Colloquium: "Classification of 3-manifolds" by Dr. Guilherme Sadovski

Speaker: Dr. Sadovski is a member of this unit (visit his page)

Abstract. In this talk, we review the classification of 2-manifolds. In particular, we focus on the concept of "connected sum" as a kind of surgery theory along open disks as well as the meaning of the Uniformization Theorem. In close analogy with the d=2 case, we start our discussion on TOP 3-manifolds by performing surgery on them along open balls. We then arrive in the result known as the Prime Decomposition Theorem. We proceed by performing surgery now along open tori, and we arrive at the Torus Decomposition Theorem. In general, a 3-manifold does not accept a single (global) geometry, as a 2-manifold does, i.e., it is not, in general, "geometrizable". Nonetheless, after cutting sufficiently along open balls and tori, its resulting pieces are! This result lead us to state Thurston's Geometrization Theorem: "Any compact 3-manifold can be split in an unique way by disjoint embedded spheres and tori, each piece accepting a single geometry. There are, however, only 8 different kinds - locally isometric to only one of the 8 Thurston's model geometries.". We finish by stating a theorem by Moise, which is a generalization of Rado's theorem for 2-manifolds: "TOP 3-manifolds accept an unique DIFF structure up to diffeomorphisms". As consequence, the full classification in the TOP category is also a full classification in the DIFF category.