Open Book Decomposition - Giroux Correspondence

Giroux Correspondence

In 2002, Emmanuel Giroux published the following result:

Theorem. Let M be a compact oriented 3-manifold. Then there is a bijection between the set of oriented contact structures on M up to isotopy and the set of open book decompositions of M up to positive stabilization.

Positive stabilization consists of modifying the page by adding a 2-dimensional 1-handle and modifying the monodromy by adding a positive Dehn twist along a curve that runs over that handle exactly once. Implicit in this theorem is that the new open book defines the same contact 3-manifold. Giroux's result has led to some breakthroughs in what is becoming more commonly called contact topology, such as the classification of contact structures on certain classes of 3-manifolds. Roughly speaking, a contact structure corresponds to an open book if, away from the binding, the contact distribution is isotopic to the tangent spaces of the pages through confoliations. One imagines smoothing the contact planes (preserving the contact condition almost everywhere) to lie tangent to the pages.