Structures of Three-Manifilds

So far, the most magnificent breakthrough in mathematics in the 21st century is the Geometrization Theorem, a bold conjecture by William Thurston (generalizing Poincar\'e's Conjecture) and proved by Grigory Perelman, based on the program suggested by Richard Hamilton. In this survey article, we will explain the statement of this result, also presenting some examples of how it can be used to obtain interesting results in differential geometry.

We give a more geometric approach to an algorithm for deciding whether two hyperbolic 3-manifolds are homeomorphic. We also give a more algebraic approach to the homeomorphism problem for geometric, but non-hyperbolic, 3-manifolds.

Geometrization says `` any closed oriented three-manifold which is prime (not a connected sum) carries one of the eight Thurston geometries OR it has incompressible torus walls whose complementary components each carry one of four particular Thurston geometries" (see Introduction and Figure 1). These geometric components have finite volume for the hyperbolic geometries (the H labeled vertices). They also have finite volume for each of the two geometries appearing as Seifert f...

After G. Perelman's solution of the Poincare Conjecture, this is a different way toward it. Given a simply connected, closed 3-manifold M, we produce a homotopy disc H, which arises from M by a finite sequence of simple modifications and, almost miraculously, can be imbedded into the ordinary space R^3. It follows that H is a disc, hence M is a sphere. In order to construct H, we use a special stratification of M, based on the fact that M is simply connected.

This paper has been withdrawn, because its material has been revised and became part of paper math.GT/0010184

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a polyhedron P with its boundary faces identified in pairs so that (\partial P)/~ is a finite number of internally flat complexe...

We state a number of open questions on 3-dimensional Poincar\'e duality groups and their subgroups, motivated by considerations from 3-manifold topology.

This survey focuses on the computational complexity of some of the fundamental decision problems in 3-manifold theory. The article discusses the wide variety of tools that are used to tackle these problems, including normal and almost surfaces, hierarchies, homomorphisms to finite groups, and hyperbolic structures.

This book provides a self-contained introduction to the topology and geometry of surfaces and three-manifolds. The main goal is to describe Thurston's geometrisation of three-manifolds, proved by Perelman in 2002. The book is divided into three parts: the first is devoted to hyperbolic geometry, the second to surfaces, and the third to three-manifolds. It contains complete proofs of Mostow's rigidity, the thick-thin decomposition, Thurston's classification of the diffeomorphi...

This survey article describes the algorithmic approaches successfully used over the time to construct hyperbolic structures on 3-dimensional topological "objects" of various types, and to classify several classes of such objects using such structures.