## Algebraic & Geometric Topology

### Lefschetz fibrations, complex structures and Seifert fibrations on $S^1 \times M^3$

Tolga Etgu

#### Abstract

We consider product 4–manifolds $S1×M$, where $M$ is a closed, connected and oriented 3–manifold. We prove that if $S1×M$ admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true:

$S1×M$ admits a symplectic structure if and only if $M$ fibers over $S1$,

under the additional assumption that $M$ has no fake 3–cells. We also discuss the relationship between the geometry of $M$ and complex structures and Seifert fibrations on $S1×M$.

#### Article information

Source
Algebr. Geom. Topol., Volume 1, Number 1 (2001), 469-489.

Dates
Accepted: 6 September 2001
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882604

Digital Object Identifier
doi:10.2140/agt.2001.1.469

Mathematical Reviews number (MathSciNet)
MR1852768

Zentralblatt MATH identifier
0977.57010

#### Citation

Etgu, Tolga. Lefschetz fibrations, complex structures and Seifert fibrations on $S^1 \times M^3$. Algebr. Geom. Topol. 1 (2001), no. 1, 469--489. doi:10.2140/agt.2001.1.469. https://projecteuclid.org/euclid.agt/1513882604

#### References

• S Baldridge, Seiberg–Witten invariants of 4–manifolds with free circle actions, preprint, 1999..
• O Biquard, Les équations de Seiberg–Witten sur une surface complexe non Kählérienne, Comm. Anal. Geom. 6 (1998), 173–197.
• W Barth, C Peters, A van de Ven, Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol.4, Springer–Verlag, 1984.
• W Chen, R Matveyev, Symplectic Lefschetz fibrations on ${S}^1 \times {M}$, Geometry and Topology, 4 (2000), 517–535.
• S Donaldson, Lefschetz fibrations in symplectic geometry, Proc. ICM, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 309–314.
• M Fernández, M Gotay, A Gray, Compact parallelizable four dimensional symplectic and complex manifolds, Proc. Amer. Math. Soc. 103 (1988), no.4, 1209–1212.
• R Filipkiewicz, Four–dimensional geometries, Ph.D. Thesis, University of Warwick, 1984.
• H Geiges, Symplectic structures on ${T}^2$–bundles over ${T}^2$, Duke Math. J. 67 (1992), no.3, 539–555.
• H Geiges, J Gonzalo, Contact geometry and complex surfaces, Invent. Math. 121 (1995), 147–209.
• R Gompf, A Stipsicz, 4–manifolds and Kirby calculus, Graduate Studies in Math., vol.20, American Mathematical Society, 1999.
• J Hempel, $3$–manifolds, Ann. of Math. Studies, Princeton University Press, 1976.
• M Kato, Topology of Hopf surfaces, J. Math. Soc. Japan 27 (1975), 222–238 and 41 (1989), 173–174.
• D Kotschick, Remarks on geometric structures on compact complex surfaces, Topology 31 (1992), no.2, 317–321.
• P Kronheimer, Embedded surfaces and gauge theory in three and four dimensions, Surveys in Differential Geometry, (Cambridge, MA, 1996), vol. III, International Press, 1998, 243–298.
• Y Matsumoto, Diffeomorphism types of elliptic surfaces, Topology 25 (1986), 549–563.
• J McCarthy, On the asphericity of a symplectic ${M}^3 \times {S}^1$, Proc. Amer. Math. Soc. 129 (2001), 257–264.
• J Milnor, A unique factorisation theorem for 3–manifolds, Amer. J. Math. 79 (1957), 623–630.
• J Morgan, The Seiberg–Witten invariants and applications to the topology of smooth four–manifolds, Mathematical Notes, vol.44, Princeton University Press, 1996.
• T Mrowka, P Ozsváth, B Yu, Seiberg–Witten monopoles on Seifert fibered spaces, Comm. Anal. Geom. 5 (1997), 685–791.
• D McDuff, D Salamon, Introduction to Symplectic Topology, Oxford Mathematical Monographs, Oxford University Press, 1995.
• W Neumann, F Raymond, Seifert manifolds, plumbing, $\mu$–invariant and orientation reversing maps, Algebraic and Geometric Topology (Santa Barbara, 1977), Lecture Notes in Mathematics, vol.664, Springer–Verlag, 1978, pp.163–196.
• P Orlik, Seifert manifolds, Lecture Notes in Mathematics, vol.291, Springer–Verlag, 1972.
• C Okonek, A Teleman, 3–dimensional Seiberg–Witten invariants and non–Kählerian geometry, Math. Ann. 312 (1998), 261–288.
• P Scott, The geometries of 3–manifolds, Bull. London Math. Soc. 15 (1983), 401–487.
• C Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809–822.
• C Taubes, More constraints on symplectic forms from the Seiberg–Witten invariants, Math. Res. Lett. 2 (1995), 9–13.
• C Taubes, The geometry of the Seiberg–Witten invariants, Proc. ICM, Vol. II (Berlin, 1998), Doc. Math., Extra Vol. II, 1998, 493–504.
• A Teleman, Projectively flat surfaces and Bogomolov's theorem on ${C}lass$ $\rm{VII}_0$ surfaces, Internat. J. Math. 5 (1994), 253–264.
• W Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no.2, 467–468.
• M Ue, Geometric 4–manifolds in the sense of Thurston and Seifert 4–manifolds I, J. Math. Soc. Japan 42 (1990), no.3, 511–540.
• M Ue, Geometric 4–manifolds in the sense of Thurston and Seifert 4–manifolds II, J. Math. Soc. Japan 43 (1991), no.1, 149–183.
• S Vidussi, The Alexander norm is smaller than the Thurston norm: a Seiberg–Witten proof, Prépublication École Polytechnique 99–6.
• C T C Wall, Geometries and geometric structures in real dimension 4 and complex dimension 2, Geometry and Topology (College Park, Maryland, 1983/84), Lecture Notes in Mathematics, vol.1167, Springer–Verlag, 1985, 268–292.
• C T C Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), no.2, 119–153.
• E Witten, Monopoles and four–manifolds, Math. Res. Lett. 1 (1994), 769–796.
• J Wood, Harmonic morphisms, conformal foliations and Seifert fibre spaces, Geometry of low–dimensional manifolds:1, (Durham, 1989), London Math. Soc. Lecture Note Series, vol.150, Cambridge University Press, 1990, 247–259.