New Exotic 4-Manifolds via Luttinger Surgery on Lefschetz Fibrations

Abstract

In [2], the first author constructed the first known examples of exotic minimal symplectic $\CP#5\CPb$ and minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to $3\CP#7\CPb$. The construction in [2] uses Y. Matsumoto's genus two Lefschetz fibrations on $M = \mathbb{T}^{2}\times \mathbb{S}^{2} #4\CPb$ over ${\mathbb{S}}^2$ along with the fake symplectic ${\mathbb{S}}^2\times {\mathbb{S}}^2$ construction given in [1]. The main goal in this paper is to generalize the construction in [2] using the higher genus versions of Matsumoto's fibration constructed by Mustafa Korkmaz and Yusuf Gurtas on $M(k,n) = Σ_{k}\times \mathbb{S}^{2} #4n\CPb$ for any $k\ge 2$ and $n=1$, and $k\ge 1$ and $n\ge 2$, respectively. Using our building blocks, we also construct symplectic 4-manifolds with the free group of rank $s\ge 1$ and various other finitely generated groups as the fundamental group.

21 pages, 7 figures. One reference added, a few extra clarifying comments added

References in corpus (2)

• Exotic smooth structures on 4-manifolds with zero signature
• Exotic Smooth Structures on Small 4-Manifolds