Non-isotopic symplectic tori in the same homology class

Authors:
Tolga Etgü and B. Doug Park

Translated by:

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3739-3750

MSC (2000):
Primary 57R17, 57R57; Secondary 53D35, 57R95

Published electronically:
December 15, 2003

MathSciNet review:
2055752

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Abstract | References | Similar Articles | Additional Information

Abstract: For any pair of integers and , we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class of an elliptic surface , where is the homology class of the fiber. We also show how such families can be non-isotopically and symplectically embedded into a more general class of symplectic -manifolds.

**[BZ]**Gerhard Burde and Heiner Zieschang,*Knots*, de Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR**808776****[FS1]**Ronald Fintushel and Ronald J. Stern,*Rational blowdowns of smooth 4-manifolds*, J. Differential Geom.**46**(1997), no. 2, 181–235. MR**1484044****[FS2]**Ronald Fintushel and Ronald J. Stern,*Knots, links, and 4-manifolds*, Invent. Math.**134**(1998), no. 2, 363–400. MR**1650308**, 10.1007/s002220050268**[FS3]**Ronald Fintushel and Ronald J. Stern,*Symplectic surfaces in a fixed homology class*, J. Differential Geom.**52**(1999), no. 2, 203–222. MR**1758295****[GS]**Robert E. Gompf and András I. Stipsicz,*4-manifolds and Kirby calculus*, Graduate Studies in Mathematics, vol. 20, American Mathematical Society, Providence, RI, 1999. MR**1707327****[Ma]**Takao Matumoto,*Extension problem of diffeomorphisms of a 3-torus over some 4-manifolds*, Hiroshima Math. J.**14**(1984), no. 1, 189–201. MR**750396****[MT]**Curtis T. McMullen and Clifford H. Taubes,*4-manifolds with inequivalent symplectic forms and 3-manifolds with inequivalent fibrations*, Math. Res. Lett.**6**(1999), no. 5-6, 681–696. MR**1739225**, 10.4310/MRL.1999.v6.n6.a8**[Mo]**H. R. Morton,*The multivariable Alexander polynomial for a closed braid*, Low-dimensional topology (Funchal, 1998) Contemp. Math., vol. 233, Amer. Math. Soc., Providence, RI, 1999, pp. 167–172. MR**1701681**, 10.1090/conm/233/03427**[Pa]**B. Doug Park,*A gluing formula for the Seiberg-Witten invariant along 𝑇³*, Michigan Math. J.**50**(2002), no. 3, 593–611. MR**1935154**, 10.1307/mmj/1039029984**[Ro]**Dale Rolfsen,*Knots and links*, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR**1277811****[ST1]**Bernd Siebert and Gang Tian,*On hyperelliptic 𝐶^{∞}-Lefschetz fibrations of four-manifolds*, Commun. Contemp. Math.**1**(1999), no. 2, 255–280. MR**1696101**, 10.1142/S0219199799000110**[ST2]**B. Siebert and G. Tian: Tian's talk at the Conference on Holomorphic Curves and Low-Dimensional Topology, Institute for Advanced Study, March 2002.**[Si]**J-C. Sikorav: The gluing construction for normally generic -holomorphic curves,*Symplectic and contact topology: interactions and perspectives*, Fields Inst. Commun., 35, pp. 175-199, Amer. Math. Soc., Providence, RI, 2003. Also available at arXiv:math.SG/0102004.**[Sm]**Ivan Smith,*Symplectic submanifolds from surface fibrations*, Pacific J. Math.**198**(2001), no. 1, 197–205. MR**1831978**, 10.2140/pjm.2001.198.197**[T1]**C.H. Taubes: The Seiberg-Witten invariants and symplectic forms,*Math. Res. Lett.***1**(1994), 809-822.**[T2]**Clifford Henry Taubes,*The Seiberg-Witten invariants and 4-manifolds with essential tori*, Geom. Topol.**5**(2001), 441–519 (electronic). MR**1833751**, 10.2140/gt.2001.5.441**[V1]**Stefano Vidussi,*Smooth structure of some symplectic surfaces*, Michigan Math. J.**49**(2001), no. 2, 325–330. MR**1852306**, 10.1307/mmj/1008719776**[V2]**S. Vidussi: Nonisotopic symplectic tori in the fiber class of elliptic surfaces,*preprint*. Available at`http://www.math.ksu.edu/~vidussi/`

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Additional Information

**Tolga Etgü**

Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1

Email:
etgut@math.mcmaster.ca

**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/S0002-9947-03-03529-3

Received by editor(s):
December 13, 2002

Received by editor(s) in revised form:
June 6, 2003

Published electronically:
December 15, 2003

Additional Notes:
The second author was partially supported by an NSERC research grant.

Article copyright:
© Copyright 2003
American Mathematical Society