Nested open books and the binding sum

TL;DR

This paper introduces nested open books and describes how to construct open book decompositions for fiber connected sums, with applications to contact geometry, providing explicit structures and compatibility conditions.

Contribution

It defines nested open books, details the construction of open books for fiber sums along bindings, and applies these results to contact open books, ensuring compatibility with contact structures.

Findings

Explicit open book decomposition for fiber connected sums.

Compatibility of open books with contact structures in fiber sums.

Construction of nested open books and push-offs of bindings.

Abstract

We introduce the notion of a nested open book, a submanifold equipped with an open book structure compatible with an ambient open book, and describe in detail the special case of a push-off of the binding of an open book. This enables us to explicitly describe a natural open book decomposition of a fibre connected sum of two open books along their bindings, provided they are diffeomorphic and admit an open book structure themselves. Furthermore, we apply the results to contact open books, showing that the natural open book structure of a contact fibre connected sum of two adapted open books along their contactomorphic bindings is again adapted to the resulting contact structure.