Virtual Betti numbers and the symplectic Kodaira dimension of fibered $4$-manifolds

Abstract

We prove that if a closed oriented 4-manifold X fibers over a 2or 3-dimensional manifold, in most cases all of its virtual Betti numbers are infinite. In turn, we show that a closed oriented 4-manifold X which is not a tower of torus bundles and fibering over a 2-or 3-dimensional manifold does not admit a torsion symplectic canonical class, nor is it of Kodaira dimension zero.

On the topology of symplectic Calabi-Yau 4-manifolds

“…In such case, vb 1 (G) = ∞. (While preparing the final version of this paper, we learned that, independently, Baykur (see [5]) and Li-Ni (see [30]) obtained, under the same assumptions, similar conclusions on the virtual Betti number. )…”

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

“…Remark 1.3. Since Euler characteristic and signature of mapping tori are both zero (and since both are multiplicative under coverings), we can conclude that, as in [4], whenever vb 1 = ∞ the virtual b + and b − are infinite as well.…”

“…On the other hand, if X admits a fibration over S 1 , in many cases (see [5]) X is finitely covered by a surface bundle over T 2 . The case of our theorem that is not covered by [4] and [14] is that Y has at least one Seifert fibered JSJ piece.…”

“…The virtual Betti numbers for 4-manifolds which fiber over 2-manifolds were subsequently computed in [4] and [14]. In [4], the virtual Betti numbers for most 4−manifolds which fiber over 3-manifolds were also shown to be ∞. In this paper, we will study the problem for 4-manifolds which fiber over S 1 , that is, 4-manifolds which are mapping tori.…”

“…Although our Theorem 1.2 considers a different class of fibered 4-manifolds from [4] and [14], there is a significant overlap. When the 4-manifold X is a surface bundle over T 2 , it also admits a fibration over S 1 .…”