Geometric topology is often split into low dimensional (4 or less) and high dimensional. This split is based upon the techniques employed, the kinds of question that can be answered, and the state of knowledge. There were enormous advances in high dimensional topology during the 60's including the solution of the high dimensional Poincare conjecture, and a good understanding of how differentiability enters into the picture, for example through the existence of exotic smooth structures on spheres.<br><br>Today a considerable effort is being made to better understand manifolds of dimensions 3 and 4. The techniques, conjectures and outlooks in these two areas are very different, although there have also been hints of various unifying themes. In the 80's it was discovered by Donaldson, Freedman and Casson that Euclidean space has exotic smooth structures only in 4 dimensions.<br><br>The theory of 3 dimensional manifolds was revolutionized in the late 70's by Thurston's Geometrization Conjecture. There are eight geometries (homogeneous Riemannian metrics) which (appear to) play a similar role in 3 dimensions to the three constant curvature geometries in two dimensions. Some problems in 3-dimensions are best studied through combinatorial and topological techniques using surfaces and their generalizations. Many problems in knot theory are of this type.