Geography of symplectic 4-manifolds admitting Lefschetz fibrations and their indecomposability

Abstract

In this paper, we show that for a given finitely presented group $G$, there exist integers $h_G \geq 0$ and $n_G \geq 4$ such that for all $h \geq h_G$ and $n \geq n_G$, and for all $0 \leq i \leq 2n - 2$, there exists a genus-$(2h + n - 1)$ Lefschetz fibration on a minimal symplectic 4-manifold with $(\chi, c_{1}^{2}) = (n, i)$ whose fundamental group is isomorphic to $G$. We also prove that such a fibration cannot be decomposed as a fiber sum for $1 \leq i \leq 2n - 2$ if $h > (5n - 3)/2$. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.