[Paper Review] Virtually symplectic fibered 4-manifolds

R. İnanç Baykur, Stefan Friedl|arXiv (Cornell University)|Oct 17, 2012

Geometric and Algebraic Topology19 references3 citations

TL;DR

This paper determines when closed, oriented 4-manifolds fibered over lower-dimensional manifolds are virtually symplectic—i.e., admit a finite cover that is symplectic. Using recent advances in 3-manifold topology, particularly the work of Agol, Przytycki, and Wise on virtually fibered and virtually symplectic structures, the authors establish precise topological conditions on the fiber and base of the fibration that guarantee virtual symplecticity or obstruction to it.

ABSTRACT

We mostly determine which closed smooth oriented 4-manifolds fibering over lower dimensional manifolds are virtually symplectic, i.e. finitely covered by symplectic 4-manifolds.

Motivation & Objective

• To classify which closed, oriented 4-manifolds that fiber over lower-dimensional manifolds are virtually symplectic.
• To determine the topological conditions on the fiber and base of a fibration that ensure a 4-manifold is virtually symplectic or not.
• To extend existing results on symplectic 4-manifolds to the broader class of virtually symplectic ones, especially in the context of fibered 4-manifolds.
• To resolve the remaining open case of 4-manifolds fibering over virtually fibered graph 3-manifolds, which had resisted prior classification.

Proposed method

• Leverages the virtual fibering theorems of Agol, Przytycki, and Wise for hyperbolic 3-manifolds and graph manifolds.
• Applies the theory of Seiberg-Witten invariants and the structure of finite covers to analyze symplectic properties of 4-manifolds.
• Uses Mayer-Vietoris sequences and homology decompositions to study Betti numbers and the positive-definite second Betti number $b_2^+$ in finite covers.
• Employs mapping torus constructions and monodromy actions to analyze the topology of fiber bundles over $S^1$.
• Applies a key lemma (Lemma 15) that rules out symplectic structures on 4-manifolds decomposed along $S^1 \times S^2$ with $b_2^+ > 1$ in each piece.
• Analyzes finite covers of fibered 4-manifolds to reduce the problem to studying symplecticity in the cover, using pullback constructions and Poincaré duality.

Experimental results

Research questions

• RQ1 When is a 4-manifold that fibers over a 3-manifold virtually symplectic?
• RQ2 What topological obstructions prevent a fibered 4-manifold from being virtually symplectic?
• RQ3 How does the JSJ decomposition of the base or fiber influence the virtual symplecticity of the total space?
• RQ4 What is the role of monodromy and finite covers in determining whether a fibered 4-manifold admits a symplectic finite cover?
• RQ5 Can the case of 4-manifolds fibering over virtually fibered graph 3-manifolds be fully classified?

Key findings

• A 4-manifold fibering over a 3-manifold $N$ is virtually symplectic if $N$ is prime and not a graph manifold.
• A 4-manifold fibering over a 3-manifold is not virtually symplectic if $N$ is not virtually fibered, such as when $N$ is not prime.
• If the fiber is a homologically essential surface or the base is a 2-torus, the 4-manifold is virtually symplectic.
• A 4-manifold fibering over a 3-manifold with only hyperbolic pieces in its JSJ decomposition is virtually symplectic.
• A 4-manifold is not virtually symplectic if the fiber is a connected sum of non-spherical 3-manifolds and the monodromy preserves a separating 2-sphere.
• For surface bundles over surfaces, a 4-manifold is virtually symplectic if and only if it is symplectic, resolving a special case completely.