A survey on Seifert fiber space theorem
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 suit many nice properties, most of them being already known since the deep work of Seifert. Nevertheless one of their main properties, the so-called Seifert fiber space theorem, has been a long standing conjecture before its proof was completed by a huge collective work involving Waldhausen, Gordon and Heil, Jaco and Shalen, Scott, Mess, Tukia, Casson and Jungreis, and Gabai, for about twenty years. It has become another example of the characteristic meaning of \(\pi_1\) for 3-manifolds. The Seifert fiber space conjecture characterizes the Seifert fiber spaces with infinite \(\pi_1\) in the class of orientable irreducible 3-manifolds in terms of a property of their fundamental groups: they contain an infinite cyclic normal subgroup. It has now become a theorem of major importance in the understanding of (compact) 3-manifolds as we further explain. We review here the motivations and applications for the understanding of 3-manifolds of the Seifert fiber space theorem, its generalizations for nonorientable 3-manifolds and PD(3) groups, and the various steps in its proof.
scientific article; zbMATH DE number 6304708
scientific article