We explicitly construct a genus-$3$ Lefschetz fibration over $\mathbb{ S}^{ 2}$ whose total space is $\mathbb{ T}^{ 2}\times \mathbb{ S}^{ 2}\# 6\overline{ \mathbb{ C} P^{ 2}}$ using the monodromy of Matsumoto's genus-$2$ Lefschetz fibration. We then construct more genus-$3$ Lefschetz fibrations whose total spaces are exotic minimal symplectic $4$-manifolds $3 \mathbb{ C} P^{ 2} \# q\overline{ \mathbb{ C} P^{ 2}}$ for $q=13,\ldots,19$. We also generalize our construction to get genus-$3k$ Lefschetz fibration structure on the $4$-manifold $\Sigma_{ k}\times \mathbb{ S}^{ 2}\# 6\overline{ \mathbb{ C} P^{ 2}}$ using the generalized Matsumoto's genus-$2k$ Lefschetz fibration. From this generalized version, we derive further exotic $4$-manifolds via Luttinger surgery and twisted fiber sum.
Categories: math.GT