Algebraic & Geometric Topology

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

Tolga Etgu

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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

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

Received: 7 August 2001
Accepted: 6 September 2001
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M50: Geometric structures on low-dimensional manifolds 57R17: Symplectic and contact topology 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 53C15: General geometric structures on manifolds (almost complex, almost product structures, etc.) 32Q55: Topological aspects of complex manifolds

product 4–manifold Lefschetz fibration symplectic manifold Seiberg–Witten invariant complex surface Seifert fibration


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.

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