In this article, we construct a family of genus 2 Lefschetz fibrations fn: Xθn → S2 with e(Xθn) = 11, b+2 (Xθn) = 1, and c21(Xθn) = 1 by applying a single lantern substitution to the twisted fiber sums of Matsumoto's genus 2 Lefschetz fibration over S2. Moreover, we compute the fundamental group of Xθn and show that it is isomorphic to the trivial group if n = −3 or −1, Z if n = −2, and Z|n+2| for all integers n = −3,−2,−1. Also, we prove that our fibrations admit −2 section, that their total spaces are symplectically minimal, and that they have symplectic Kodaira dimension κ = 2. In addition, using techniques developed over the past decade with other authors, we also construct the genus 2 Lefschetz fibrations over S2 with c21 = 1,2 and χ = 1 via the fiber sums of Matsumoto's and Xiao's genus 2 Lefschetz fibrations, and present some applications in constructing exotic smooth structures on small 4-manifolds with b+2 = 1 and b+2 = 3.
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