Small Lefschetz fibrations and exotic 4-manifolds

TL;DR

This paper constructs explicit genus-2 Lefschetz fibrations with minimal symplectic 4-manifolds homeomorphic to certain rational surfaces, proving non-existence for others, and introduces a reverse engineering method for positive Dehn twist factorizations.

Contribution

It provides explicit constructions of genus-2 Lefschetz fibrations for specific 4-manifolds and introduces a reverse engineering approach for positive Dehn twist factorizations.

Findings

Constructed minimal symplectic 4-manifolds homeomorphic to rational surfaces.

Proved non-existence of certain genus-2 Lefschetz fibrations for other 4-manifolds.

Produced new genus-2 Lefschetz fibrations with minimal critical points.

Abstract

We explicitly construct genus-2 Lefschetz fibrations whose total spaces are minimal symplectic 4-manifolds homeomorphic to complex rational surfaces CP^2 # p (-CP^2) for p=7, 8, 9, and to 3 CP^2 #q (-CP^2) for q =12,...,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^+ at most 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 CP^2 # 4 (-CP^2) and 3 CP^2 # 6 (-CP^2) from small Lefschetz fibrations over surfaces of higher genera.