Sections of surface bundles and Lefschetz fibrations

Abstract

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.

Families of contact 3-manifolds with arbitrarily large Stein fillings

“…For any section of a genus g Lefschetz fibration over a genus h ≥ 1 surface, its self-intersection number is determined by the number of critical points when g = 1, and is bounded above by 2h − 2, when g ≥ 2 and h ≥ 1, as shown in [6]. So the triples (g, h, n) realized in the theorem above are all one can possibly get.…”

“…All the relations below should be understood to take place in Γ 1 g . Our key input is the following family of relations obtained in [6] (See proof of Theorem 21; relations [12][13][14][15][16][17][18][19][20]. See Figure 8 for the curves that appear below.…”

“…We will also show that the Stein fillings of (Y (2, h, n), ξ 2,h,n ) can be chosen so that they have arbitrarily small (negative) signatures. All these examples are derived from special families of Lefschetz fibrations on closed 4-manifolds (Theorem 4.1), which are built using relations in the mapping class groups of surfaces with boundaries after [6]. Lastly, we outline how to get similar families of Stein fillings of a fixed contact structure on more general 3-manifolds, so as to illustrate that the contact 3-manifolds (Y (g, h, n), ξ g,h,n ) above are nowhere close to being special in this sense.…”

Small Lefschetz fibrations and exotic 4-manifolds

“…Let Σ The positive factorization in (1) above gives rise to a genus-g Lefschetz fibration (X, f ) with l critical points and m disjoint sections S j of self-intersection S 2 j = −k j [11]. Identifying the regular fiber F with Σ g , we can view the vanishing cycles of f as c i .…”

Smoothings of singularities and symplectic surgery

“…Next, we construct a symplectic manifold which contains the curve configuration (A, B) described above. According to [2,Theorem 1] there is a surface bundle X → Σ N −2 with fiber genus 15N 2 −47N +34 over the surface with genus N − 2 such that there is a section with self-intersection −N . Let M denote the blow-up of X in a fiber.…”