Constructing Lefschetz fibrations via daisy substitutions

Abstract

We construct new families of nonhyperelliptic Lefschetz fibrations by applying the daisy substitutions to the families of words $(c_1{c}_2\cdots c_{2g-1}{c}_{2g}{c_{2g+1}}^2{c}_{2g}\cdot c_{2g-1}\cdots c_2{c}_1{)}^2=1$, $(c_1{c}_2\cdots c_{2g}{c}_{2g+1}{)}^{2g+2}=1$, and $(c_1{c}_2\cdots c_{2g-1}{c}_{2g}{)}^{2(2g+1)}=1$ in the mapping class group ${\Gamma }_g$ of the closed orientable surface of genus $g$, and we study the sections of these Lefschetz fibrations. Furthermore, we show that the total spaces of some of these Lefschetz fibrations are irreducible exotic $4$-manifolds, and we compute their Seiberg–Witten invariants. By applying the knot surgery to the family of Lefschetz fibrations obtained from the word $(c_1{c}_2\cdots c_{2g}{c}_{2g+1}{)}^{2g+2}=1$ via daisy substitutions, we also construct an infinite family of pairwise nondiffeomorphic irreducible symplectic and nonsymplectic $4$-manifolds homeomorphic to $(g^2-g+1){\mathbb{CP}}^2\#(3g^2-g(k-3)+2k+3){\overline{\mathbb{CP}}}^2$ for any $g\ge 3$ and $k=2,\dots ,g+1$.