Broken Lefschetz fibrations and smooth structures on 4–manifolds

Abstract

Broken Lefschetz fibrations are generalizations of Lefschetz fibrations on smooth 4–manifolds, which are allowed to have indefinite fold singularities along embedded circles in addition to Lefschetz type singularities on a discrete set. In the recent past, there has been a flurry of activity around broken Lefschetz fibrations, extending ideas stemmed in symplectic geometry and gauge theory on one end by Auroux–Donaldson– Katzarkov, and Perutz [2; 27], and employing handlebody and singularity theories to suggest new ways to study the topology of 4–manifolds on the other end by the author, Saeki, Gay–Kirby, Lekili, Akbulut–Karakurt, Williams, Hayano, and others [1; 4; 5; 6; 7; 8; 9; 14; 15; 17; 18; 19; 20; 23; 28; 29]. Given any surjective continuous map from a closed oriented 4–manifold X to the 2–sphere, there exists a rather special broken Lefschetz fibration on X within the same homotopy class, with only connected fibers and no exceptional spheres contained on the fibers, with at most one circle of indefinite fold singularities whose image in the base is embedded, and where all the Lefschetz critical points lie on fibers with the highest genus. These are called simplified broken Lefschetz fibrations (SBLF in short), and constitute an important subfamily of broken Lefschetz fibrations, allowing one to study the underlying topology effectively. (See for instance [4; 5; 6; 7; 17; 18; 19; 20].) The underlying topology of a simplified broken Lefschetz fibrations is rather simple: It is either a relatively minimal genus g Lefschetz fibration over the 2–sphere, or it decomposes as a relatively minimal genus g Lefschetz fibration over a 2–disk, a trivial genus g 1 bundle over a 2–disk, and a fibered cobordism in between prescribed by a single round handle [6].