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.
For each pair (e, σ) of integers satisfying 2e + 3σ ≥ 0, σ ≤ −2, and e + σ ≡ 0 (mod 4), with four exceptions, we construct a minimal, simply connected symplectic 4-manifold with Euler characteristic e and signature σ. We also produce simply connected, minimal symplectic 4-manifolds with signature zero (resp. signature −1) with Euler characteristic 4k (resp. 4k + 1) for all k ≥ 46 (resp. k ≥ 49).
Abstract. The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to CP 2 #(2k + 1)CP 2 for k = 1, . . . , 4, or to 3CP 2 #(2l + 3)CP 2 for l = 1, . . . , 6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4-manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on CP 2 #3CP 2 , 3CP 2 #5CP 2 and 3CP 2 #7CP 2 .
Abstract. We investigate the possible self-intersection numbers for sections of surface bundles and Lefschetz fibrations over surfaces. When the fiber genus g and the base genus h are positive, we prove that the adjunction bound 2h − 2 is the only universal bound on the self-intersection number of a section of any such genus g bundle and fibration. As a side result, in the mapping class group of a surface with boundary, we calculate the precise value of the commutator lengths of all powers of a Dehn twist about a boundary component, concluding that the stable commutator length of such a Dehn twist is 1/2. We furthermore prove that there is no upper bound on the number of critical points of genus-g Lefschetz fibrations over surfaces with positive genera admitting sections of maximal self-intersection, for g ≥ 2.
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