Constructing Lefschetz fibrations via Daisy Substitutions

TL;DR

This paper develops new non-hyperelliptic Lefschetz fibrations using daisy substitutions in mapping class groups, explores their sections, and constructs exotic 4-manifolds with computed Seiberg-Witten invariants, including infinite families of non-diffeomorphic manifolds.

Contribution

It introduces a novel method of constructing Lefschetz fibrations via daisy substitutions and studies their properties, including exotic 4-manifolds and Seiberg-Witten invariants.

Findings

Constructed new families of non-hyperelliptic Lefschetz fibrations.

Identified some total spaces as irreducible exotic 4-manifolds.

Generated infinite families of non-diffeomorphic 4-manifolds with specific topological types.

Abstract

We construct new families of non-hyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words $(c_1c_2 \cdots c_{2g-1}c_{2g}{c_{2g+1}}^2c_{2g}c_{2g-1} \cdots c_2c_1)^2 = 1$, $(c_1c_2 \cdots c_{2g}c_{2g+1})^{2g+2} = 1$, and $(c_1c_2 \cdots c_{2g-1}c_{2g})^{2(2g+1)} = 1$ in the mapping class group $\Gamma_{g}$ of the closed orientable surface of genus $g$, and study the sections of these Lefschetz fibrations. Furthemore, we show that the total spaces of some of these Lefschetz fibraions are irreducible exotic $4$-manifolds, and compute their Seiberg-Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word $(c_1c_2 \cdots c_{2g}c_{2g+1})^{2g+2} = 1$ via daisy substitutions, we also construct an infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic $4$-manifolds homeomorphic to $(g^2 - g + 1){\mathbb{CP}}{}^{2} \# (3g^{2} - g(k-3) + 2k + 3)\overline{\mathbb{CP}}{}^{2}$ for any $g \geq 3$, and $k = 2, \cdots, g+1$.