Embedding all contact 3-manifolds in a fixed contact 5-manifold

TL;DR

This paper demonstrates that all contact 3-manifolds can be embedded into specific fixed contact 5-manifolds, using techniques like spun embeddings and Lefschetz fibrations, expanding understanding of contact topology in higher dimensions.

Contribution

It shows that all contact 3-manifolds embed into particular fixed contact 5-manifolds, providing new methods and constructions in contact topology.

Findings

All contact 3-manifolds embed into a Stein fillable contact structure on a twisted S^3-bundle over S^2.

All contact 3-manifolds embed into a unique overtwisted contact structure on S^3×S^2.

Uses spun embeddings and Lefschetz fibrations for the embeddings.

Abstract

In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted $S^3$-bundle over $S^2$ and also into a unique overtwisted contact structure on $S^3\times S^2$. These results are proven using "spun embeddings" and Lefschetz fibrations.