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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 \(S^1\) 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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Notes

  1. The curves \(\alpha _i\) may not be supported in \(F_1\). Some of them may be components of \(\mathcal C\) which are adjacent to \(F_2\) on both sides.

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Acknowledgments

The proof of Theorem 1.6 was written in 2010. Inspired by recent progress on related topics [5, 6, 15], we expanded this note to the current version. We wish to thank Chung-I Ho, Yi Liu and Stefano Vidussi for interesting discussion. We are also grateful to Anar Akhmedov, Inanc Baykur, Nikolai Saveliev and Stefano Vidussi for comments on earlier versions of this paper. The first author was supported by NSF grant numbers DMS-1065927, DMS-1207037. The second author was supported by an AIM Five-Year Fellowship, NSF grant numbers DMS-1021956, DMS-1103976, and an Alfred P. Sloan Research Fellowship.

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  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Tian-Jun Li

  2. Department of Mathematics, Caltech, MC 253-37, 1200 E California Blvd, Pasadena, CA, 91125, USA

    Yi Ni

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  1. Tian-Jun Li
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Correspondence to Yi Ni.

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Li, TJ., Ni, Y. Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori. Math. Z. 277, 195–208 (2014). https://doi.org/10.1007/s00209-013-1250-x

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