Lagrangian tori in homotopy elliptic surfaces

Etgü, Tolga; McKinnon, David; Park, B.

doi:10.1090/S0002-9947-05-03757-8

Lagrangian tori in homotopy elliptic surfaces
HTML articles powered by AMS MathViewer

by Tolga Etgü, David McKinnon and B. Doug Park PDF

Trans. Amer. Math. Soc. 357 (2005), 3757-3774 Request permission

Abstract:

Let $E(1)_K$ denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in $E(1)_K$ and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of $K$ has some irreducible factor which does not divide $t^n-1$ for any positive integer $n$. We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic $4$-manifolds.

References

D. Auckly: Topologically knotted Lagrangians in simply connected four manifolds, Proc. Amer. Math. Soc. 133 (2005), 885–889.
Gerhard Burde, Alexanderpolynome Neuwirthscher Knoten, Topology 5 (1966), 321–330 (German). MR 199858, DOI 10.1016/0040-9383(66)90023-1
Gerhard Burde and Heiner Zieschang, Knots, 2nd ed., De Gruyter Studies in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR 1959408
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
—: Homologous non-isotopic symplectic tori in a $K3$-surface, Commun. Contemp. Math. (to appear), math.GT/0305201.
—: Homologous non-isotopic symplectic tori in homotopy rational elliptic surfaces, Math. Proc. Cambridge Philos. Soc. (to appear), math.GT/0307029.
—: Symplectic tori in rational elliptic surfaces, math.GT/0308276.
—: A note on fundamental groups of symplectic torus complements in $4$-manifolds, preprint.
—: Homologous non-isotopic Lagrangian tori in symplectic 4-manifolds, in preparation.
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, Invariants for Lagrangian tori, Geom. Topol. 8 (2004), 947–968. MR 2087074, DOI 10.2140/gt.2004.8.947
Ralph H. Fox, Free differential calculus. II. The isomorphism problem of groups, Ann. of Math. (2) 59 (1954), 196–210. MR 62125, DOI 10.2307/1969686
R. H. Fox, A quick trip through knot theory, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
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
Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks, The first 1,701,936 knots, Math. Intelligencer 20 (1998), no. 4, 33–48. MR 1646740, DOI 10.1007/BF03025227
Taizo Kanenobu, The augmentation subgroup of a pretzel link, Math. Sem. Notes Kobe Univ. 7 (1979), no. 2, 363–384. MR 557309
Daniel A. Marcus, Number fields, Universitext, Springer-Verlag, New York-Heidelberg, 1977. MR 0457396
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
B. D. Park and S. Vidussi: Non-isotopic Lagrangian tori in elliptic surfaces, preprint.
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
C. L. Stewart: Private communication.
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
—: Hyperbolic structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle, math.GT/9801045.
S. Vidussi: Lagrangian surfaces in a fixed homology class: Existence of knotted Lagrangian tori, J. Differential Geom. (to appear), math.GT/0311174.