3-manifolds without any embedding in symplectic 4-manifolds

TL;DR

This paper proves that infinitely many closed 3-manifolds cannot embed into any closed symplectic 4-manifold, using advanced invariants to disprove a previous conjecture.

Contribution

It introduces new examples of 3-manifolds that do not embed in symplectic 4-manifolds, challenging existing beliefs in the field.

Findings

Existence of infinitely many 3-manifolds not embeddable in symplectic 4-manifolds

Construction of L-spaces that cannot bound definite 4-manifolds

Use of Heegaard Floer and instanton invariants to establish non-embeddability

Abstract

We show that there exist infinitely many closed 3-manifolds that do not embed in closed symplectic 4-manifolds, disproving a conjecture of Etnyre-Min-Mukherjee. To do this, we construct L-spaces that cannot bound positive or negative definite manifolds. The arguments use Heegaard Floer correction terms and instanton moduli spaces.