Geography of Genus 2 Lefschetz fibrations

2018/11/08 by Nakamura, Kai · 1 citation
#FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1811.03708

Abstract

Questions of geography of various classes of 4-manifolds have been a central motivating question in 4-manifold topology. Baykur and Korkmaz asked which small, simply connected, minimal 4-manifolds admit a genus 2 Lefschetz fibration. They were able to classify all the possible homeomorphism types and realize all but one with the exception of a genus 2 Lefschetz fibration on a symplectic 4-manifold homeomorphic, but not diffeomorphic to 3 \mathbbCP² # 11\mathbbCP². We give a positive factorization of type (10,10) that corresponds to such a genus 2 Lefschetz fibration. Furthermore, we observe two restrictions on the geography of genus 2 Lefschetz fibrations, we find that they satisfy the Noether inequality and a BMY like inequality. We then find positive factorizations that describe genus 2 Lefschetz fibrations on simply connected, minimal symplectic 4-manifolds for many of these points.

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• A note on the topology of Lefschetz fibrations

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