Handle decompositions and stabilizations of open books

TL;DR

This paper develops handle decomposition techniques for open book decompositions of manifolds, enabling stabilization and revealing new open books, with applications to simplifying open books with trivial monodromy.

Contribution

It introduces handle decompositions that encode open books and describes handle slides that produce new open books, extending stabilization methods for manifolds of dimension at least 3.

Findings

Recovered known stabilization operations for open books.

Showed that any open book with trivial monodromy can be stabilized to a boundary connected sum of trivial disk bundles.

Provided a method to encode open books via handle decompositions.

Abstract

We build handle decompositions of n-manifolds that encode given open book decompositions and describe handle slides that reveal new open book decompositions on the same underlying manifold, for $n \geq 3$. This recovers known stabilization operations for open books. As an application, we show that any open book with trivial monodromy can be stabilized to an open book whose page is a boundary connected sum of trivial disk bundles over spheres.