Wrinkled fibrations on near-symplectic manifolds

By Yankı Lekili. 2007

75 citations recorded by the source corpus.

Motivated by the programmes initiated by Taubes and Perutz, we study the geometry of near-symplectic 4-manifolds, i.e., manifolds equipped with a closed 2-form which is symplectic outside a union of embedded 1-dimensional submanifolds, and broken Lefschetz fibrations on them. We present a set of four moves which allow us to pass from any given fibration to any other broken fibration which is deformation equivalent to it. Moreover, we study the change of the near-symplectic geometry under each of these moves. The arguments rely on the introduction of a more general class of maps, which we call wrinkled fibrations and which allow us to rely on classical singularity theory.Finally, we illustrate these constructions by showing how one can merge components of the zero-set of the near-symplectic form. We also disprove a conjecture of Gay and Kirby by showing that any achiral broken Lefschetz fibration can be turned into a broken Lefschetz fibration by applying a sequence of our moves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/736022b575b6ece664c02e2e72dc963339a77cd9

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In Geometry and Topology

1. R. Baykur
2. R. Kirby
3. David T. Gay
4. Chris Wendl
5. Yankı Lekili
6. Clifford H. Taubes

In Mathematical Physics

1. R. Baykur
2. R. Kirby
3. David T. Gay
4. Chris Wendl
5. Yankı Lekili
6. Clifford H. Taubes

Citations in the corpus

• Existence of broken Lefschetz fibrations (2008, background): “…the next section, combining different topological modifications of singularities ([6, 9, 4]), we will show how one can effectively eliminate the definite fold singularities and cusps so to homotope a generic map to a broken Lefschetz fibration, implying the abundance of broken Lefschetz fibrations.”
• Trisecting 4–manifolds (2012, background): “Figure 23 introduces a swallowtail at the vertical tangency of the kink; this move has been discussed extensively elsewhere [7] and is a standard singularity that occurs in a homotopy between homotopies between Morse functions.”