On open books and embedding of smooth and contact manifolds

TL;DR

This paper explores embedding theorems for manifolds within open books and contact structures, extending classical results and providing new embedding and contact embedding constructions for high-dimensional manifolds.

Contribution

It proves an open book version of the Haefliger--Hirsch theorem and establishes new embedding results for manifolds bounding achiral Lefschetz fibrations.

Findings

Every k-connected closed n-manifold (n≥7, k<(n-4)/2) admits an open book embedding in a sphere.

Closed manifolds bounding achiral Lefschetz fibrations embed in standard spheres.

Contact structures on certain manifolds can be embedded into standard contact structures on Euclidean spaces.

Abstract

We discuss embedding of manifolds in the category of open books, contact manifolds and contact open books. We prove an open book version of the Haefliger--Hirsch embedding theorem by showing that every $k$-connected closed $n$-manifold ($n\geq 7$, $k < \frac{n-4}{2}$) admits an open book embedding in the trivial open book of $\mathbb{S}^{2n-k}$. We then prove that every closed manifold $M^{2n+1}$ that bounds an achiral Lefschetz fibration, admits open book embedding in the trivial open book of $\mathbb{S}^{2\lfloor\frac{3n}{2}\rfloor + 3}$. We also prove that every closed manifold $M^{2n+1}$ bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on $\mathbb{R}^{2n+3}.$ Finally, we give various examples of contact open book embeddings of contact $(2n+1)$-manifolds in the trivial supporting open book of the standard contact structure on $\mathbb{S}^{4n+1}.$