Small Lefschetz fibrations and exotic 4-manifolds

Abstract

We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces $${\mathbb {CP}}^{2}\# p\, \overline{\mathbb {CP}}^{2}$$ for $$p=7, 8, 9$$ , and to $$3 {\mathbb {CP}}^{2}\#q\, \overline{\mathbb {CP}}^{2}$$ for $$q =12, \ldots ,19$$ . Complementarily, we prove that there are no minimal genus-2 Lefschetz fibrations whose total spaces are homeomorphic to any other simply-connected 4-manifold with $$b^+ \le 3$$ , with one possible exception when $$b^+=3$$ . Meanwhile, we produce positive Dehn twist factorizations for several new genus-2 Lefschetz fibrations with small number of critical points, including the smallest possible example, which follow from a reverse engineering procedure we introduce for this setting. We also derive exotic minimal symplectic 4-manifolds in the homeomorphism classes of $${\mathbb {CP}}^{2}\# 4 \overline{\mathbb {CP}}^{2}$$ and $$3 {\mathbb {CP}}^{2}\# 6 \overline{\mathbb {CP}}^{2}$$ from small Lefschetz fibrations over surfaces of higher genera.