11/28/2006

William Jaco
IAS and Oklahoma State

On The Homeomorphism Problem: Classification of 3-manifolds

While it may not be widely known, the Thurston Geometrization Conjecture gives a theoretical proof of the Homeomorphism Problem and, consequentially, that 3-manifolds can be classified. In the announcement of G. Perelman (and subsequent work by others), the Thurston Geometrization Conjecture has been claimed to be true. We shall give the steps to an algorithm that determines if two given 3-manifolds are homeomorphic (The Homeomorphism Problem). For 3-manifolds this is equivalent to the existence of a classification. Namely, a list of 3-manifolds can be constructed so that each 3-manifold appears on the list precisely once and given a 3-manifold it can be decided where it is placed on the list (Classification of 3-manifolds).