Genus two Lefschetz fibrations with $b^{+}_{2}=1$ and ${c_1}^{2}=1,2$

TL;DR

This paper constructs specific genus two Lefschetz fibrations with controlled topological invariants, computes their fundamental groups, and explores their symplectic properties, contributing new examples and techniques in 4-manifold topology.

Contribution

It introduces a family of genus two Lefschetz fibrations with particular invariants using lantern substitutions, and extends methods to construct fibrations with various invariants and applications.

Findings

Fundamental groups are computed explicitly.

Fibrations admit -2 sections and are symplectically minimal.

Constructs new examples with exotic smooth structures.

Abstract

In this article we construct a family of genus two Lefschetz fibrations $f_{n}: X_{\theta_n} \rightarrow \mathbb{S}^{2}$ with $e(X_{\theta_n})=11$, $b^{+}_{2}(X_{\theta_n})=1$, and $c_1^{2}(X_{\theta_n})=1$ by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus two Lefschetz fibration over $\mathbb{S}^2$. Moreover, we compute the fundamental group of $X_{\theta_n}$ and show that it is isomorphic to the trivial group if $n = -3$ or $-1$, $\mathbb{Z}$ if $n =-2$, and $\mathbb{Z}_{|n+2|}$ for all integers $n\neq -3, -2, -1$. Also, we prove that our fibrations admit $-2$ section, show that their total space are symplectically minimal, and have the symplectic Kodaira dimension $\kappa = 2$. In addition, using the techniques developed in \cite{A, AP1, ABP, AP2, AZ, AO}, we also construct the genus two Lefschetz fibrations over $\mathbb{S}^2$ with $c_1^{2} = 1, 2$ and $\chi = 1$ via the fiber sums of Matsumoto's and Xiao's genus two Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small $4$-manifolds with $b^{+}_{2} = 1$ and $b^{+}_{2} = 3$.