Lefschetz fibrations, complex structures and Seifert fibrations on S1 × M3

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

S^{1} \times 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 $S^{1} \times M$ .

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

References

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Publication

Received: 7 August 2001

Accepted: 6 September 2001

Published: 9 September 2001

Authors

Tolga Etgu
Department of Mathematics University of California at Berkeley Berkeley CA 94720 USA

Volume 1, issue 1 (2001)

Tolga Etgu