Open book embeddings of closed non-orientable $3$-manifolds

Abstract

In this note, we discuss open book embeddings of closed non-orientable $3$-manifolds in $5$-manifolds. We also show that a huge class of closed orientable $3$-manifolds, namely the orientation double covers of certain closed non-orientable $3$-manifolds, open book embeds in the $5$-sphere $S^5$. Finally, we give a new proof of a well-known theorem which states that every closed non-orientable $3$-manifold smoothly embeds in $S^5$.