New Exotic 4-Manifolds via Luttinger Surgery on Lefschetz Fibrations

TL;DR

This paper generalizes previous constructions of exotic symplectic 4-manifolds using Luttinger surgery on higher genus Lefschetz fibrations, producing new examples with diverse fundamental groups.

Contribution

It extends the construction of exotic minimal symplectic 4-manifolds to higher genus Lefschetz fibrations on complex surfaces, enabling the creation of manifolds with various fundamental groups.

Findings

Constructed new exotic symplectic 4-manifolds with specified fundamental groups.

Generalized the previous genus two fibration construction to higher genus cases.

Produced manifolds homeomorphic but not diffeomorphic to known complex surfaces.

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 $M(k,n) = \Sigma_{k}\times \mathbb{S}^{2} #4n\CPb$ for any $k \geq 2$ and $n = 1$, and $k \geq 1$ and $n \geq 2$, respectively. Using our building blocks, we also construct symplectic 4-manifolds with the free group of rank $s \geq 1$ and various other finitely generated groups as the fundamental group.