A note on codimension $2$ spun embedding

TL;DR

This paper proves that spun embeddings of open book decompositions of certain manifolds always embed in the ambient manifold, with specific results for simply connected spin 5-manifolds and 3-dimensional open books.

Contribution

It introduces a general theorem on spun embeddings of open books and applies it to specific classes of manifolds, including spin 5-manifolds and Morse open books.

Findings

Open book decompositions of certain manifolds spin embed in the ambient manifold.

Every open book of a simply connected spin 5-manifold spun embeds in S^7.

3-dimensional open books spun embed in S^5.

Abstract

We prove that if a closed manifold $B$ is a connected component of the binding of an open book decomposition of a manifold $M$, then every open book decomposition of $B$ spun embeds in $M$. As an application, we prove that every open book decomposition of a simply connected spin $5$-manifold spun embeds in $S^7$ and every $3$-dimensional open book spun embeds in $S^5$. We also define a notion of spun embedding for Morse open books.