Lecture 1 In this course, we focus on compact and orientable 3-manifolds. We ask the following fundamental questions: What do all 3-manifolds look like? Can we classify them? Consider the case of closed orientable surfaces. They are characterized by the genus. The case g = 0 is the 2-sphere. One can equip it with the round metric. The Figure 1. Surfaces with genus 0, 1 and 2 case g = 1 corresponds to the two dimensional torus ∼ R2/Γ, where Γ is a lattice subgroup of R2. It admits a naturally induced flat metric from R2. For g ≥ 2 we can equip the surface Σg of genus g with the hyperbolic metric induced from the Poincar´ e disk model of H2 that is to say there is a discrete, torsion-free subgroup Γg of the isometries of H2 with Σg ∼ H2/Γg. Geometric manifolds Definition (Homogeneous metric). Let (M, g) be a Riemannian manifold. The metric g is called homogeneous if the action Isom(M)×M → M is transitive. Definition (Locally homogeneous metric). A Riemannian metric g on a man- ifold M is called locally homogeneous if its lifted metric ˜ on the universal cover ˜ is homogeneous. The round metric of the 2-sphere (i.e. g = 0) is homogeneous, but the hyper- bolic metric of the genus-2 Riemann surface is not. However, the latter is locally homogeneous. Definition (Geometric manifolds). A manifold is called geometric if it ad- mits a finite volume complete locally homogeneous Riemannian metric. Here is the list of Geometric 3-manifolds by type (see [19]): (1) Round: S3 and its finite quotients – lens spaces, dodecahedron spaces – classification was completed by Hopf. (2) Flat: T 3 and its finite quotients – these are completely classified. 3 http://dx.doi.org/10.1090/ulect/053/01

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