Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Symplectic $ \mathbf{S}^{1} \times N^3$, subgroup separability, and vanishing Thurston norm

Authors: Stefan Friedl and Stefano Vidussi
Journal: J. Amer. Math. Soc. 21 (2008), 597-610
MSC (2000): Primary 57R17, 57M27
Published electronically: August 28, 2007
MathSciNet review: 2373361
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ N$ be a closed, oriented $ 3$-manifold. A folklore conjecture states that $ S^{1} \times N$ admits a symplectic structure if and only if $ N$ admits a fibration over the circle. We will prove this conjecture in the case when $ N$ is irreducible and its fundamental group satisfies appropriate subgroup separability conditions. This statement includes $ 3$-manifolds with vanishing Thurston norm, graph manifolds and $ 3$-manifolds with surface subgroup separability (a condition satisfied conjecturally by all hyperbolic $ 3$-manifolds). Our result covers, in particular, the case of 0-framed surgeries along knots of genus one. The statement follows from the proof that twisted Alexander polynomials decide fiberability for all the $ 3$-manifolds listed above. As a corollary, it follows that twisted Alexander polynomials decide if a knot of genus one is fibered.

References [Enhancements On Off] (What's this?)

  • [AS05] C. Adams, E. Schoenfeld, Totally Geodesic Seifert Surfaces in Hyperbolic Knot and Link Complements. I, Geom. Dedicata 116: 237-247 (2005). MR 2195448 (2006j:57008)
  • [Bo02] F. Bonahon, Geometric structures on 3-manifolds, Handbook of geometric topology, 93-164, North-Holland, Amsterdam (2002). MR 1886669 (2003b:57021)
  • [BZ67] G. Burde, H. Zieschang, Neuwirthsche Knoten und Flächenabbildungen, Abh. Math. Sem. Univ. Hamburg 31: 239-246 (1967). MR 0229229 (37:4803)
  • [BKS87] R. Burns, A. Karrass, D. Solitar, A note on groups with separable finitely generated subgroups, Bull. Austral. Math. Soc. 36 , no. 1: 153-160 (1987). MR 897431 (88g:20057)
  • [EN85] D. Eisenbud, W. Neumann, Three-dimensional link theory and invariants of plane curve singularities, Annals of Mathematics Studies, 110. Princeton University Press, Princeton, NJ (1985). MR 817982 (87g:57007)
  • [FK06] S. Friedl, T. Kim, Thurston norm, fibered manifolds and twisted Alexander polynomials, Topology, Vol. 45: 929-953 (2006). MR 2263219 (2007g:57020)
  • [FV06a] S. Friedl, S. Vidussi, Twisted Alexander polynomials and symplectic structures, American Journal of Mathematics, to appear.
  • [FV06b] S. Friedl, S. Vidussi, Nontrivial Alexander polynomials of knots and links, Bull. Lond. Math. Soc. 39: 614-622 (2007).
  • [FV07] S. Friedl, S. Vidussi, Symplectic $ 4$-manifolds with a free circle action, Preprint (2007).
  • [Ga83] D. Gabai, Foliations and the topology of 3-manifolds, J. Differential Geometry 18, no. 3: 445-503 (1983). MR 723813 (86a:57009)
  • [Ga87] D. Gabai, Foliations and the topology of 3-manifolds. III, J. Differential Geometry 26, no. 3: 479-536 (1987). MR 910018 (89a:57014b)
  • [Gi99] R. Gitik, Doubles of groups and hyperbolic LERF 3-manifolds, Ann. of Math. (2) 150, no. 3: 775-806 (1999). MR 1740992 (2001a:20044)
  • [Ha01] E. Hamilton, Abelian Subgroup Separability of Haken $ 3$-manifolds and Closed Hyperbolic $ n$-orbifolds, Proc. London Math. Soc. 83 no. 3: 626-646 (2001). MR 1851085 (2002g:57033)
  • [Hat] A. Hatcher, Basic Topology of 3-Manifolds, notes available at http://www.math. hatcher.
  • [He76] J. Hempel, $ 3$-Manifolds, Ann. of Math. Studies, No. 86. Princeton University Press, Princeton, N. J. (1976). MR 0415619 (54:3702)
  • [He87] J. Hempel, Residual finiteness for $ 3$-manifolds, Combinatorial group theory and topology (Alta, Utah, 1984), 379-396, Ann. of Math. Stud., 111, Princeton Univ. Press, Princeton, NJ (1987). MR 895623 (89b:57002)
  • [Ko87] S. Kojima, Finite covers of $ 3$-manifolds containing essential surfaces of Euler characteristic $ =0$, Proc. Amer. Math. Soc. 101, no. 4: 743-747 (1987). MR 911044 (89b:57010)
  • [Kr98] P. Kronheimer, Embedded surfaces and gauge theory in three and four dimensions, Surveys in differential geometry, Vol. III (Cambridge, MA, 1996), 243-298, Int. Press, Boston, MA (1998). MR 1677890 (2000a:57086)
  • [Kr99] P. Kronheimer, Minimal genus in $ S\sp 1\times M\sp 3$, Invent. Math. 135, no. 1: 45-61 (1999). MR 1664695 (2000c:57071)
  • [Ja80] W. Jaco, Lectures on three-manifold topology, CBMS Regional Conference Series in Mathematics, 43. American Mathematical Society, Providence, R.I. (1980). MR 565450 (81k:57009)
  • [Li01] X. S. Lin, Representations of knot groups and twisted Alexander polynomials, Acta Math. Sin. (Engl. Ser.) 17, no. 3: 361-380 (2001). MR 1852950 (2003f:57018)
  • [LN91] D. Long, G. Niblo, Subgroup separability and $ 3$-manifold groups, Math. Z. 207, no. 2: 209-215 (1991). MR 1109662 (92g:20047)
  • [LR05] D. Long, A. W. Reid, Surface subgroups and subgroup separability in 3-manifold topology, Publicacoes Matematicas do IMPA. 25 $ \sp {\rm o}$ Coloquio Brasileiro de Matematica (2005). MR 2164951 (2006f:57018)
  • [Lu88] J. Luecke, Finite covers of $ 3$-manifolds containing essential tori, Trans. Amer. Math. Soc. 310: 381-391 (1988). MR 965759 (90c:57011)
  • [McC01] J. McCarthy, On the asphericity of a symplectic $ M^{3} \times S^{1}$, Proc. Amer. Math. Soc. 129: 257-264 (2001). MR 1707526 (2001c:57024)
  • [McM02] C. T. McMullen, The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology, Ann. Sci. Ecole Norm. Sup. (4) 35, no. 2: 153-171 (2002). MR 1914929 (2003d:57044)
  • [MeT96] G. Meng, C. H. Taubes, SW = Milnor torsion, Math. Res. Lett. 3: 661-674 (1996). MR 1418579 (98j:57049)
  • [NW01] G. A. Niblo, D. T. Wise, Subgroup separability, knot groups and graph manifolds, Proc. Amer. Math. Soc. 129, no. 3: 685-693 (2001). MR 1707529 (2001f:20075)
  • [Sc78] P. Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2) 17, no. 3: 555-565 (1978). MR 0494062 (58:12996)
  • [St62] J. Stallings, On fibering certain 3-manifolds, 1962 Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), pp. 95-100, Prentice-Hall, Englewood Cliffs, N.J. (1962). MR 0158375 (28:1600)
  • [Ta94] C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1: 809-822 (1994). MR 1306023 (95j:57039)
  • [Ta95] C. H. Taubes, More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res. Lett. 2: 9-13 (1995). MR 1312973 (96a:57075)
  • [Th76] W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55, no. 2: 467-468 (1976). MR 0402764 (53:6578)
  • [Th82] W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. 6 (1982). MR 648524 (83h:57019)
  • [Th86] W. P. Thurston, A norm for the homology of 3-manifolds, Mem. Amer. Math. Soc. 339: 99-130 (1986). MR 823443 (88h:57014)
  • [Tu01] V. Turaev, Introduction to combinatorial torsions, Birkhäuser, Basel (2001). MR 1809561 (2001m:57042)
  • [Vi99] S. Vidussi, The Alexander norm is smaller than the Thurston norm; a Seiberg-Witten proof, Prepublication Ecole Polytechnique 6 (1999).
  • [Vi03] S. Vidussi, Norms on the cohomology of a 3-manifold and SW theory, Pacific J. Math. 208, no. 1: 169-186 (2003). MR 1979378 (2004f:57038)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 57R17, 57M27

Retrieve articles in all journals with MSC (2000): 57R17, 57M27

Additional Information

Stefan Friedl
Affiliation: Département de Mathématiques, Université du Québec à Montréal, Montréal, Québec, H3C 3P8, Canada

Stefano Vidussi
Affiliation: Department of Mathematics, University of California, Riverside, California 92521

Received by editor(s): August 2, 2006
Published electronically: August 28, 2007
Additional Notes: The second author was partially supported by NSF grant #0629956.
Dedicated: Dedicated to the memory of Xiao-Song Lin
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society