On sections of elliptic fibrations

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.

On sections of genus two Lefschetz fibrations

“…, 5. In [Korkmaz and Ozbagci 2008], similar relations were found in the mapping class group 1,n of a genus 1 surface with n boundary components for n = 4, . .…”

Fillings of Genus–1 Open Books and 4–Braids

“…This follows from the following relation in Γ 3 1 , known as the star relation [20]: (t a1 t a2 t a3 t b2 ) 3 = t δ1 t δ2 t δ3 , which can be derived by applying the lantern relation and braid relations to the well-known 3-chain relation [23]. For ∆ = t δ1 t δ2 t δ3 the boundary multitwist, and η = t a1 t a2 t a3 , we can rewrite it as η t b2 η t b2 η t b2 = ∆…”

“…The star relation prescribes a genus-1 elliptic fibration with three (−1)-sections [23]. The lemma therefore implies that we can construct the Stein filling of T 3 as the complement of the union of an I 9 fiber (which one gets by clustering the nodal singularities induced by all the vanishing cycles a i ) and three (−1)-sections.…”

Elliptic open books on torus bundles over the circle

“…So the monodromy is φ = δ 1 δ 2 α −3 1 α −3 2 (note that since the curves are pairwise disjoint, all these Dehn twists commute with each other). We have the relation δ 1 δ 2 = (α 1 α 2 β) 4 , where β is the right-handed Dehn twists around the essential curve disjoint from the δ i 's and intersecting each α i geometrically once (see [12] for this and other such relations on punctured tori). It is crucial that we have no more than 4 parallel left-handed Dehn twists on a twice-punctured FIGURE 4.…”

“…On the other hand, when k ≤ 2, we have right-handed Dehn twists around boundary components. In particular, when k = 0 or 1, by using certain relations [12] in the mapping class group of torus with 8 and 7 punctures, respectively , the monodromy can be written as a product of right-handed Dehn twists implying that ob ′ k is compatible with a Stein fillable contact structure. Elliptic open books compatible with For k = 2 case we weren't able to verify that the monodromy of ob ′ 2 can be written as a product of right-handed Dehn twists, but obtained another elliptic open book on Y 2 which is compatible with a Stein fillable contact structure.…”