On open books and embedding of smooth and contact manifolds

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 ($n\ge 7$, $k<\frac{n-4}{2}$) 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 ${\mathbb{S}}^{4n+1}.$

arXiv comment19 pages, 6 figures

References in corpus (7)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Lectures on open book decompositions and contact structures
• Lecture notes on stabilization of contact open books
• Subcritical Stein manifolds are split
• Braided embeddings of contact 3-manifolds in the standard contact 5-sphere
• The Legendrian Whitney trick
• Embeddings of $3$--manifolds via open books