On four-manifolds fibering over surfaces
type: Item
instance of: scholarly article
title: On four-manifolds fibering over surfaces
zbMATH Open document ID: 0926.57024
author: Friedrich Hegenbarth
author: Alberto Cavicchioli
author: Dušan Repovš
published in: Tsukuba Journal of Mathematics
publication date: 1999-11-22T00:00:00Z
review text: This is a paper regarding topological and smooth 4-manifolds that fiber over surfaces. In [\textit{J. A. Hillman}, Topology Appl. 40, No. 3, 275-286 (1991; Zbl 0754.57014); The algebraic characterization of geometric 4-manifolds, Lond. Math. Soc. Lect. Note Ser. 198 (1994; Zbl 0812.57001)] necessary and sufficient conditions for a 4-manifold to have the homotopy type of a surface bundle over a surface are given. A 4-manifold \(X\) is a homotopy surface bundle if and only if \(\chi(X) = \chi(F_1) \chi(F_2)\) and \(\pi_1 (X)\) is an extension of \(\pi_1 (F_1)\) by \(\pi_1 (F_2)\). This paper is aimed at the question of how many fiber structures exist for a given 4-manifold and pair of surfaces. The paper begins with an analysis of the classifying space of the mapping class group. We should remark that the cohomology class of this classifying space corresponding to the signature of the 4-manifold is related to Casson's invariant. After the analysis of this classifying space, the authors prove that the spectral sequence of the fibration collapses. One interesting application of this result is that Rokhlin's theorem holds for this class of 4-manifolds (even the topological 4-manifolds).
reviewed by: David Auckly
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 1247880
zbMATH Keywords: surface bundle
zbMATH Keywords: Rokhlin's theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163584
OpenAlex ID: W652788014
DOI: 10.21099/TKBJM/1496163584
description: scientific article; zbMATH DE number 1247880