Milnor open books and milnor fillable contact structures

It was proven by Caubel, N emethi, Popescu-Pampu in 2004 that any 3- dimensional Milnor fillable manifold inherits a unique contact structure from any of its fi llings up to contactomorphism. The proof runs as follows: in the contact boundary of a germ of an isolated singularity, to every holomorphic function f with an isolated singularity, a so-called Milnor open book decomposition is assigned. Then it is shown that for any choice of f, this open book is compatible with the contact structure filled by the singularity. The key fact is that in a Milnor fillable 3-manifold, there is a Milnor open book which is determined by the topology of the manifold. In other words, di ffeomorphic Milnor fillable 3-manifolds admit isomorphic open book decompositions compatible with the filled contact structures on them. Following Giroux's result, it is deduced that the Milnor fi llable contact structures on a Milnor fillable 3-manifold are contactomorphic. In this thesis, we give this proof in detail.