Geometric decompositions of almost contact manifolds

TL;DR

This paper introduces approximately holomorphic techniques in almost contact geometry, demonstrating how to construct geometric decompositions like open books and Lefschetz pencils to study contact structures.

Contribution

It develops the framework of approximately holomorphic geometry for almost contact manifolds and sketches the existence of key geometric decompositions.

Findings

Existence of open book decompositions for almost contact manifolds

Existence of Lefschetz pencils in contact geometry

Application of decompositions to contact structure existence

Abstract

These notes are intended to be an introduction to the use of approximately holomorphic techniques in almost contact and contact geometry. We develop the setup of the approximately holomorphic geometry. Once done, we sketch the existence of the two main geometric decompositions available for an almost contact or contact manifold: open books and Lefschetz pencils. The use of the two decompositions for the problem of existence of contact structures is mentioned.