Embeddings of $3$--manifolds via open books

TL;DR

This paper demonstrates that all closed orientable 3-manifolds can be embedded into specific 5-manifolds using open book decompositions, providing a new perspective on 3-manifold embeddings.

Contribution

It shows that every open book of a closed orientable 3-manifold admits an embedding into certain 5-manifolds with specified open book structures, and reestablishes that all such 3-manifolds embed in S^5.

Findings

Every open book of a closed orientable 3-manifold embeds in open books of S^2×S^3 and its twisted version.

All closed orientable 3-manifolds embed in S^5.

Open book embeddings can be used to reprove known embedding results.

Abstract

In this note, we discuss embeddings of $3$--manifolds via open books. First we show that every open book of every closed orientable $3$--manifold admits an open book embedding in any open book decompistion of $S^2 \times S^3$ and $S^2 \widetilde{\times} S^3$ with the page a disk bundle over $S^2$ and monodromy the identity. We then use open book embeddings to reprove that every closed orientable $3$--manifold embeds in $S^5.$