New exotic 4-manifolds via Luttinger surgery on Lefschetz fibrations

Abstract

In [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794], the first author constructed the first known example of exotic minimal symplectic[Formula: see text] and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to [Formula: see text]. The construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] uses Yukio Matsumoto's genus two Lefschetz fibrations on [Formula: see text] over 𝕊2 along with the fake symplectic 𝕊2 × 𝕊2 construction given in [Construction of symplectic cohomology 𝕊2 × 𝕊2, Proc. Gökova Geom. Topol. Conf.14 (2007) 36–48]. The main goal in this paper is to generalize the construction in [Small exotic 4-manifolds, Algebr. Geom. Topol.8 (2008) 1781–1794] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on [Formula: see text] for any k ≥ 2 and n = 1, and k ≥ 1 and n ≥ 2, respectively. Using our symplectic building blocks, we also construct new symplectic 4-manifolds with the free group of rank s ≥ 1, the free product of the finite cyclic groups, and various other finitely generated groups as the fundamental group.