Open books and embeddings of smooth and contact manifolds

Abstract We discuss some embedding results in the category of open books, Lefschetz fibrations, contact manifolds and contact open books. First we prove an open book version of the Haefliger–Hirsch embedding theorem by showing that every k-connected closed n-manifold (n ≥ 7, k < (n − 4)/2) with signature zero admits an open book embedding in the trivial open book of 𝕊2n−k. We then prove that every closed manifold M2n+1 that bounds an achiral Lefschetz fibration admits an open book embedding in the trivial open book of 𝕊2⌊3n/2⌋+3. We also prove that every closed manifold M2n+1 bounding an achiral Lefschetz fibration admits a contact structure that isocontact embeds in the standard contact structure on ℝ2n+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 𝕊4n+1.