Symplectic Surgeries Along Certain Singularities and New Lefschetz Fibrations

Abstract

We define a new 4-dimensional symplectic cut and paste operations arising from the generalized star relations $(t_{a_0}t_{a_1}t_{a_2} \cdots t_{a_{2g+1}})^{2g+1} = t_{b_1} t_{b_2}^{g}t_{b_3}$, also known as the trident relations, in the mapping class group $Γ_{g,3}$ of an orientable surface of genus $g\geq1$ with $3$ boundary components. We also construct new families of Lefschetz fibrations by applying the (generalized) star relations and the chain relations to the families of words $(t_{c_1}t_{c_2} \cdots t_{c_{2g-1}}t_{c_{2g}}t_{c_{2g+1}}^2t_{c_{2g}}t_{c_{2g-1}} \cdots t_{c_2}t_{c_1})^{2n} = 1$, $(t_{c_1}t_{c_2} \cdots t_{c_{2g}}t_{c_{2g+1}})^{(2g+2)n} = 1$ and $(t_{c_1}t_{c_2} \cdots t_{c_{2g-1}}t_{c_{2g}})^{2(2g+1)n} = 1$ in the mapping class group $Γ_{g}$ of the closed orientable surface of genus $g \geq 1$ and $n \geq 1$. Furthemore, we show that the total spaces of some of these Lefschetz fibraions are irreducible exotic symplectic $4$-manifolds. Using the degenerate cases of the generalized star relations, we also realize all elliptic Lefschetz fibrations and genus two Lefschetz fibrations over $\mathbb{S}^{2}$ with non-separating vanishing cycles.