Vincent Colin

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.

Reeb vector fields and open book decompositions

We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient

Sutures and contact homology I

c={k\over n}

Une infinité de structures de contact tendues sur les variétés toroïdales

, where

Constructions controlees de champs de Reeb et applications

n

Homologie de contact des variétés toroïdales

is the number of prongs along the boundary. If

The discriminant and oscillation lengths for contact and Legendrian isotopies

k\geq 2

Chirurgies d'indice un et isotopies de sphères dans les variétés de contact tendues

, then there is a well-defined linearized contact homology group. If

Recollement de variétés de contact tendues

k\geq 3

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

, then the linearized contact homology is exponentially growing with respect to the action, and every Reeb vector field of the corresponding contact structure admits an infinite number of simple periodic orbits.