Homologous non-isotopic symplectic surfaces of higher genus
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- by B. Doug Park, Mainak Poddar and Stefano Vidussi PDF
- Trans. Amer. Math. Soc. 359 (2007), 2651-2662 Request permission
Abstract:
We construct an infinite family of homologous, non-isotopic, symplectic surfaces of any genus greater than one in a certain class of closed, simply connected, symplectic four-manifolds. Our construction is the first example of this phenomenon for surfaces of genus greater than one.References
- D. Auroux, S. K. Donaldson, and L. Katzarkov, Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves, Math. Ann. 326 (2003), no. 1, 185–203. MR 1981618, DOI 10.1007/s00208-003-0418-9
- Steven A. Bleiler, Craig D. Hodgson, and Jeffrey R. Weeks, Cosmetic surgery on knots, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 23–34. MR 1734400, DOI 10.2140/gtm.1999.2.23
- David Eisenbud and Walter Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, vol. 110, Princeton University Press, Princeton, NJ, 1985. MR 817982
- Tolga Etgü and B. Doug Park, Homologous non-isotopic symplectic tori in homotopy rational elliptic surfaces, Math. Proc. Cambridge Philos. Soc. 140 (2006), no. 1, 71–78. MR 2197576, DOI 10.1017/S0305004105008790
- T. Etgü and B. D. Park: A note on fundamental groups of symplectic torus complements. Preprint, http://www.math.uwaterloo.ca/~bdpark/preprint.html
- 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, Nonsymplectic 4-manifolds with one basic class, Pacific J. Math. 194 (2000), no. 2, 325–333. MR 1760784, DOI 10.2140/pjm.2000.194.325
- 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, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
- 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
- William Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, vol. 43, American Mathematical Society, Providence, R.I., 1980. MR 565450, DOI 10.1090/cbms/043
- B. Doug Park, Doubling homotopy $K3$ surfaces, J. Knot Theory Ramifications 12 (2003), no. 3, 347–354. MR 1983090, DOI 10.1142/S0218216503002469
- Bernd Siebert and Gang Tian, On the holomorphicity of genus two Lefschetz fibrations, Ann. of Math. (2) 161 (2005), no. 2, 959–1020. MR 2153404, DOI 10.4007/annals.2005.161.959
- 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
- Ronald Fintushel and Ronald J. Stern, Symplectic surfaces in a fixed homology class, J. Differential Geom. 52 (1999), no. 2, 203–222. MR 1758295
- W. P. Thurston: The Geometry and Topology of Three-Manifolds. Lecture notes, Princeton University, 1980. http://www.msri.org/publications/books/gt3m/
- S. Vidussi: Lagrangian surfaces in a fixed homology class: Existence of knotted Lagrangian tori. J. Differential Geom. 74 (2006), 507–522.
- Stefano Vidussi, Symplectic tori in homotopy $E(1)$’s, Proc. Amer. Math. Soc. 133 (2005), no. 8, 2477–2481. MR 2138891, DOI 10.1090/S0002-9939-05-07527-1
Additional Information
- B. Doug Park
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Email: bdpark@math.uwaterloo.ca
- Mainak Poddar
- Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
- Email: mpoddar@math.uwaterloo.ca
- Stefano Vidussi
- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- Email: svidussi@math.ucr.edu
- Received by editor(s): February 21, 2005
- Published electronically: January 4, 2007
- Additional Notes: The first author was partially supported by NSERC and CFI/OIT grants.
The third author was partially supported by NSF grant #0306074. - © Copyright 2007 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 359 (2007), 2651-2662
- MSC (2000): Primary 57R17, 57M05; Secondary 53D35, 57R95
- DOI: https://doi.org/10.1090/S0002-9947-07-04168-2
- MathSciNet review: 2286049
Dedicated: Dedicated to Ron Fintushel on the occasion of his sixtieth birthday