Causas

In order to apply Lemma 7.2.4 we cut off h so that it has support in the chosen neigh- bourhood U of K and is equal to h on some possibly smaller neighbourhood. By abuse of notation we continue to call this function h. We then add a long neck toX and consider the subset

Xh =X∪ {(p, t) | 0≤t ≤h(p)}.

This is again a strong symplectic filling of M with symplectic dilation ∂

∂t. We next identify

∂X with ∂Xh ={(p, h(p))} by sendingp7→(p, h(p)). Under this identification the pullback

of the contact structure is exactly ehα.

Hence, by using a Darboux chart to identifyU withW and matching inward and outward pointing symplectic dilations, we may smoothly attach H to X in a way that is compatible with both symplectic structures at the expense of multiplying ω0 by some large constant.

Moreover, if K ×(−,0] were a product piece of a symplectic spanning surface, then the extension of this to Xh is again symplectic with transverse boundary and under the gluing

map described above such a surface will attach smoothly to the symplectic core discr2 = 0 in H. Thus, if Σ was a symplectic spanning surface with boundary K, then we have described how to cap it off to get a closed symplectic surface ˆΣ = Σ∪D2 _{in the interior of a symplectic}

manifold ˆX. Furthermore, by ([Gay], Theorem 1.1) we may assume that ˆX is a convex symplectic filling. With this construction we may now prove Theorem 7.2.3.

Proof of Theorem 7.2.3. We let L be a transverse link. Then by attaching handles along

each component as above we obtain an embedded symplectic surface ˆ

Σ = Σ∪k_i=1D2_i

in the interior of some convex filling ˆX. By Theorem 1.3 in [Eli2] we may symplectically embed ˆX in a closed symplectic manifoldY so that the Symplectic Thom conjecture implies that ˆΣ minimises genus in its homology class inY. Now let Σ0 be any other spanning surface ofL, then we can close this up in ˆX to a surface ˆΣ0 , which we then embed in Y. IfH2(X) is

trivial, then [ ˆΣ] = [ ˆΣ0] as homology classes inY. Thus, as ˆΣ minimises genus in its homology class, we have

−(χ(Σ) +k) =−χ( ˆΣ)≤ −χ( ˆΣ0) =−(χ(Σ0) +k).

We conclude that−χ(Σ) ≤ −χ(Σ0) for all spanning surfaces of L and hence

−χ(Σ)≤χX_min(L).

The opposite inequality is obvious by the definition ofχX

min and henceχXmin(L) = −χ(Σ).

The main examples for which the hypotheses of Theorem 7.2.3 hold are S3 _{and S}2_×S1

with their Stein fillings B4 and B3×S1 respectively.

7.3

Symplectic links are quasipositive in

S

In this section we shall consider the special case of symplectic links in (S3, ξst) considered as

116 7. Symplectic cobordism and transverse knots

are symplectic has been considered in [BO], where it is shown that the symplectic links are precisely those that arequasipositive. Before stating this result we need to recall the notion of a braid and Rudolph’s definition of quasipositivity.

The braid group onn-strandsBnis defined as the fundamental group of the configuration

space Cn = (Cn \ ∆n)/Sn, where ∆n is the diagonal subvariety consisting of all vectors

(z1, z2, ... , zn) for which at least two distinct entries are equal and Sn acts by permutations

on the coordinate vectors. Inherent in this definition is the choice of base point, which can be solved by considering free homotopy classes of loops, or equivalently conjugacy classes of braids. Now the braid group is generated by positive half twists σ1, ..., σn−1 and has the

following presentation

Bn=hσ1, ..., σn−1 |σiσjσi =σjσiσj if |i−j|= 1 andσiσj =σjσi if |i−j| ≥2i.

An important fact that we shall need is that the braid group is isomorphic to the mapping class groupM CGc_(D2_,{p

1, p2..., pn}) of compactly supported diffeomorphisms ofD2 that fix

n marked points{p1, p2..., pn}as a set (cf. [Bir]). Given an elementβ ∈Bnwe can construct

its closure inS1_×D2 _{that we denote by ˆβ. This is obtained by taking the associated element} φ ∈M CGc_(D2_,{p

1, p2..., pn}) and defining ˆβ to be the image of [0,1]× {p1, p2..., pn} in the

mapping torus defined by φ.

Definition 7.3.1 (Quasipositivity, [Rud1]). A braid β is called quasipositive if it has a factorisation of the form

β =

k

Y

i=1

Qiσ1Q−1i ,

where Qi ∈Bn are arbitrary braids.

Any link in S3 _{is isotopic to a link in braided position (cf. [Bir]), thus when we speak}

of a quasipositive link we will mean that it is quasipositive after being braided. With this notion of quasipositivity we may state the following result.

Theorem 7.3.2 ([BO]). A link L⊂S3 _{is symplectic if and only if it is quasipositive.}

Remark 7.3.3. In fact Boileau and Orevkov show that if Σ is any symplectic spanning surface

inB4_{, then the pair (B}4_{,Σ) is diffeomorphic to (B}4_,Σ

alg), where Σalg is a piece of algebraic

curve that is quasipositive (cf. Example 7.1.11). We also note that the proof of Theorem 7.3.2 in [BO] requires only that the link in question is positively transverse with respect to the contact structure without any additional assumption on the asymptotics of the spanning surface.

Thus, when studying transverse links from the perspective of symplectic cobordism, it is reasonable to single out the set of null-bordant links, which in view of Theorem 7.3.2 is just the set of quasipositive links and we will denote this set by QP. There is an obvious map onQP given by χmin =χB

4

min, which by Theorem 7.2.3 is order preserving.

Proposition 7.3.4. Let L1, L2 ∈ QP and assume that L1 ≺sL2 via a connected symplectic

cobordism Σ, then

χmin(L1) = −χ(Σ) +χmin(L2).