A note on embeddings of 3-manifolds in symplectic 4-manifolds

Mukherjee, Anubhav

doi:10.2140/agt.2025.25.3251

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ISSN (electronic): 1472-2739

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A note on embeddings of $3$-manifolds in symplectic $4$-manifolds

Anubhav Mukherjee

Algebraic & Geometric Topology 25 (2025) 3251–3270

DOI: 10.2140/agt.2025.25.3251

Abstract

We show that any closed oriented 3-manifold can be topologically embedded in some simply connected closed symplectic 4-manifold and that it can be made a smooth embedding after one stabilization. As a corollary of the proof, we show that the homology cobordism group is generated by Stein fillable 3-manifolds. We also find obstructions on smooth embeddings: there exist 3-manifolds that cannot smoothly embed in a way that appropriately respects orientations in any symplectic manifold with a weakly convex boundary.

Keywords

embeddings, 3-manifolds, contact structure, symplectic, surgery

Mathematical Subject Classification

Primary: 57K43

Secondary: 53D05

References

Publication

Received: 12 March 2022

Revised: 15 March 2024

Accepted: 19 May 2024

Published: 1 October 2025

Authors

Anubhav Mukherjee
Department of Mathematics Princeton University Princeton, NJ United States

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Volume 25, issue 6 (2025)