Volume 16, issue 2 (2012)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 23
Issue 3, 1085–1619
Issue 2, 541–1084
Issue 1, 1–540

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Ethics Statement
Author Index
To Appear
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Other MSP Journals
Lagrangian spheres, symplectic surfaces and the symplectic mapping class group

Tian-Jun Li and Weiwei Wu

Geometry & Topology 16 (2012) 1121–1169
1 M Abreu, D McDuff, Topology of symplectomorphism groups of rational ruled surfaces, J. Amer. Math. Soc. 13 (2000) 971 MR1775741
2 M Alberich-Carramiñana, Geometry of the plane Cremona maps, Lecture Notes in Math. 1769, Springer (2002) MR1874328
3 M Audin, Lagrangian skeletons, periodic geodesic flows and symplectic cuttings, Manuscripta Math. 124 (2007) 533 MR2357797
4 J F Barraud, Nodal symplectic spheres in $\mathbf C\mathrm{P}^2$ with positive self-intersection, Internat. Math. Res. Notices (1999) 495 MR1692591
5 P Biran, Symplectic packing in dimension $4$, Geom. Funct. Anal. 7 (1997) 420 MR1466333
6 P Biran, A stability property of symplectic packing, Invent. Math. 136 (1999) 123 MR1681101
7 P Biran, K Cieliebak, Symplectic topology on subcritical manifolds, Comment. Math. Helv. 76 (2001) 712 MR1881704
8 P Biran, O Cornea, Quantum Structures for Lagrangian submanifolds arXiv:0708.4221
9 F Bourgeois, A Morse–Bott approach to contact homology, PhD thesis, New York University (2002)
10 F Bourgeois, Y Eliashberg, H Hofer, K Wysocki, E Zehnder, Compactness results in symplectic field theory, Geom. Topol. 7 (2003) 799 MR2026549
11 J Coffey, Symplectomorphism groups and isotropic skeletons, Geom. Topol. 9 (2005) 935 MR2140995
12 C Conley, E Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37 (1984) 207 MR733717
13 J D Evans, Lagrangian spheres in del Pezzo surfaces, J. Topol. 3 (2010) 181 MR2608481
14 J D Evans, Symplectic topology of some Stein and rational surfaces, PhD thesis, University of Cambridge (2010)
15 J D Evans, Symplectic mapping class groups of some Stein and rational surfaces, J. Symplectic Geom. 9 (2011) 45 MR2787361
16 R Fintushel, R J Stern, Invariants for Lagrangian tori, Geom. Topol. 8 (2004) 947 MR2087074
17 R Friedman, J W Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988) 297 MR925124
18 H Z Gao, Representing homology classes of almost definite $4$–manifolds, Topology Appl. 52 (1993) 109 MR1241187
19 R E Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995) 527 MR1356781
20 M Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307 MR809718
21 R Hind, Lagrangian isotopies in Stein manifolds arXiv:math/0311093
22 R Hind, Lagrangian spheres in $S^2\times S^2$, Geom. Funct. Anal. 14 (2004) 303 MR2060197
23 R Hind, A Ivrii, Ruled $4$–manifolds and isotopies of symplectic surfaces, Math. Z. 265 (2010) 639 MR2644314
24 R Hind, M Pinsonnault, W Wu, Symplectomorphism groups of non-compact manifolds and space of Lagrangians, preprint (2011)
25 H Hofer, V Lizan, J C Sikorav, On genericity for holomorphic curves in four-dimensional almost-complex manifolds, J. Geom. Anal. 7 (1997) 149 MR1630789
26 S Ivashkovich, V Shevchishin, Structure of the moduli space in a neighborhood of a cusp-curve and meromorphic hulls, Invent. Math. 136 (1999) 571 MR1695206
27 K Kikuchi, Positive $2$–spheres in $4$–manifolds of signature $(1,n)$, Pacific J. Math. 160 (1993) 245 MR1233354
28 F Lalonde, Isotopy of symplectic balls, Gromov's radius and the structure of ruled symplectic $4$–manifolds, Math. Ann. 300 (1994) 273 MR1299063
29 F Lalonde, D McDuff, $J$–curves and the classification of rational and ruled symplectic $4$–manifolds, from: "Contact and symplectic geometry (Cambridge, 1994)" (editor C B Thomas), Publ. Newton Inst. 8, Cambridge Univ. Press (1996) 3 MR1432456
30 E Lerman, Symplectic cuts, Math. Res. Lett. 2 (1995) 247 MR1338784
31 B H Li, T J Li, Symplectic genus, minimal genus and diffeomorphisms, Asian J. Math. 6 (2002) 123 MR1902650
32 T J Li, Existence of symplectic surfaces, from: "Geometry and topology of manifolds" (editors H U Boden, I Hambleton, A J Nicas), Fields Inst. Commun. 47, Amer. Math. Soc. (2005) 203 MR2189933
33 T J Li, The Kodaira dimension of symplectic $4$–manifolds, from: "Floer homology, gauge theory, and low-dimensional topology" (editors D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 249 MR2249257
34 T J Li, The space of symplectic structures on closed $4$–manifolds, from: "Third International Congress of Chinese Mathematicians. Part 1, 2" (editors K S Lau, Z P Xin, S T Yau), AMS/IP Stud. Adv. Math., 42, pt. 1 2, Amer. Math. Soc. (2008) 259 MR2409637
35 T J Li, A K Liu, The equivalence between $\mathrm{SW}$ and $\mathrm{Gr}$ in the case where $b^+=1$, Internat. Math. Res. Notices (1999) 335 MR1683312
36 T J Li, A K Liu, Uniqueness of symplectic canonical class, surface cone and symplectic cone of $4$–manifolds with $B_+=1$, J. Differential Geom. 58 (2001) 331 MR1913946
37 T J Li, M Usher, Symplectic forms and surfaces of negative square, J. Symplectic Geom. 4 (2006) 71 MR2240213
38 T J Li, W Wu, On homotopy types of symplectomorphism groups on close symplectic $4$–manifolds, in preparation
39 D McDuff, The structure of rational and ruled symplectic $4$–manifolds, J. Amer. Math. Soc. 3 (1990) 679 MR1049697
40 D McDuff, From symplectic deformation to isotopy, from: "Topics in symplectic $4$–manifolds (Irvine, CA, 1996)" (editor R J Stern), First Int. Press Lect. Ser. I, Int. Press (1998) 85 MR1635697
41 D McDuff, D Salamon, Introduction to symplectic topology, Oxford Math. Monogr., The Clarendon Press, Oxford Univ. Press (1998) MR1698616
42 D McDuff, D Salamon, $J$–holomorphic curves and symplectic topology, Amer. Math. Soc. Colloquium Publ. 52, Amer. Math. Soc. (2004) MR2045629
43 D McDuff, F Schlenk, The embedding capacity of $4$–dimensional symplectic ellipsoids arXiv:0912.0532
44 M Pinsonnault, Maximal compact tori in the Hamiltonian group of $4$–dimensional symplectic manifolds, J. Mod. Dyn. 2 (2008) 431 MR2417479
45 D Salamon, E Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992) 1303 MR1181727
46 P Seidel, Symplectic automorphisms of $T^*S^2$ arXiv:math/9803084
47 P Seidel, Lectures on four-dimensional Dehn twists, from: "Symplectic $4$–manifolds and algebraic surfaces" (editors F Catanese, G Tian), Lecture Notes in Math. 1938, Springer (2008) 231 MR2441414
48 P Seidel, I Smith, The symplectic topology of Ramanujam's surface, Comment. Math. Helv. 80 (2005) 859 MR2182703
49 V Shevchishin, Secondary Stiefel–Whitney class and diffeomorphisms of rational and ruled symplectic $4$–manifolds arXiv:0904.0283v2
50 J C Sikorav, The gluing construction for normally generic $J$–holomorphic curves, from: "Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001)" (editors Y Eliashberg, B Khesin, F Lalonde), Fields Inst. Commun. 35, Amer. Math. Soc. (2003) 175 MR1969276
51 C H Taubes, Counting pseudo-holomorphic submanifolds in dimension $4$, J. Differential Geom. 44 (1996) 818 MR1438194
52 C H Taubes, $\mathrm{GR}=\mathrm{SW}$: counting curves and connections, J. Differential Geom. 52 (1999) 453 MR1761081
53 S Vidussi, Lagrangian surfaces in a fixed homology class: existence of knotted Lagrangian tori, J. Differential Geom. 74 (2006) 507 MR2269786
54 J Y Welschinger, Effective classes and Lagrangian tori in symplectic four-manifolds, J. Symplectic Geom. 5 (2007) 9 MR2371182
55 C Wendl, Automatic transversality and orbifolds of punctured holomorphic curves in dimension four, Comment. Math. Helv. 85 (2010) 347 MR2595183