429Lens spaces, rational balls and the ribbon conjecture PAOLO LISCAWe apply Donaldson's 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. 57M25 IntroductionIt is a well-known fact that every ribbon knot is smoothly slice. The ribbon conjecture states that, conversely, a smoothly slice knot is ribbon. In this paper we prove that the ribbon conjecture holds for 2-bridge knots, deducing this result from a characterization of the 3-dimensional lens spaces which smoothly bound rational homology 4-dimensional balls (Theorem 1.2 below).A link in S 3 is called 2-bridge if it can be isotoped until it has exactly two local maxima with respect to a standard height function. Figure 1 represents the 2-bridge link L.c 1 ; : : : ; c n /, where c i 2 ‫,ޚ‬ i D 1; : : : ; n. Given coprime integers p > q > 0 Definition 1.1 Let ‫ޑ‬ >0 denote the set of positive rational numbers, and define maps f; gW ‫ޑ‬ >0 ! ‫ޑ‬ >0 by setting, forwhere p > q 0 > 0 and qq 0 Á 1 .mod p/. Define R ‫ޑ‬ >0 to be the smallest subset of ‫ޑ‬ >0 such that f .R/ Â R, g.R/ Â R and R contains the set of rational numbers According to Siebenmann [11], Casson, Gordon and Conway showed that every knot of the form K.p; q/ with p q 2 R is ribbon. The interior of any ribbon disk can be radially pushed inside the 4-ball B 4 to obtain a smoothly embedded disk, and the 2-fold cover of B 4 branched along a slicing disk for K.p; q/ is a smooth rational homology ball with boundary the lens space L.p; q/. Therefore if K.p; q/ is a knot (ie if p is odd)Geometry & Topology, Volume 11 (2007) Lens spaces, rational balls and the ribbon conjecture

Abstract. 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.

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.

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We characterize L -spaces which are Seifert fibered over the 2-sphere in terms of taut foliations, transverse foliations and transverse contact structures. We give a sufficient condition for certain contact Seifert fibered 3-manifolds with e 0 = −1 to have nonzero contact Ozsváth-Szabó invariants. This yields an algorithm for deciding whether a given small Seifert fibered L -space carries a contact structure with nonvanishing contact Ozsváth-Szabó invariant. As an application, we prove the existence of tight contact structures on some 3-manifolds obtained by integral surgery along a positive torus knot in the 3-sphere. Finally, we prove planarity of every contact structure on small Seifert fibered L -spaces with e 0 ≥ −1 , and we discuss some consequences.

Let S 3 r (K) be the oriented 3-manifold obtained by rational r-surgery on a knot K ⊂ S 3 . Using the contact Ozsváth-Szabó 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.

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

We classify tight contact structures on the small Seifert fibered 3-manifold M (−1; r 1 , r 2 , r 3 ) with r i ∈ (0, 1) ∩ Q and r 1 , r 2 ≥ 1 2 . The result is obtained by combining convex surface theory with computations of contact Ozsváth-Szabó invariants. We also show that some of the tight contact structures on the manifolds considered are nonfillable, justifying the use of Heegaard Floer theory.