Citation Context

...) is diffeomorphic to S 3 p(U) (where here U is the unknot) under an orientationpreserving diffeomorphism, then K is the unknot. The first ingredient in the proof of Theorem 1.1 is a theorem of Gabai =-=[8]-=- which expresses the minimal genus problem in terms of taut foliations. This result, togetherHOLOMORPHIC DISKS AND GENUS BOUNDS 3 with a theorem of Eliashberg and Thurston [5] gives a reformulation i...

Citation Context

... can be viewed as equivalence classes of nowhere vanishing vector fields over Y , where two vector fields are considered equivalent if they are homotopic in the complement of a ball in Y , c.f. [40], =-=[12]-=-. Dually, an oriented two-plane distribution gives rise to an equivalence class of nowhere vanishing vector fields (which are transverse to the distribution, and form a positive basis for TY ). Now, t...

Citation Context

...etching the construction, we describe a refinement which lives in Floer homology with twisted coefficients. The contact invariant is constructed with the help of some work of Giroux. Specifically, in =-=[11]-=-, Giroux shows that contact structures over Y are in one-to-one correspondence with equivalence classes of open book decompositions of Y , under an equivalence relation given by a suitable notion of s...

Citation Context

...ield-theoretic properties of Heegaard Floer homology, together with the suitable handle-decomposition of an arbitrary symplectic four-manifold induced from the Lefschetz pencils provided by Donaldson =-=[2]-=-. (The non-vanishing result from [32] is analogous to a non-vanishing theorem for the Seiberg-Witten invariants of symplectic manifolds proved by Taubes, c.f. [36] and [37].) 1.1. Contact structures. ...

Citation Context

...that ω extends smoothly over X = W ∪Y V . Although Eliashberg’s is the construction we need, concave fillings have been constructed previously in a number of different contexts, see for example [22], =-=[1]-=-, [7], [10], [25]. Indeed, since the first posting of the present article, Etnyre pointed out to us an alternate proof of Eliashberg’s theorem [6], see also [25]. In the construction, V is given as th...

Citation Context

...nd Thurston [5] gives a reformulation in terms of certain symplectically semi-fillable contact structures. The final breakthrough which makes this paper possible is an embedding theorem of Eliashberg =-=[3]-=-, see also [6] and [25], which shows that a symplectic semi-filling of a three-manifold can be embedded in a closed, symplectic four-manifold. From this, we then appeal to a theorem [32], which implie...

Citation Context

... boundary Y . A fourmanifold W is said to have convex boundary if there is a contact structure ξ over Y with the property that the restriction of ω to the two-planes of ξ is everywhere positive, c.f. =-=[3]-=-. Indeed, if we fix the contact structure Y over ξ, we say that W is a convex weak symplectic filling of (Y, ξ). If W is a convex weak symplectic filling of a possibly disconnected three-manifold Y ′ ...

Citation Context

...e (Y, ξ) admits a weak convex symplectic filling, it is called weakly fillable. Note that every contact structure (Y, ξ) can be realized as the concave boundary of some symplectic four-manifold (c.f. =-=[7]-=-, [10], and [3]). This is one justification for dropping the modifier “convex” from the terminology “weakly fillable”. If a contactHOLOMORPHIC DISKS AND GENUS BOUNDS 7 structure (Y, ξ) admits a weak ...

Citation Context

...result in turn leads to an alternate proof of a theorem proved jointly by Kronheimer, Mrowka, and us [19], first conjectured by Gordon [13] (the cases where p = 0 and ±1 follow from theorems of Gabai =-=[9]-=- and Gordon and Luecke [14] respectively): Corollary 1.3. [19] Let K ⊂ S 3 be a knot with the property that for some integer p, S 3 p(K) is diffeomorphic to S 3 p(U) (where here U is the unknot) under...

Citation Context

...the largest integer s for which the group ̂HFK∗(K, s) ̸= 0. This result in turn leads to an alternate proof of a theorem proved jointly by Kronheimer, Mrowka, and us [19], first conjectured by Gordon =-=[13]-=- (the cases where p = 0 and ±1 follow from theorems of Gabai [9] and Gordon and Luecke [14] respectively): Corollary 1.3. [19] Let K ⊂ S 3 be a knot with the property that for some integer p, S 3 p(K)...

Citation Context

... ξ) admits a weak convex symplectic filling, it is called weakly fillable. Note that every contact structure (Y, ξ) can be realized as the concave boundary of some symplectic four-manifold (c.f. [7], =-=[10]-=-, and [3]). This is one justification for dropping the modifier “convex” from the terminology “weakly fillable”. If a contactHOLOMORPHIC DISKS AND GENUS BOUNDS 7 structure (Y, ξ) admits a weak symple...

Citation Context

...] gives a reformulation in terms of certain symplectically semi-fillable contact structures. The final breakthrough which makes this paper possible is an embedding theorem of Eliashberg [3], see also =-=[6]-=- and [25], which shows that a symplectic semi-filling of a three-manifold can be embedded in a closed, symplectic four-manifold. From this, we then appeal to a theorem [32], which implies the non-vani...