Geography of Genus 2 Lefschetz fibrations

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 homeom...

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Bibliographic Details
Published inarXiv.org
Main Author Nakamura, Kai
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 08.11.2018
Subjects
Geography
Topology
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Summary: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 \(3 \mathbb{CP}^2 \# 11\overline{\mathbb{CP}}^2\). 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.
ISSN:2331-8422