Milnor open books and milnor fillable contact structures

Abstract

It was proven by Caubel, N emethi, Popescu-Pampu [1] in 2004 that any 3- dimensional Milnor llable manifold inherits a unique contact structure from any of its 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 lled by the singularity. The key fact is that in a Milnor llable 3-manifold, there is a Milnor open book which is determined by the topology of the manifold. In other words, di eomorphic Milnor llable 3-manifolds admit isomorphic open book decompositions compatible with the lled contact structures on them. Following Giroux's result, it is deduced that the Milnor llable contact structures on a Milnor llable 3-manifold are contactomorphic. In this thesis, we give this proof in detail.