Seifert fibered spaces in 3-manifolds

Publisher Summary This chapter describes Seifert Fibered Spaces in 3-Manifolds. There exist finitely many disjoint, non-contractible, pairwise non-parallel, embedded 2-spheres in M, whose homotopy classes generate π2 (M) as a π2 (M)-module; and modulo the Poincare conjecture, these 2-spheres are unique up to ambient homeomorphism. Thus, all singular 2-spheres in M, that is, maps of S2 into M, may be described, up to homotopy, in terms of a geometric picture in M. The strong version of the sphere theorem presented in the chapter gives a great deal of information about fundamental groups of compact 3-manifolds, for example that they are finite free products of torsion-free groups and finite groups. It also provides in a slightly refined version a reduction of the classification problem for compact, oriented 3-manifolds to the classification problem for compact, irreducible, 3-manifolds.