Symplectic Lefschetz fibrations with arbitrary fundamental groups. With an appendix by Ivan Smith
type: Item
instance of: scholarly article
title: Symplectic Lefschetz fibrations with arbitrary fundamental groups. With an appendix by Ivan Smith
zbMATH Open document ID: 1031.57021
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: The authors give an explicit construction of a symplectic Lefschetz fibration (SLF) whose total space is a smooth compact \(4\)-manifold with a prescribed fundamental group, following the existence result by \textit{S. Donaldson} [Doc. Math., J. DMV, Extra Vol. ICM Berlin 1998, vol. II, 309-314 (1998; Zbl 0909.53018)] and the result on the realization of any finitely presentable group as a fundamental group by \textit{R. Gompf} [Ann. Math., II. Ser. 142, 527-595 (1995; Zbl 0849.53027)]. The construction in the present article is based on an algebraic geometric method for creating positive relations among right-handed Dehn twists. The method is provided by a careful study of structures of the mapping class group \(M^n_{g,r}\) of a smooth genus \(g\) surface with \(n\) punctures and \(r\) boundary components, and symplectic connected sums for symplectic \(4\)-manifolds. Proofs of the main theorems are purely topological, based on exploiting the correspondence between subgroups of \(M^n_g(=M^n_{g,0})\) and graphs of vanishing cycles, and are achieved by showing comprehensible illustrations. In the last section, the authors prove that the mapping class group \(M^1_1\) is isomorphic to \(\widetilde{SL}(2, \mathbb Z)\). This fact is used to study numerical properties of Dehn twist relations in elliptic SLFs and the authors prospect further study of \(M^1_g\) for SLFs with fibers of higher genus, besides I. Smith's result on the non-existence of SLFs with monodromy contained in the Torelli group.
Mathematics Subject Classification ID: 57R17
Mathematics Subject Classification ID: 53D35
zbMATH DE Number: 1782645
zbMATH Keywords: symplectic Lefschetz fibration
zbMATH Keywords: fundamental group
zbMATH Keywords: mapping class group
zbMATH Keywords: Dehn twists
Wikidata QID: Q115194556
author: Jaume Amorós
author: Fedor Bogomolov
author: Ludmil Katzarkov
author: Tony Pantev
reviewed by: Haruo S. Suzuki
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214339791
OpenAlex ID: W1883489583
DOI: 10.4310/JDG/1214339791
description: scientific article; zbMATH DE number 1782645