Lagrangian tori in homotopy elliptic surfaces
Authors:
Tolga Etgü, David McKinnon and B. Doug Park
Journal:
Trans. Amer. Math. Soc. 357 (2005), 37573774
MSC (2000):
Primary 53D12, 57M05, 57R17; Secondary 57R52
Published electronically:
March 31, 2005
MathSciNet review:
2146648
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let denote the symplectic fourmanifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot in constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in and prove that infinitely many of these tori have complements with mutually nonisomorphic fundamental groups if the Alexander polynomial of has some irreducible factor which does not divide for any positive integer . We also show how these tori can be nonisotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic manifolds.
 1.
D. Auckly: Topologically knotted Lagrangians in simply connected four manifolds, Proc. Amer. Math. Soc. 133 (2005), 885889.
 2.
Gerhard
Burde, Alexanderpolynome Neuwirthscher Knoten, Topology
5 (1966), 321–330 (German). MR 0199858
(33 #7998)
 3.
Gerhard
Burde and Heiner
Zieschang, Knots, 2nd ed., de Gruyter Studies in Mathematics,
vol. 5, Walter de Gruyter & Co., Berlin, 2003. MR 1959408
(2003m:57005)
 4.
Tolga
Etgü and B.
Doug Park, Nonisotopic symplectic tori in the
same homology class, Trans. Amer. Math.
Soc. 356 (2004), no. 9, 3739–3750 (electronic). MR 2055752
(2005e:57067), http://dx.doi.org/10.1090/S0002994703035293
 5.
: Homologous nonisotopic symplectic tori in a surface, Commun. Contemp. Math. (to appear), math.GT/0305201.
 6.
: Homologous nonisotopic symplectic tori in homotopy rational elliptic surfaces, Math. Proc. Cambridge Philos. Soc. (to appear), math.GT/0307029.
 7.
: Symplectic tori in rational elliptic surfaces, math.GT/0308276.
 8.
: A note on fundamental groups of symplectic torus complements in manifolds, preprint.
 9.
: Homologous nonisotopic Lagrangian tori in symplectic 4manifolds, in preparation.
 10.
Ronald
Fintushel and Ronald
J. Stern, Knots, links, and 4manifolds, Invent. Math.
134 (1998), no. 2, 363–400. MR 1650308
(99j:57033), http://dx.doi.org/10.1007/s002220050268
 11.
Ronald
Fintushel and Ronald
J. Stern, Invariants for Lagrangian tori, Geom. Topol.
8 (2004), 947–968 (electronic). MR 2087074
(2005h:57046), http://dx.doi.org/10.2140/gt.2004.8.947
 12.
Ralph
H. Fox, Free differential calculus. II. The isomorphism problem of
groups, Ann. of Math. (2) 59 (1954), 196–210.
MR
0062125 (15,931e)
 13.
R.
H. Fox, A quick trip through knot theory, Topology of
3manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961)
PrenticeHall, Englewood Cliffs, N.J., 1962, pp. 120–167. MR 0140099
(25 #3522)
 14.
Robert
E. Gompf and András
I. Stipsicz, 4manifolds and Kirby calculus, Graduate Studies
in Mathematics, vol. 20, American Mathematical Society, Providence,
RI, 1999. MR
1707327 (2000h:57038)
 15.
Jim
Hoste, Morwen
Thistlethwaite, and Jeff
Weeks, The first 1,701,936 knots, Math. Intelligencer
20 (1998), no. 4, 33–48. MR 1646740
(99i:57015), http://dx.doi.org/10.1007/BF03025227
 16.
Taizo
Kanenobu, The augmentation subgroup of a pretzel link, Math.
Sem. Notes Kobe Univ. 7 (1979), no. 2, 363–384.
MR 557309
(81a:57008)
 17.
Daniel
A. Marcus, Number fields, SpringerVerlag, New
YorkHeidelberg, 1977. Universitext. MR 0457396
(56 #15601)
 18.
Takao
Matumoto, Extension problem of diffeomorphisms of a 3torus over
some 4manifolds, Hiroshima Math. J. 14 (1984),
no. 1, 189–201. MR 750396
(86b:57016)
 19.
B. D. Park and S. Vidussi: Nonisotopic Lagrangian tori in elliptic surfaces, preprint.
 20.
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
(95c:57018)
 21.
C. L. Stewart: Private communication.
 22.
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
(89k:57023), http://dx.doi.org/10.1090/S027309791988156856
 23.
: Hyperbolic structures on 3manifolds, II: Surface groups and 3manifolds which fiber over the circle, math.GT/9801045.
 24.
S. Vidussi: Lagrangian surfaces in a fixed homology class: Existence of knotted Lagrangian tori, J. Differential Geom. (to appear), math.GT/0311174.
 1.
 D. Auckly: Topologically knotted Lagrangians in simply connected four manifolds, Proc. Amer. Math. Soc. 133 (2005), 885889.
 2.
 G. Burde: Alexanderpolynome Neuwirthscher Knoten, Topology 5 (1966), 321330. MR 0199858 (33:7998)
 3.
 G. Burde and H. Zieschang: Knots. Second edition. de Gruyter Studies in Mathematics, 5. Walter de Gruyter & Co., Berlin, 2003. MR 1959408 (2003m:57005)
 4.
 T. Etgü and B. D. Park: Nonisotopic symplectic tori in the same homology class, Trans. Amer. Math. Soc. 356 (2004), 37393750. MR 2055752
 5.
 : Homologous nonisotopic symplectic tori in a surface, Commun. Contemp. Math. (to appear), math.GT/0305201.
 6.
 : Homologous nonisotopic symplectic tori in homotopy rational elliptic surfaces, Math. Proc. Cambridge Philos. Soc. (to appear), math.GT/0307029.
 7.
 : Symplectic tori in rational elliptic surfaces, math.GT/0308276.
 8.
 : A note on fundamental groups of symplectic torus complements in manifolds, preprint.
 9.
 : Homologous nonisotopic Lagrangian tori in symplectic 4manifolds, in preparation.
 10.
 R. Fintushel and R. J. Stern: Knots, links and 4manifolds, Invent. Math. 134 (1998), 363400. MR 1650308 (99j:57033)
 11.
 : Invariants for Lagrangian tori, Geom. Topol. 8 (2004), 947968. MR 2087074
 12.
 R. H. Fox: Free differential calculus. II. The isomorphism problem of groups. Ann. of Math. 59 (1954), 196210. MR 0062125 (15:931e)
 13.
 : A quick trip through knot theory, in Topology of Manifolds and Related Topics (Proc. The Univ. of Georgia Institute, 1961), pp. 120167, PrenticeHall, Englewood Cliffs, 1962. MR 0140099 (25:3522)
 14.
 R. E. Gompf and A. I. Stipsicz:Manifolds and Kirby Calculus, Graduate Studies in Mathematics 20, Amer. Math. Soc., Providence, 1999. MR 1707327 (2000h:57038)
 15.
 J. Hoste, M. Thistlethwaite and J. Weeks: The first knots, Math. Intelligencer 20 (1998), 3348.MR 1646740 (99i:57015)
 16.
 T. Kanenobu: The augmentation subgroup of a pretzel link, Math. Sem. Notes Kobe Univ. 7 (1979), 363384. MR 0557309 (81a:57008)
 17.
 D. A. Marcus: Number Fields. Universitext. SpringerVerlag, New YorkHeidelberg, 1977. MR 0457396 (56:15601)
 18.
 T. Matumoto: Extension problem of diffeomorphisms of a torus over some manifolds, Hiroshima Math. J. 14 (1984), 189201. MR 0750396 (86b:57016)
 19.
 B. D. Park and S. Vidussi: Nonisotopic Lagrangian tori in elliptic surfaces, preprint.
 20.
 D. Rolfsen: Knots and Links. Second printing, with corrections. Mathematics Lecture Series, 7. Publish or Perish Inc., Houston, 1990. MR 1277811 (95c:57018)
 21.
 C. L. Stewart: Private communication.
 22.
 W. P. Thurston: On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. 19 (1988), 417431. MR 0956596 (89k:57023)
 23.
 : Hyperbolic structures on 3manifolds, II: Surface groups and 3manifolds which fiber over the circle, math.GT/9801045.
 24.
 S. Vidussi: Lagrangian surfaces in a fixed homology class: Existence of knotted Lagrangian tori, J. Differential Geom. (to appear), math.GT/0311174.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
53D12,
57M05,
57R17,
57R52
Retrieve articles in all journals
with MSC (2000):
53D12,
57M05,
57R17,
57R52
Additional Information
Tolga Etgü
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada L8S 4K1
Address at time of publication:
Department of Mathematics, Koç University, Istanbul, 34450, Turkey
Email:
tetgu@ku.edu.tr
David McKinnon
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
dmckinnon@math.uwaterloo.ca
B. Doug Park
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
Email:
bdpark@math.uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0002994705037578
PII:
S 00029947(05)037578
Received by editor(s):
March 21, 2004
Published electronically:
March 31, 2005
Additional Notes:
The second author was partially supported by an NSERC research grant.
The third author was partially supported by NSERC and CFI/OIT grants.
Article copyright:
© Copyright 2005
American Mathematical Society
