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Transactions of the American Mathematical Society

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Non-isotopic symplectic surfaces in product 4-manifolds


Authors: Christopher S. Hays and B. Doug Park
Journal: Trans. Amer. Math. Soc. 360 (2008), 5771-5788
MSC (2000): Primary 57R17; Secondary 20F36, 57R52, 57R95
DOI: https://doi.org/10.1090/S0002-9947-08-04717-X
Published electronically: June 4, 2008
MathSciNet review: 2425690
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \Sigma_g$ be a closed Riemann surface of genus $ g$. Generalizing Ivan Smith's construction, we give the first examples of an infinite family of homotopic but pairwise non-isotopic symplectic surfaces of even genera inside the product symplectic $ 4$-manifolds $ \Sigma_g \times \Sigma_h$, where $ g\geq 1$ and $ h\geq 0$.


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  • 1. J. S. Birman: On braid groups, Comm. Pure Appl. Math. 22 (1969), 41-72. MR 0234447 (38:2764)
  • 2. J. S. Birman: Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, No. 82, Princeton University Press, Princeton, NJ, 1974. MR 0375281 (51:11477)
  • 3. S. K. Donaldson: Lefschetz pencils on symplectic manifolds, J. Differential Geom. 53 (1999), 205-236. MR 1802722 (2002g:53154)
  • 4. A. L. Edmonds, R. S. Kulkarni and R. E. Stong: Realizability of branched coverings of surfaces, Trans. Amer. Math. Soc. 282 (1984), 773-790. MR 732119 (85k:57005)
  • 5. Y. Eliashberg and L. Polterovich: Unknottedness of Lagrangian surfaces in symplectic $ 4$-manifolds, Internat. Math. Res. Notices 1993, 295-301. MR 1248704 (94j:58061)
  • 6. P. Erdös and J. Lehner: The distribution of the number of summands in the partitions of a positive integer, Duke Math. J. 8 (1941), 335-345. MR 0004841 (3:69a)
  • 7. T. Etgü: Symplectic and Lagrangian surfaces in $ 4$-manifolds, preprint. http://portal. ku.edu.tr/˜tetgu/papers/survey.pdf
  • 8. E. Fadell and J. Van Buskirk: The braid groups of $ E^{2}$ and $ S^{2}$, Duke Math. J. 29 (1962), 243-257. MR 0141128 (25:4539)
  • 9. R. Fintushel and R. J. Stern: Symplectic surfaces in a fixed homology class, J. Differential Geom. 52 (1999), 203-222. MR 1758295 (2001j:57036)
  • 10. R. Gillette and J. Van Buskirk: The word problem and consequences for the braid groups and mapping class groups of the $ 2$-sphere, Trans. Amer. Math. Soc. 131 (1968), 277-296. MR 0231894 (38:221)
  • 11. R. E. Gompf: A new construction of symplectic manifolds, Ann. of Math. 142 (1995), 527-595. MR 1356781 (96j:57025)
  • 12. R. E. Gompf and A. I. Stipsicz: $ 4$-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, Vol. 20, Amer. Math. Soc., Providence, RI, 1999. MR 1707327 (2000h:57038)
  • 13. G. H. Hardy and E. M. Wright: An Introduction to the Theory of Numbers. Fifth edition. Oxford University Press, New York, NY, 1979. MR 568909 (81i:10002)
  • 14. J. A. Hillman: Four-Manifolds, Geometries and Knots. Geometry & Topology Monographs, Vol. 5, Geometry & Topology Publications, Coventry, UK, 2002. MR 1943724 (2003m:57047)
  • 15. R. Hind and A. Ivrii: Isotopies of high genus Lagrangian surfaces, math.SG/0602475.
  • 16. F. Hirzebruch: The signature of ramified coverings, in Global Analysis (Papers in Honor of K. Kodaira), 253-265, Univ. Tokyo Press, Tokyo, 1969. MR 0258060 (41:2707)
  • 17. D. H. Husemoller: Ramified coverings of Riemann surfaces, Duke Math. J. 29 (1962), 167-174. MR 0136726 (25:188)
  • 18. C. Labruère and L. Paris: Presentations for the punctured mapping class groups in terms of Artin groups, Algebr. Geom. Topol. 1 (2001), 73-114. MR 1805936 (2002a:57003)
  • 19. H.-V. Lê: Realizing homology classes by symplectic submanifolds, math.SG/0505562.
  • 20. T.-J. Li: Existence of symplectic surfaces, in Geometry and Topology of Manifolds, 203-217, Fields Institute Communications, Vol. 47, Amer. Math. Soc., Providence, RI, 2005. MR 2189933 (2007c:57042)
  • 21. J. D. McCarthy: E-mail communication to the second author on May 1, 2006.
  • 22. L. Mosher: Train track expansions of measured foliations, preprint. http://newark.rutgers. edu/˜mosher/
  • 23. L. Paris and D. Rolfsen: Geometric subgroups of surface braid groups, Ann. Inst. Fourier, Grenoble 49 (1999), 417-472. MR 1697370 (2000f:20059)
  • 24. L. Paris and D. Rolfsen: Geometric subgroups of mapping class groups, J. Reine Angew. Math. 521 (2000), 47-83. MR 1752295 (2001b:57035)
  • 25. B. D. Park, M. Poddar and S. Vidussi: Homologous non-isotopic symplectic surfaces of higher genus, Trans. Amer. Math. Soc. 359 (2007), 2651-2662. MR 2286049 (2007m:57031)
  • 26. E. Pervova and C. Petronio: On the existence of branched coverings between surfaces with prescribed branch data, I, Algebr. Geom. Topol. 6 (2006), 1957-1985. MR 2263056
  • 27. B. Siebert and G. Tian: On the holomorphicity of genus two Lefschetz fibrations, Ann. of Math. 161 (2005), 959-1020. MR 2153404 (2006g:53141)
  • 28. I. Smith: Symplectic submanifolds from surface fibrations, Pacific J. Math. 198 (2001), 197-205. MR 1831978 (2002b:57029)
  • 29. O. Zariski: The topological discriminant group of a Riemann surface of genus $ p$, Amer. J. Math. 59 (1937), 335-358. MR 1507244

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Additional Information

Christopher S. Hays
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: cshays@math.msu.edu

B. Doug Park
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email: bdpark@math.uwaterloo.ca

DOI: https://doi.org/10.1090/S0002-9947-08-04717-X
Received by editor(s): June 5, 2006
Published electronically: June 4, 2008
Article copyright: © Copyright 2008 American Mathematical Society

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