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Symplectic Lefschetz fibrations on $S^1 \times M^3$

Weimin Chen and Rostislav Matveyev

Geometry & Topology 4 (2000) 517–535

arXiv: math.DG/0002022

Abstract

In this paper we classify symplectic Lefschetz fibrations (with empty base locus) on a four-manifold which is the product of a three-manifold with a circle. This result provides further evidence in support of the following conjecture regarding symplectic structures on such a four-manifold: if the product of a three-manifold with a circle admits a symplectic structure, then the three-manifold must fiber over a circle, and up to a self-diffeomorphism of the four-manifold, the symplectic structure is deformation equivalent to the canonical symplectic structure determined by the fibration of the three-manifold over the circle.

Keywords
4–manifold, symplectic structure, Lefschetz fibration, Seiberg–Witten invariants
Mathematical Subject Classification 2000
Primary: 57M50
Secondary: 57R17, 57R57
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Publication
Received: 12 April 2000
Revised: 8 December 2000
Accepted: 17 December 2000
Published: 21 December 2000
Proposed: Dieter Kotschick
Seconded: Robion Kirby, Yasha Eliashberg
Authors
Weimin Chen
University of Wisconsin at Madison
Madison
Wisconsin 53706
USA
Rostislav Matveyev
SUNY at Stony Brook
Stony Brook
New York 11794
USA