Constructing Lefschetz-type fibrations on four-manifolds

2007/01/03 by David Gay, David T. Gay, Robion Kirby · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Geometric and Algebraic Topology #Geometry #Gravitational singularity #Homotopy #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Pure mathematics #SPHERES #Surface (topology) #Type (biology) #math.GT #math.SG #msc:57M50 #msc:57R17

paper · pdf · doi:10.2140/gt.2007.11.2075

published as Geom. Topol. 11 (2007) 2075-2115 · 42 pages, 13 figures

arxiv created 2007/01/03 · openalex publication_date 2007/10/15 · arxiv updated 2014/11/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We show how to construct broken, achiral Lefschetz fibrations on arbitrary smooth, closed, oriented 4-manifolds. These are generalizations of Lefschetz fibrations over the 2-sphere, where we allow Lefschetz singularities with the non-standard orientation as well as circles of singularities corresponding to round 1-handles. We can also arrange that a given surface of square 0 is a fiber. The construction is easier and more explicit in the case of doubles of 4-manifolds without 3-and 4-handles, such as the homotopy 4-spheres arising from nontrivial balanced presentations of the trivial group.

Citations

• The topology of four-dimensional manifolds
• Lefschetz pencils on symplectic manifolds
• Akbulut's corks and h-cobordisms of smooth simply connected 4-manifolds
• Lagrangian matching invariants for fibred four-manifolds: II
• Classification of overtwisted contact structures on 3-manifolds
• Killing the Akbulut-Kirby 4-sphere, with relevance to the Andrews-Curtis and Schoenflies problems
• Lagrangian matching invariants for fibred four-manifolds: I

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