Small Lefschetz fibrations and exotic 4-manifolds

R. Inanc Baykur, Mustafa Korkmaz

Published 2015-10-01Version 1

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