We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsv\'ath and Szab\'o's correction terms in Heegaard Floer homology to obtain general constraints on the surgery coefficients. We then turn our attention to the case of integral surgeries, with particular emphasis on positive torus knots. Finally, combining these results with a lattice-theoretic obstruction based on Donaldon's theorem, we classify which integral surgeries along torus knots of the form $T_{kq\pm 1,q}$ bound rational homology balls.Comment: 32 pages, many figures. Several minor improvements and corrections. Comments are welcome

We consider a large family MJX-tex-caligraphicscriptF of torus bundles over the circle, and we use recent work of Li–Mak to construct, on each Y∈F, a Stein fillable contact structure ξY. We prove that (i) each Stein filling of (Y,ξY) has vanishing first Chern class and first Betti number, (ii) if Y∈F is elliptic, then all Stein fillings of (Y,ξY) are pairwise diffeomorphic and (iii) if Y∈F is parabolic or hyperbolic, then all Stein fillings of (Y,ξY) share the same Betti numbers and fall into finitely many diffeomorphism classes. Moreover, for infinitely many hyperbolic torus bundles Y∈F we exhibit non‐homotopy equivalent Stein fillings of (Y,ξY).

Abstract. We give bounds on the gap functions of the singularities of a cuspidal plane curve of arbitrary genus, generalising recent work of Borodzik and Livingston. We apply these inequalities to unicuspidal curves whose singularity has one Puiseux pair: we prove two identities tying the parameters of the singularity, the genus, and the degree of the curve; we improve on some degree-multiplicity asymptotic inequalities; finally, we prove some finiteness results, we construct infinite families of examples, and in some cases we give an almost complete classification.

In this note we use Heegaard Floer homology to study smooth cobordisms of algebraic knots and complex deformations of cusp singularities of curves. The main tool will be the concordance invariant ν + : we study its behaviour with respect to connected sums, providing an explicit formula in the case of L-space knots and proving subadditivity in general.

An invariant of orientable 3-manifolds is defined by taking the minimum n such that a given 3-manifold embeds in the connected sum of n copies of S 2 × S 2 , and we call this n the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.