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International Mathematics Research Notices (2006) Vol. 2006 : article ID 70272, 21 pages, doi:10.1155/IMRN/2006/70272
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Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

Realizing 4-manifolds as achiral Lefschetz fibrations

John B. Etnyre and Terry Fuller

School of Mathematics, Georgia Institute of Technology Atlanta, GA 30332-0160, USA E-mail address: etnyre{at}math.gatech.edu
Department of Mathematics, California State University Northridge, CA 91330, USA E-mail address: terry.fuller{at}csun.edu

We show that any 4-manifold, after surgery on a curve, admits an achiral Lefschetz fibration. In particular, if X is a simply connected 4-manifold, we show that X#S2xS2 and $$X\#{S}^{2}\tilde{\times }{S}^{2}$$ both admit achiral Lefschetz fibrations. We also show that these surgered manifolds admit near-symplectic structures, and prove more generally that achiral Lefschetz fibrations with sections have near-symplectic structures. As a corollary to our proof, we obtain an alternate proof of Gompf's result on the existence of symplectic structures on Lefschetz pencils.


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