Paolo Lisca

Lens spaces, rational balls and the ribbon conjecture

We apply Donaldsons theorem on the intersection forms of definite 4-manifolds to characterize the lens spaces which smoothly bound rational homology 4-dimensional balls. Our result implies, in particular, that every smoothly slice 2-bridge knot is ribbon, proving the ribbon conjecture for 2-bridge knots.

Ozsváth–Szábo invariants and tight contact three-manifolds I

Let S 3 r (K) be the oriented 3-manifold obtained by rational r-surgery on a knot K ⊂ S 3 . Using the contact Ozsvath-SzabO invariants we prove, for a class of knots K containing all the algebraic knots, that S 3 r (K)carries positive, tight contact structures for every r ≠ 2g s (K) - 1, where g s (K) is the slice genus of K. This implies, in particular, that the Brieskorn spheres -Σ(2,3,4) and -Σ(2,3,3) carry tight, positive contact structures. As an application of our main result we show that for each m ∈ N there exists a Seifert fibered rational homology 3-sphere M m carrying at least m pairwise non-isomorphic tight, nonfillable contact structures.

Heegard Floer invariants of Legendrian knots in contact three-manifolds

We define invariants of null–homologous Legendrian and transverse knots in contact 3–manifolds. The invariants are determined by elements of the knot Floer homology of the underlying smooth knot. We compute these invariants, and show that they do not vanish for certain non–loose knots in overtwisted 3–spheres. Moreover, we apply the invariants to find transversely non–simple knot types in many overtwisted contact 3–manifolds.

Symplectic fillings and positive scalar curvature

Let X be a 4-manifold with contact boundary. We prove that the monopole invariants of X introduced by Kronheimer and Mrowka vanish under the following assumptions: (i) a connected component of the boundary of X carries a metric with positive scalar curvature and (ii) either b_2^+(X)>0 or the boundary of X is disconnected. As an application we show that the Poincare homology 3-sphere, oriented as the boundary of the positive E_8 plumbing, does not carry symplectically semi-fillable contact structures. This proves, in particular, a conjecture of Gompf, and provides the first example of a 3-manifold which is not symplectically semi-fillable. Using work of Froyshov, we also prove a result constraining the topology of symplectic fillings of rational homology 3-spheres having positive scalar curvature metrics.

Stein 4-manifolds with boundary and contact structures

Abstract We discuss several applications of Seiberg-Witten theory in conjunction with an embedding theorem (proved elsewhere) for complex 2-dimensional Stein manifolds with boundary. We show that a closed, real 2-dimensional surface smoothly embedded in the interior of such a manifold satisfies an adjunction inequality, regardless of the sign of its self-intersection. This inequality gives constraints on the minimum genus of a smooth surface representing a given 2-homology class. We also discuss consequences for the contact structures existing on the boundaries of these Stein manifolds. We prove a slice version of the Bennequin-Eliashberg inequality for holomorphically fillable contact structures, and we show that there exist families of homology 3-spheres with arbitrarily large numbers of homotopic, nonisomorphic tight contact structures. Another result we mention is that the canonical class of a complex 2-dimensional Stein manifold with boundary is invariant under self-diffeomorphisms fixing the boundary.

On symplectic fillings of lens spaces

Le 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 three-sphere 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.

On the existence of tight contact structures on Seifert fibered 3-manifolds

We determine the closed, oriented Seifert fibered 3-manifolds which carry positive tight contact structures. Our main tool is a new non-vanishing criterion for the contact Ozsvath-Szabo invariant.

Sums of lens spaces bounding rational balls

We classify connected sums of three-dimensional lens spaces which smoothly bound rational homology balls. We use this result to determine the order of each lens space in the group of rational homology 3‐spheres up to rational homology cobordisms, and to determine the concordance order of each 2‐bridge knot. 57M99; 57M25

Transverse contact structures on Seifert 3–manifolds

We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.

Seifert fibered contact three-manifolds via surgery

Using contact surgery we dene families of contact structures on certain Seifert bered three{manifolds. We prove that all these contact structures are tight using contact Ozsv ath{Szab o invariants. We use these examples to show that, given a natural number n, there exists a Seifert bered three{manifold carrying at least n pairwise non{isomorphic tight, not llable contact structures.