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

Tian-Jun Li; Yi Ni

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arXiv · 1211.4245

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

Tian-Jun Li·Yi Ni

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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.

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Tian-Jun Li, Yi Ni. 2012-11-18. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori. https://arxiv.org/abs/1211.4245

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