#### Algebraic and Geometric Topology 1 (2001),
paper no. 24, pages 469-489.

## Lefschetz fibrations, complex structures and Seifert fibrations on S^1 X M^3

### Tolga Etgu

**Abstract**.
We consider product 4--manifolds S^1 X M, where M is a closed,
connected and oriented 3-manifold. We prove that if S^1 X M admits a
complex structure or a Lefschetz or Seifert fibration, then the
following statement is true: S^1 X 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 X
M.
**Keywords**.
Product 4-manifold, Lefschetz fibration, symplectic manifold,
Seiberg-Witten invariant, complex surface, Seifert fibration

**AMS subject classification**.
Primary: 57M50, 57R17, 57R57.
Secondary: 53C15, 32Q55.

**DOI:** 10.2140/agt.2001.1.469

**E-print:** `arXiv:math.SG/0109150`

Submitted: 7 August 2001.
Accepted: 6 September 2001.
Published: 9 September 2001.

Notes on file formats
Tolga Etgu

Department of Mathematics

University of California at Berkeley

Berkeley, CA 94720, USA

Email: tolga@math.berkeley.edu

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