Embeddings of $3$--manifolds via open books

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 $S^2\tilde{\times }{S}^3$ 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 $S^5.$

11 pages, 5 figures. Fixed typos, added references and revised using comments and suggestions of an anonymous referee