An Introduction to 3-manifolds

Introduction In these lecture notes we will give a quick introduction to 3–manifolds, with a special emphasis on their fundamental groups. The lectures were held at the summer school 'groups and manifolds' held in Münster July 18 to 21 2011. In the first section we will show that given k ≥ 4 any finitely presented group is the fundamental group of a closed k–dimensional man-ifold. This is not the case for 3–manifolds, we will for example see that Z, Z/n, Z ⊕ Z/2 and Z 3 are the only abelian groups which arise as fundamental groups of closed 3–manifolds. In the second section we recall the classification of surfaces via their geometry and outline the proofs for several basic properties of surface groups. We will furthermore summarize the Thurston classification of diffeomorphisms of surfaces. We will then shift our attention to 3–manifolds. In the third section we will first introduce various examples of 3–manifolds, e.g. lens spaces, Seifert fibered spaces, fibered 3–manifolds and exteriors of knots and links, we will furthermore see that new examples can be constructed by connected sum and by gluing along tori. The goal in the remainder of the lecture notes will then be to bring some order into the world of 3– manifolds. The prime decomposition theorem of Kneser and Thurston stated in Section 4.1 will allow us to restrict ourselves to prime 3– manifolds. In Section 4.2 we will state Dehn's lemma and the sphere theorem, the combination of these two theorems shows that most prime 3–manifolds are aspherical and that most of their topology is controlled by the fundamental group. In Section 2 we had seen that 'most' surfaces are hyperbolic, in Section 5 we will therefore study properties hyperbolic 3–manifolds. The justification for studying hyperbolic 3–manifolds comes from the Ge-ometrization Theorem conjectured by Thurston and proved by Perel-man. The theorem says that any prime manifold can be constructed by gluing Seifert fibered spaces and hyperbolic manifolds along incom-pressible 3–manifolds. Acknowledgment. We would like to thank Arthur Bartels and Michael Weiss for organizing the summer school. We also wish thank the audience for pointing out various inaccuracies during the talk.