Diffeomorphism types of genus 2 Lefschetz fibrations
type: Item
instance of: scholarly article
title: Diffeomorphism types of genus 2 Lefschetz fibrations
zbMATH Open document ID: 0905.57014
author: Terry Fuller
published in: Mathematische Annalen
publication date: 1998-10-18T00:00:00Z
review text: A genus \(g\) Lefschetz fibration is a smooth mapping \(f:M\to \mathbb{S}^2\) \((M\) being a smooth oriented 4-manifold) so that the regular fiber is diffeomorphic to an oriented genus \(g\) surface \(\Sigma_g\), and \(f\) is locally given -- in suitable complex coordinate neighborhoods of each critical point of \(M\) -- by \(f(z,w) =z^2 +w^2\). According to \textit{A. Kas} [Pac. J. Math. 89, 89-104 (1980; Zbl 0457.14011)], for \(g\geq 2\) every such fibration is uniquely determined by a trivial element in the mapping class group of \(\Sigma_g\). The present paper takes into account, in the case \(g=2\), the genus two Lefschetz fibrations \(f_1:M_1 \to\mathbb{S}^2\) and \(f_2: M_2\to \mathbb{S}^2\) induced by two known relations in the mapping class group of \(\Sigma_2\) [\textit{J. S. Birman}, Braids, links, and mapping class groups, Ann. Math. Stud. 82 (1975; Zbl 0305.57013)] and \textit{Y. Matsumoto} [in: Topology and Teichmüller spaces. Proceedings of the 37th Taniguchi symposium, Katinkulta, Finland, July, 24--28, 1995. Singapore: World Scientific. 123--148 (1996; Zbl 0921.57006)], where \(M_1\) and \(M_2\) were proved to be both homeomorphic to \(5\mathbb{C} \mathbb{P}^2 \neq \overline {\mathbb{C}\mathbb{P}^2})\). By means of double branched covering techniques, the smooth 4-manifolds \(M_1\) and \(M_2\) are now entirely identified as complex surfaces; as a consequence, it is proved that \(M_1\), \(M_2\) and \(5\mathbb{C} \mathbb{P}^2 \neq\overline {\mathbb{C} \mathbb{P}^2}\) are mutually non-diffeomorphic. This provides a complete negative answer to a question contained in the Kirby problem list [\textit{Kirby}, Problems in low-dimensional topology (in: Geometric topology, 1993 Georgia international topology conference (1997; Zbl 0882.00041)]) and originally raised by Matsumoto in the quoted paper.
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R55
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57M12
zbMATH DE Number: 1179886
zbMATH Keywords: smooth 4-manifold
zbMATH Keywords: mapping class group
zbMATH Keywords: Lefschetz fibration
zbMATH Keywords: complex surface
zbMATH Keywords: handle-decomposition
zbMATH Keywords: branched cover
reviewed by: Maria Rita Casali
MaRDI profile type: Publication
full work available at URL: s002080050182
OpenAlex ID: W2068508067
DOI: 10.1007/S002080050182
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description: scientific article; zbMATH DE number 1179886