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

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Abstract

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over S1 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 S1. 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.

Original languageEnglish (US)
Pages (from-to)195-208
Number of pages14
JournalMathematische Zeitschrift
Volume277
Issue number1-2
DOIs
StatePublished - Jun 2014

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