The fundamental group of symplectic $4$-manifolds with $b^+=1$

TL;DR

This paper investigates the fundamental groups of symplectic 4-manifolds with $b^+=1$, using Luttinger surgery to explore their topology and construct examples with specific invariants.

Contribution

It applies Luttinger surgery to analyze the fundamental groups and provides new constructions of symplectic 4-manifolds with $b^+=1$ and prescribed $c_1^2$.

Findings

Topological analysis of symplectic 4-manifolds with $b^+=1$

Construction methods for manifolds with specific invariants

Application of Luttinger surgery to fundamental group study

Abstract

In this article we apply the technique of Luttinger surgery to study the complexity of the fundamental group of symplectic $4$-manifolds with holomorphic Euler number $\chi_h=1$. We discuss the topology of symplectic $4$-manifolds with $b^+=1$ and provide various constructions of symplectic $4$-manifolds with $b^+=1$ and prescribed $c_1^2$.