#### Volume 1, issue 1 (2001)

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

### Tolga Etgu

Algebraic & Geometric Topology 1 (2001) 469–489
 arXiv: math.SG/0109150
##### Abstract

We consider product 4–manifolds ${S}^{1}×M$, where $M$ is a closed, connected and oriented 3–manifold. We prove that if ${S}^{1}×M$ admits a complex structure or a Lefschetz or Seifert fibration, then the following statement is true:

${S}^{1}×M$ admits a symplectic structure if and only if $M$ fibers over ${S}^{1}$,

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 ${S}^{1}×M$.

##### Keywords
product 4–manifold, Lefschetz fibration, symplectic manifold, Seiberg–Witten invariant, complex surface, Seifert fibration
##### Mathematical Subject Classification 2000
Primary: 57M50, 57R17, 57R57
Secondary: 53C15, 32Q55