On symplectic fillings of lens spaces

Abstract

Abstract. Let ξ st be the contact structure naturally induced on the lens space L(p, q) = S 3 /Z/pZ by the standard contact structure ξ st on the threesphere S 3 . We obtain a complete classification of the symplectic fillings of (L(p, q), ξ st ) up to orientation-preserving diffeomorphisms. In view of our results, we formulate a conjecture on the diffeomorphism types of the smoothings of complex two-dimensional cyclic quotient singularities.

Planar open books, monodromy factorizations and symplectic fillings

“…Note that we do not check that the nonstandard filling is unique: indeed, the images of all the Dehn twists in Ab Map D n are uniquely determined, but we do not have an appropriate analog of Lemma 2.3 for this case. work [18], but our technique gives a slightly stronger result: uniqueness of filling up to symplectic deformation, not just diffeomorphism.…”

Strongly fillable contact manifolds and J-holomorphic foliations

“…McDuff argued by compactification in order to apply her classification results for rational and ruled symplectic 4-manifolds, and several other uniqueness and finiteness results have since been obtained using similar ideas, e.g. [Lis08,OO05]. Many of these uniqueness results can be recovered, and some of them strengthened or generalized, using the punctured holomorphic curve techniques introduced here (cf.…”

The symplectic topology of some rational homology balls

“…It is known that K(p 2 , pq − 1) is slice (in fact ribbon) for p > q > 0 relatively prime (see for ex. [27]). Fintushel-Stern's rational homology balls ( [15]) are given by the double branched cover of the four-ball branched over the slice disk for K(p 2 , pq − 1).…”

“…Up to contact isomorphism, it is known that there is a unique universally tight contact structure on L(p 2 , pq−1). Furthermore, Lisca has given a classification result for the diffeomorphism types of the fillings of the tight contact structures on lens spaces ( [27]). It follows from this classification that in the case of (L(p 2 , pq − 1), ξ p,q ), there are two possibilities for the diffeomorphism types of symplectic fillings, and these classes are realized by the manifolds C p,q and the double branched cover of D 4 branched along the slice disk bounding K(p 2 , pq − 1).…”