Symplectic tori in homotopy 𝐄(1)’s

Vidussi, Stefano

doi:10.1090/S0002-9939-05-07527-1

Symplectic tori in homotopy $\mathbf {E}(1)$’s
HTML articles powered by AMS MathViewer

by Stefano Vidussi PDF

Proc. Amer. Math. Soc. 133 (2005), 2477-2481 Request permission

Abstract:

This short note presents a simple construction of nonisotopic symplectic tori representing the same primitive homology class in the symplectic $4$-manifold $E(1)_{K}$, obtained by knot surgery on the rational elliptic surface $E(1) = \mathbb {P}^{2} \# 9 {\overline {\mathbb {P}}^{2}}$ with the left-handed trefoil knot $K$. $E(1)_{K}$ has the simplest homotopy type among simply-connected symplectic $4$-manifolds known to exhibit such a property.

References

Tolga Etgü and B. Doug Park, Non-isotopic symplectic tori in the same homology class, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3739–3750. MR 2055752, DOI 10.1090/S0002-9947-03-03529-3
T. Etgü, B.D. Park, Homologous Non-isotopic Symplectic Tori in a K3 Surface, preprint (2003).
T. Etgü, B.D. Park, Homologous Non-isotopic Symplectic Tori in Homotopy Rational Elliptic Surfaces, to appear, Math. Proc. Cambridge Philos. Soc..
T. Etgü, B.D. Park, Symplectic Tori in Rational Elliptic Surfaces, preprint (2003).
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
Ronald Fintushel and Ronald J. Stern, Invariants for Lagrangian tori, Geom. Topol. 8 (2004), 947–968. MR 2087074, DOI 10.2140/gt.2004.8.947
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
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
Gang Tian, Symplectic isotopy in four dimension, First International Congress of Chinese Mathematicians (Beijing, 1998) AMS/IP Stud. Adv. Math., vol. 20, Amer. Math. Soc., Providence, RI, 2001, pp. 143–147. MR 1830170, DOI 10.1090/amsip/020/09
Stefano Vidussi, Smooth structure of some symplectic surfaces, Michigan Math. J. 49 (2001), no. 2, 325–330. MR 1852306, DOI 10.1307/mmj/1008719776
Stefano Vidussi, Nonisotopic symplectic tori in the fiber class of elliptic surfaces, J. Symplectic Geom. 2 (2004), no. 2, 207–218. MR 2108374
S. Vidussi, Lagrangian Surfaces in a Fixed Homology Class: Existence of Knotted Lagrangian Tori, to appear, J. Differential Geom.