Non-isotopic symplectic tori in the same homology class

Etgü, Tolga; Park, B.

doi:10.1090/S0002-9947-03-03529-3

Non-isotopic symplectic tori in the same homology class
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by Tolga Etgü and B. Doug Park PDF

Trans. Amer. Math. Soc. 356 (2004), 3739-3750 Request permission

Abstract:

For any pair of integers $n\geq 1$ and $q\geq 2$, we construct an infinite family of mutually non-isotopic symplectic tori representing the homology class $q[F]$ of an elliptic surface $E(n)$, where $[F]$ 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 $4$-manifolds.

References

Gerhard Burde and Heiner Zieschang, Knots, De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1985. MR 808776
Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
Ronald Fintushel and Ronald J. Stern, Knots, links, and $4$-manifolds, Invent. Math. 134 (1998), no. 2, 363–400. MR 1650308, DOI 10.1007/s002220050268
Ronald Fintushel and Ronald J. Stern, Symplectic surfaces in a fixed homology class, J. Differential Geom. 52 (1999), no. 2, 203–222. MR 1758295
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, DOI 10.1090/gsm/020
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
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, DOI 10.4310/MRL.1999.v6.n6.a8
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, DOI 10.1090/conm/233/03427
B. Doug Park, A gluing formula for the Seiberg-Witten invariant along $T^3$, Michigan Math. J. 50 (2002), no. 3, 593–611. MR 1935154, DOI 10.1307/mmj/1039029984
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
Bernd Siebert and Gang Tian, On hyperelliptic $C^\infty$-Lefschetz fibrations of four-manifolds, Commun. Contemp. Math. 1 (1999), no. 2, 255–280. MR 1696101, DOI 10.1142/S0219199799000110
B. Siebert and G. Tian: Tian’s talk at the Conference on Holomorphic Curves and Low-Dimensional Topology, Institute for Advanced Study, March 2002.
J-C. Sikorav: The gluing construction for normally generic $J$-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.
Ivan Smith, Symplectic submanifolds from surface fibrations, Pacific J. Math. 198 (2001), no. 1, 197–205. MR 1831978, DOI 10.2140/pjm.2001.198.197
C.H. Taubes: The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809–822.
Clifford Henry Taubes, The Seiberg-Witten invariants and 4-manifolds with essential tori, Geom. Topol. 5 (2001), 441–519. MR 1833751, DOI 10.2140/gt.2001.5.441
Stefano Vidussi, Smooth structure of some symplectic surfaces, Michigan Math. J. 49 (2001), no. 2, 325–330. MR 1852306, DOI 10.1307/mmj/1008719776
S. Vidussi: Nonisotopic symplectic tori in the fiber class of elliptic surfaces, preprint. Available at http://www.math.ksu.edu/~vidussi/