Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

TL;DR

This paper investigates the virtual Betti numbers and symplectic properties of 4-manifolds fibering over the circle, revealing conditions under which these manifolds are virtually symplectic or have specific fiber types.

Contribution

It computes the virtual first Betti numbers for certain 4-manifolds and establishes new criteria linking fiber types to virtual symplecticity and Betti numbers.

Findings

Virtual first Betti numbers are computed for 4-manifolds over S^1.

Symplectic 4-manifolds with nonpositive Kodaira dimension have sphere or torus fiber.

Virtually fibered 3-dimensional fibers imply the 4-manifold is virtually symplectic unless Betti number is 1.

Abstract

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$. In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.