Constructing Lefschetz fibrations via Daisy Substitutions

Anar Akhmedov; Naoyuki Monden

paper

Constructing Lefschetz fibrations via Daisy Substitutions

arXiv:1405.6669

Abstract

We construct new families of non-hyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words $(c_{1} c_{2} \dots c_{2 g - 1} c_{2 g} c_{2 g + 1}^{2} c_{2 g} c_{2 g - 1} \dots c_{2} c_{1})^{2} = 1$ , $(c_{1} c_{2} \dots c_{2 g} c_{2 g + 1})^{2 g + 2} = 1$ , and $(c_{1} c_{2} \dots c_{2 g - 1} c_{2 g})^{2 (2 g + 1)} = 1$ in the mapping class group $Γ_{g}$ 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 $(c_{1} c_{2} \dots c_{2 g} c_{2 g + 1})^{2 g + 2} = 1$ via daisy substitutions, we also construct an infinite family of pairwise non-diffeomorphic irreducible symplectic and non-symplectic $4$ -manifolds homeomorphic to $(g^{2} - g + 1) CP^{2} # (3 g^{2} - g (k - 3) + 2 k + 3) CP^{2}$ for any $g \geq 3$ , and $k = 2, \dots, g + 1$ .

arXiv comment27 pages, 6 figures. minor revisions for publication