A survey on Seifert fiber space theorem
type: Item
instance of: scholarly article
title: A survey on Seifert fiber space theorem
zbMATH Open document ID: 1293.57001
DOI: 10.1155/2014/694106
author: Jean-Philippe Préaux
published in: ISRN Geometry
publication date: 2014-06-16T00:00:00Z
review text: Summary: We review the history of the proof of the Seifert fiber space theorem, as well as its motivations in 3-manifold topology and its generalizations. From the introduction: In the topology of low-dimensional manifolds (dimension at most 3) the fundamental group, or \(\pi_1\), plays a central key role. On the one hand several of the main topological properties of 2 and 3-manifolds can be rephrased in term of properties of the fundamental group and on the other hand in the generic cases \(\pi_1\) fully determines their homeomorphism type. That \(\pi_1\) generally determines their homotopy type follows from the fact that they are generically \(K(\pi, 1)\) (i.e. their universal covering space is contractible); that the homotopy type determines generically the homeomorphism type appears as a rigidity property, or informally because the lack of dimension prevents the existence of too many manifolds, which contrasts with higher dimensions. This has been well known for a long time for surfaces; advances in the study of 3-manifolds have shown that it remains globally true in dimension 3. For example, one can think of the Poincaré conjecture, the Dehn and sphere theorems of Papakyriakopoulos, the torus theorem, the rigidity theorem for Haken manifolds, or Mostow's rigidity theorem for hyperbolic 3-manifolds, and so forth. It provides a somehow common paradigm for their study, much linked to combinatorial and geometrical group theory, which has developed into an independent discipline: low-dimensional topology, among the more general topology of manifolds. The first reference book on the subject originated in the annotated notes that the young student Seifert took during the courses of algebraic topology given by Threlfall. In 1933, Seifert introduced a particular class of 3-manifolds, known as Seifert manifolds or Seifert fiber spaces. They have since then been widely studied, well understood, and have given a great impact on the modern understanding of 3-manifolds. They…
Mathematics Subject Classification ID: 57-02
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 6304708
zbMATH Keywords: Seifert fiber spaces
zbMATH Keywords: fundamental group
zbMATH Keywords: cyclic normal subgroup
zbMATH Keywords: geometrization conjecture
zbMATH Keywords: Poincaré conjecture
Wikidata QID: Q59048896
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/694106
OpenAlex ID: W2032142743
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description: scientific article; zbMATH DE number 6304708