Open books and embeddings of smooth and contact manifolds
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Arijit Nath
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.
Funding statement: Arijit Nath was supported by a CSIR India Fellowship with Ref. no. 09/084(0688)/2016-EMR-I.
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Communicated by: T. Leistner
Acknowledgements
We thank Dishant M. Pancholi for suggesting the problem of open book embedding. We also thank John B. Etnyre for many comments and suggestions that have helped to improve this article. The authors would like to thank the referee for the detailed report which has helped to improve this article.
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© 2023 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Pseudoholomorphic curves on the LCS-fication of contact manifolds
- Shellable tilings on relative simplicial complexes and their h-vectors
- On projections of free semialgebraic sets
- Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi–Yau varieties
- Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids
- Open books and embeddings of smooth and contact manifolds
- Fano fourfolds having a prime divisor of Picard number 1
- Characterizations of symplectic polar spaces
- Chow groups of Gushel–Mukai fivefolds
Articles in the same Issue
- Frontmatter
- Pseudoholomorphic curves on the LCS-fication of contact manifolds
- Shellable tilings on relative simplicial complexes and their h-vectors
- On projections of free semialgebraic sets
- Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi–Yau varieties
- Mean surface and volume particle tensors under L-restricted isotropy and associated ellipsoids
- Open books and embeddings of smooth and contact manifolds
- Fano fourfolds having a prime divisor of Picard number 1
- Characterizations of symplectic polar spaces
- Chow groups of Gushel–Mukai fivefolds
