Pdf on jun 18, 2017, garret sobczyk and others published geometrization of the real number system find, read and cite all the research you need on researchgate. Thurston 89 sought to decompose any compact 3manifold into pieces, each of which admits canonical riemannian metrics, the models of which are inspired by a thorough understanding of possible locally homogeneous spaces. Completion of the proof of the geometrization conjecture. Pdf geometrization of three manifolds and perelmans proof. This work depends on the accumulative works of many geometric analysts in the past thirty years. We make precise the notion of a geometric structure. The main technique for this study is the theory of alexandrov spaces. We then discuss the two dimensional geometries including a brief proof of the uniformization theorem. In this paper, we give a complete proof of the poincar. This book gives a complete proof of the geometrization conjecture, which describes all compact 3manifolds in terms of geometric pieces, i. We begin by discussing group actions, covering space topology, fiber bundles, and seifert fiber spaces.
In mathematics, thurstons geometrization conjecture states that certain threedimensional topological spaces each have a unique geometric structure that. Introduction the uniformization theorem tells us that every compact surface without boundary, or twomanifold, admits a geometric structure, and further, one of only three possible geometric structures. From this and previous results, geometrization follows easily. The method is to understand the limits as time goes to infinity of ricci flow with surgery.
Preliminaries throughout this talk, a manifold is a connected, orientable, smoothmanifold, possibly withboundary. The eight geometries of the geometrization conjecture. It is an analogue of the uniformization theorem for twodimensional surfaces, which states that every simply connected riemann surface can be given one of three geometries euclidean, spherical, or hyperbolic. This is a survey about thurstons geometrization conjecture of three manifolds and perelmans proof with the ricci flow. The existence of ricci flow with surgery has application to 3manifolds far beyond the poincare conjecture.
It forms the heart of the proof via ricci flow of thurstons geometrization conjecture. This theory gives local models for the collapsed part of the manifold. In 1982 william thurston presented the geometrization conjecture, which. Start with a finite collection of graph manifolds with incompressible boundary if there is any and finite volume hyperbolic manifolds. These local models can be glued together to prove that the collapsed part of the manifold is a graph manifold with incompressible boundary. In mathematics, thurstons geometrization conjecture states that each of certain threedimensional topological spaces has a unique geometric structure that can be associated with it. Conjecture and the closely related 3dimensional spherical spaceform conjecture are then immediate. Every closed, oriented 3d manifold m3 can be obtained as follows.