Symplectic mapping class group relations generalizing the chain relation

Abstract

Abstract. In this paper, we examine mapping class group relations of some symplectic manifolds. For each n ě 1 and k ě 1, we show that the 2n-dimensional Weinstein domain W " tf " δu X B 2n`2 , determined by the degree k homogeneous polynomial f P Crz0, . . . , zns, has a Boothby-Wang type boundary and a right-handed fibered Dehn twist along the boundary that is symplectically isotopic to a product of right-handed Dehn twists along Lagrangian spheres. We also present explicit descriptions of the symplectomorphisms in the case n " 2 recovering the classical chain relation for the torus with two boundary components.

Lefschetz-Bott fibrations on line bundles over symplectic manifolds

“…One can interpret the above relation from the point of view of symplectic fillings of the link of the singularity 0 ∈ X n+1 d as follows. As Acu and Avdeck pointed out in [1], the product of Dehn twists comes from a Lefschetz fibration on a smoothing of the singularity, i.e., a Milnor fiber of the singularity. It serves as a Stein filling of the link of the singularity.…”

“…[18, Section 2.5]). Let π : (E, Ω) → D be a symplectic Lefschetz-Bott fibration, (V, dλ) = (π −1 (1), Ω| π −1 (1) ) and φ ∈ Symp c (V, dλ) the monodromy of π along ∂D. Suppose Ω is non-degenerate on E and exact near ∂E and on each regular fiber of π.…”

“…We will apply the above theorem to construct various strong symplectic fillings of contact manifolds. In their paper, Acu and Avdek [1] (cf. [2]) gave relations in symplectic mapping class groups of Milnor fibers.…”

The quantum Johnson homomorphism and symplectomorphism of 3-folds

“…For homology classes of degree = 3, this is immediate since the monodromy is compactly supported near the vanishing cycles (say near T ). But in fact, we can do better -the monodromy is the product of two β = 6 powers of the total monodromy of an A 5 -singularities (which are a special case of weighted homogeneous singularities) which is symplectically isotopic to the right-handed fibered Dehn twist that comes from the circle action on the contact type boundary (See: [Sei00, Lemma 4.16], and the recent [AA16].) In particular, it is compactly supported near ∂T and can be made disjoint on the chain-level from the vanishing cycles themselves.…”