Paolo Ghiggini

Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions

We sketch the proof of the equivalence between the hat versions of Heegaard Floer homology and embedded contact homology (ECH). The key point is to express these two Floer homology theories in terms of an open book decomposition of the ambient manifold.

Sutures and contact homology I

We define a relative version of contact homology for contact manifolds with convex boundary, and prove basic properties of this relative contact homology. Similar considerations also hold for embedded contact homology.

Linear Legendrian curves in T^3

Using convex surfaces and Kandas classification theorem, we classify Legendrian isotopy classes of Legendrian linear curves in all tight contact structures on

Subcritical contact surgeries and the topology of symplectic fillings

T^3

The vanishing of the contact invariant in the presence of torsion

. Some of the knot types considered in this article provide new examples of non transversally simple knot types.

The equivalence of Heegaard Floer homology and embedded contact homology III: from hat to plus

By a result of Eliashberg, every symplectic filling of a three-dimensional contact connected sum is obtained by performing a boundary connected sum on another symplectic filling. We prove a partial generalization of this result for subcritical contact surgeries in higher dimensions: given any contact manifold that arises from another contact manifold by subcritical surgery, its belt sphere is zero in the oriented bordism group Ω∗^{SO}(W ) of any symplectically aspherical filling W , and in dimension five, it will even be nullhomotopic. More generally, if the filling is not aspherical but is semipositive, then the belt sphere will be trivial in H∗(W ). Using the same methods, we show that the contact connected sum decomposition for tight contact structures in dimension three does not extend to higher dimensions: in particular, we exhibit connected sums of manifolds of dimension at least five with Stein fillable contact structures that do not arise as contact connected sums. The proofs are based on holomorphic disk-filling techniques, with families of Legendrian open books (so-called “Lobs”) as boundary conditions.