Every closed, oriented 3d manifold m3 can be obtained as follows. Start with a finite collection of graph manifolds with incompressible boundary if there is any and finite volume hyperbolic manifolds. In 1982 william thurston presented the geometrization conjecture, which. This work depends on the accumulative works of many geometric analysts in the past thirty years. We begin by discussing group actions, covering space topology, fiber bundles, and seifert fiber spaces. The eight geometries of the geometrization conjecture. 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. 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 existence of ricci flow with surgery has application to 3manifolds far beyond the poincare conjecture. In this paper we explore the eight geometries of thurstons geometrization conjecture. Preliminaries throughout this talk, a manifold is a connected, orientable, smoothmanifold, possibly withboundary.
In mathematics, thurstons geometrization conjecture states that certain threedimensional topological spaces each have a unique geometric structure that. In mathematics, thurstons geometrization conjecture states that each of certain threedimensional topological spaces has a unique geometric structure that can be associated with it. Completion of the proof of the geometrization conjecture. Pdf geometrization of three manifolds and perelmans proof. 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. This theory gives local models for the collapsed part of the manifold. We make precise the notion of a geometric structure. We then discuss the two dimensional geometries including a brief proof of the uniformization theorem. 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. The main technique for this study is the theory of alexandrov spaces. This is a survey about thurstons geometrization conjecture of three manifolds and perelmans proof with the ricci flow. The method is to understand the limits as time goes to infinity of ricci flow with surgery. It forms the heart of the proof via ricci flow of thurstons geometrization conjecture.1168 958 510 1002 423 1257 1171 401 983 1295 1406 657 354 1432 1097 263 1637 483 192 898 1380 1061 1392 1448 1477 705 226 1307 1007 1106 758 561 209 643 1319