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.
Bibliographical noteFunding Information:
The proof of Theorem 1.6 was written in 2010. Inspired by recent progress on related topics [, , ], 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.