Marco Golla

A note on cobordisms of algebraic knots

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

Ozsváth–Szabó invariants of contact surgeries

\nu^+

Cuspidal curves and Heegaard Floer homology

: 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.

On Stein fillings of contact torus bundles

We give new tightness criteria for positive surgeries along knots in the 3-sphere, generalising results of Lisca and Stipsicz, and Sahamie. The main tools will be Honda, Kazez and Matiand Ozsvath and Szabos Floer-theoretic contact invariants. We compute Ozsvath-Szabo contact invariant of positive contact surgeries along Legendrian knots in the 3-sphere in terms of the classical invariants of the knot. We also combine a Legendrian cabling construction with contact surgeries to get results about rational contact surgeries. 57R17; 57R57

Dehn surgeries and rational homology balls

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.

Embedding 3-manifolds in spin 4-manifolds

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

Pair of pants decomposition of 4–manifolds

We consider the question of which Dehn surgeries along a given knot bound rational homology balls. We use Ozsvath and Szabos correction terms in Heegaard Floer homology to obtain general constrain ...

Heegaard Floer correction terms, with a twist

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 S2 ×S2, 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.

Comparing invariants of Legendrian knots

Using tropical geometry, Mikhalkin has proved that every smooth complex hypersurface in

Heegaard Floer homology and concordance bounds on the Thurston norm.

\mathbb{CP}^{n+1}