Topology of broken Lefschetz fibrations and near-symplectic four-manifolds

Abstract

The topology of broken Lefschetz fibrations is studied by means of handle decompositions. We consider a slight generalization of round handles and describe the handle diagrams for all that appear in dimension four. We establish simplified handlebody and monodromy representations for a certain subclass of broken Lefschetz fibrations and pencils, showing that all near-symplectic closed 4-manifolds can be supported by such objects, paralleling a result of Auroux, Donaldson and Katzarkov. Various constructions of broken Lefschetz fibrations and a generalization of the symplectic fiber sum operation to the near-symplectic setting are given. Extending the study of Lefschetz fibrations, we detect certain constraints on the symplectic fiber sum operation to result in a 4-manifold with nontrivial Seiberg-Witten invariant, as well as the self-intersection numbers that sections of broken Lefschetz fibrations can acquire. IntroductionIn the last decade, symplectic topology has been extensively used to explore the world of smooth 4-manifolds, where Donaldson's work providing a description of symplectic 4-manifolds in terms of Lefschetz fibrations up to blow-ups played a remarkable role. Auroux, Donaldson and Katzarkov [Auroux et al. 2005] extended this result, establishing a correspondence between the larger class of nearsymplectic 4-manifolds and an appropriate generalization of Lefschetz fibrations up to blow-ups. A detailed topological study of these fibrations, called broken Lefschetz fibrations herein, and generalization of various ideas from the study of symplectic 4-manifolds to this broader setting are the main themes of our article.Our goal is to give handlebody descriptions of broken Lefschetz fibrations to assist with identifying the total spaces of these fibrations and with calculating smooth invariants. Although we only refer to the Seiberg-Witten invariant in this paper, two other invariants motivate our studies very much. One is the Heegaard-Floer invariant of Ozsváth and Szabó which fits in a TQFT and makes use of handle MSC2000: primary 57M50, 57R65; secondary 57R17.

Wrinkled fibrations on near-symplectic manifolds

“…An important such deformation is provided by the Luttinger-Simpson model given on D 4 R 4 where the birth (or death) of a circle can be observed explicitly [12]:…”

“…The Figure 18 drawn in the most symmetric fashion presents a different choice of three vanishing cycles D 1 , D 2 and D 3 on the twice punctured torus, and through similar arguments as above one can see that this picture stands for a broken Lefschetz fibration that can be used to replace a positive Lefschetz singularity locally. This contains the nontrivial part of Perutz's example of a broken Lefschetz fibration (Example 1.3 in [12]; also see Example 3.2 in [4]), and corresponds to the broken Lefschetz fibration that Lekili obtains after perturbing a positive Lefschetz singularity in his paper. One can then draw the curves C 1 , C 2 and C 3 by joining the vertices of the hexagon formed by D 1 , D 2 , D 3 in the center as in Figure 18, and get the picture we had above (Figure 16, on the left) up to isotopy.…”

“…The 2-handles corresponding to Lefschetz handles have fiber framing minus one, and the 2-handle of the round 2-handle given in red has fiber framing zero. The reader can turn to [4] for the conventions we use to depict broken Lefschetz fibrations in this appendix.…”

Maps on 3-manifolds given by surgery

“…In a slightly different direction, [Costantino and Thurston 2008] gives a lower bound for the number of crossing singularities of stable maps from a 3-manifold to ‫ޒ‬ 2 in terms of the Gromov norm of the 3-manifold. Recently Baykur [2008;2009] and Gay and Kirby [2007] studied the topology of 4-manifolds through the singularities of their maps into surfaces.…”

“…(2) By taking an embedding ‫ޒ‬ 2 ⊂ S 2 we get for every closed orientable 3-manifold a stable map into S 2 as well without singular points of types (C) and (E). Then by using the method of [Saeki 2006], for example, for eliminating the singular points of type (A), we get a stable map, which is a direct analogue of the indefinite generic maps appearing in [Baykur 2008;2009;Gay and Kirby 2007].…”