No abstract

ABSTRACT. We find a new relation among right-handed Dehn twists in the mapping class group of a k-holed torus for 4 ≤ k ≤ 9. This relation induces an elliptic Lefschetz fibration on the complex elliptic surface E(1) → S 2 with twelve singular fibers and k disjoint sections. More importantly we can locate these k sections in a Kirby diagram of the induced elliptic Lefschetz fibration. The n-th power of our relation gives an explicit description for k disjoint sections of the induced elliptic Lefschetz fibration on the complex elliptic surface E(n) → S 2 for n ≥ 2.

We construct examples of Lefschetz fibrations with prescribed singular fibers. By taking differences of pairs of such fibrations with the same singular fibers, we obtain new examples of surface bundles over surfaces with non-zero signature. From these we derive new upper bounds for the minimal genus of a surface representing a given element in the second homology of a mapping class group.Comment: 20 pages, 7 figures, accepted for publication in Topolog

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.

Abstract. We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP 2 #p CP 2 for p = 7, 8, 9, and to 3CP 2 #q CP 2 for q = 12, . . . , 19. Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with b + ≤ 3, with one possible exception when b + = 3. Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of CP 2 #4CP 2 and 3CP 2 #6CP 2 from small Lefschetz fibrations over surfaces of higher genera.