Let M be a compact Stein surface with boundary. We show that M admits infinitely many pairwise nonequivalent positive allowable Lefschetz fibrations over D 2 with bounded fibers.

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Abstract. In this note we define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book).

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