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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 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.

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
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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