A Survey on Seifert Fiber Space Theorem

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. 1. Introduction The reader is supposed to be familiar with the topology of 3-manifolds. Basic courses can be found in reference books (cf. [1–3]). In the topology of low-dimensional manifolds (dimension at most 3) the fundamental group, or , 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 the fully determines their homeomorphic type. That the generally determines their homotopy type follows from the fact that they are generically (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 since 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 manifold, 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 introduces a particular class of 3-manifolds, known as Seifert manifolds or Seifert fiber spaces. They have been since 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 the for 3-manifolds. The Seifert fiber space conjecture