mathbb R)\), which in turn is a consequence of Hamiltonian displacement of one from the other.

Reviewer: Yong-Geun Oh (Pohang)

MSC:

53D42	Symplectic field theory; contact homology
53D40	Symplectic aspects of Floer homology and cohomology
53D12	Lagrangian submanifolds; Maslov index

Keywords:

Legendrian contact homology; wrapped Floer homology; exact Lagrangian fillings

Citations:

Zbl 1254.57024

References:

[1]	C. Abbas, Pseudoholomorphic strips in symplectizations. II. Fredholm theory and transversality. Comm. Pure Appl. Math. 57 (2004), no. 1, 1–58.MR 2007355 Zbl 1073.53104 · Zbl 1073.53104
[2]	A. Abbondandolo and M. Schwarz, On the Floer homology of cotangent bundles. Comm. Pure Appl. Math. 59(2006), no. 2, 254–316.MR 2190223 Zbl 1084.53074 · Zbl 1084.53074
[3]	M. Abouzaid, On the wrapped Fukaya category and based loops. J. Symplectic Geom. 10(2012), no. 1, 27–79.MR 2904032 Zbl 1298.53092 · Zbl 1298.53092
[4]	M. Abouzaid and P. Seidel, An open string analogue of Viterbo functoriality. Geom. Topol. 14(2010), no. 2, 627–718.MR 2602848 Zbl 1195.53106 · Zbl 1195.53106
[5]	F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder, Compactness results in symplectic field theory. Geom. Topol. 7 (2003), 799–888.MR 2026549 Zbl 1131.53312 · Zbl 1131.53312
[6]	Y. Chekanov, Differential algebra of Legendrian links. Invent. Math. 150 (2002), n,. 3, 441–483.MR 1946550 Zbl 1029.57011 · Zbl 1029.57011
[7]	D. L. Dragnev, Fredholm theory and transversality for noncompact pseudoholomorphic maps in symplectizations. Comm. Pure Appl. Math. 57 (2004), no. 6, 726–763. MR 2038115 Zbl 1063.53086
[8]	T. Ekholm, Rational symplectic field theory over Z2for exact Lagrangian cobordisms. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 3, 641–704.MR 2421157 Zbl 1154.57020
[9]	T. Ekholm, Rational SFT, linearized Legendrian contact homology, and Lagrangian Floer cohomology. In I. Itenberg, B. Jöricke, and M. Passare (eds.), Perspectives in analysis, geometry, and topology.On the occasion of the 60thbirthday of O. Viro. Progress in Mathematics, 296. Birkhäuser/Springer, New York, 2012, 109–145. MR 2884034 MR 2867634(collection)Zbl 1254.57024 Zbl 1230.00045(collection)
[10]	T. Ekholm, J. Etnyre, and M. Sullivan, The contact homology of Legendrian submanifolds in R2nC1. J. Differential Geom. 71 (2005), no. 2 177–305.MR 2197142 Zbl 1103.53048
[11]	T. Ekholm, J. Etnyre, and M. Sullivan, Non-isotopic Legendrian submanifolds in R2nC1. J. Differential Geom. 71 (2005), no. 1, 85–128.MR 2191769 Zbl 1098.57013 · Zbl 1098.57013
[12]	T. Ekholm, J. Etnyre, and M. Sullivan, Orientations in Legendrian contact homology and exact Lagrangian immersions. Internat. J. Math. 16 (2005), no. 5, 453–532. MR 2141318 Zbl 1076.53099 · Zbl 1076.53099
[13]	T. Ekholm, J. Etnyre, and M. Sullivan, Legendrian contact homology in P R. Trans. Amer. Math. Soc. 359(2007), no. 7, 3301–3335.MR 2299457 Zbl 1119.53051 · Zbl 1119.53051
[14]	T. Ekholm, J. B. Etnyre, and J. M. Sabloff, A duality exact sequence for Legendrian contact homology. Duke Math. J. 150 (2009), no. 1, 1–75.MR 2560107 Zbl 1193.53179 Lifting pseudo-holomorphic discs and applications105 · Zbl 1193.53179
[15]	Y. Eliashberg, A. Givental, and H. Hofer, Introduction to symplectic field theory. Geom. Funct. Anal.(2000), Special Volume, Part II, 560–673. GAFA 2000 (Tel Aviv, 1999).MR 1826267 Zbl 1821865 · Zbl 0989.81114
[16]	J. B. Etnyre, L. L. Ng, and J. M. Sabloff, Invariants of Legendrian knots and coherent orientations. J. Symplectic Geom. 1 (2002), no. 2, 321–367.MR 1959585 Zbl 1024.57014 · Zbl 1024.57014
[17]	A. Floer, Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3, 513–547.MR 965228 Zbl 0674.57027 · Zbl 0674.57027
[18]	K. Fukaya, P. Seidel, and I. Smith, The symplectic geometry of cotangent bundles from a categorical viewpoint. In A. Kapustin, M. Kreuzer and K.-G. Schlesinger (eds.), Homological mirror symmetry. Lecture Notes in Physics, 757. Springer, Berlin, 2009, 1–26.MR 2596633 Zbl 1163.53344 Zbl 1151.81001(collection) · Zbl 1163.53344
[19]	L. Lazzarini, Existence of a somewhere injective pseudo-holomorphic disc. Geom. Funct. Anal. 10(2000), 829–862.MR 1791142 Zbl 1003.32004 · Zbl 1003.32004
[20]	D. McDuff and D. Salamon, J -holomorphic curves and symplectic topology. American Mathematical Society Colloquium Publications 52, 2nded., American Mathematical Society, Providence, R.I., 2012.MR 2954391 Zbl 1272.53002 · Zbl 1272.53002
[21]	J. W. Robbin and D. A. Salamon, Asymptotic behaviour of holomorphic strips. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(2001), no. 5, 573–612.MR 1849689 Zbl 0999.53048
[22]	J. M. Sabloff and L. Traynor, Obstructions to Lagrangian cobordisms between Legendrians via generating families. Algebr. Geom. Topol. 13 (2013), no. 5, 2733–2797.MR 3116302 Zbl 1270.53096 · Zbl 1270.53096
[23]	J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds. In Michèle Audin and Jacques Lafontaine (eds.), Holomorphic curves in symplectic geometry.Progress in Mathematics, 117. Birkhäuser, Basel, 1994, 165–189. MR 1274929 MR 1274923(collection)
[24]	D. V. Widder, Functions harmonic in a strip. Proc. Amer. Math. Soc. 12 (1961), 67–72. MR 0132838 Zbl 0096.07703 Received Mai 20, 2013 Georgios Dimitroglou Rizell, Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, U.K. e-mail:[e-mail] · Zbl 0096.07703