9704833 Jaco This project involves the development and use of efficient triangulations and normal surface theory in the study and understanding of 3-manifolds. The primary objective is to develop an algorithm to decide if two 3-manifolds are homeomorphic, thereby providing a complete classification of 3-manifolds. Efficient triangulations lead to combinatorial methods that yield results analogous to those obtained when a 3-manifold is known to have a rich geometry (hyperbolic structure). All 3-manifolds admit triangulations, those which are least understood are precisely the class which admit efficient triangulations; hence, this project provides new methods to study the least understood cases of 3-manifolds. Low-dimensional manifolds provide the geometric models of most physical phenomena. It is natural to ask for a description of all possible such models. For nearly a century, the classification (precise listing of all possible models) of 2-manifolds has been understood. Following this work by fifty years, it was learned that it is not possible to classify all 4-dimensional models. This project attacks the problem in dimension 3, which remains open. ***
