Constructions of small symplectic 4-manifolds using Luttinger surgery

TL;DR

This paper constructs small, exotic symplectic 4-manifolds using Luttinger surgery, providing new examples with specific topological properties and fundamental groups, expanding the landscape of known minimal symplectic 4-manifolds.

Contribution

It introduces novel constructions of minimal symplectic 4-manifolds with prescribed fundamental groups and topological features using Luttinger surgery techniques.

Findings

Constructed a minimal symplectic 4-manifold homeomorphic but not diffeomorphic to CP^2#3(-CP^2) with a genus 2 surface and Lagrangian torus.

Built a minimal symplectic 4-manifold homeomorphic but not diffeomorphic to 3CP^2#5(-CP^2) with two Lagrangian tori.

Produced the smallest known minimal symplectic manifolds with abelian fundamental groups, including finite and infinite cyclic groups.

Abstract

In this article we use the technique of Luttinger surgery to produce small examples of simply connected and non-simply connected minimal symplectic 4-manifolds. In particular, we construct: (1) An example of a minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to CP^2#3(-CP^2) which contains a symplectic surface of genus 2, trivial normal bundle, and simply connected complement and a disjoint nullhomologous Lagrangian torus with the fundamental group of the complement generated by one of the loops on the torus. (2) A minimal symplectic 4-manifold that is homeomorphic but not diffeomorphic to 3CP^2#5(-CP^2) which has two essential Lagrangian tori with simply connected complement. These manifolds can be used to replace E(1) in many known theorems and constructions. Examples in this article include the smallest known minimal symplectic manifolds with abelian fundamental groups including symplectic manifolds with finite and infinite cyclic fundamental group and Euler characteristic 6.