Geometric topology
In mathematics, geometric topology is the study of manifolds and their embeddings, with representative topics being knot theory and braid groups. It has come over time to be almost synonymous with low-dimensional topology, concerning in particular objects of two, three, or four dimensions.
In the rapid development of topology after 1945, a distinction was drawn between the fields of algebraic topology typefied by homotopy theory, geometric topology with the Poincaré conjecture as its biggest unsolved problem, and differential topology as the study mostly of differential structures, with Morse theory as its natural technique. These fields all rested on general topology, which was the study of the general topological space. This classification would come to seem more artificial, with the passing of years.
The methods of geometric topology being combinatorial, and difficulties arising (for example in Dehn's lemma), the field was something of a poor relation. The solution by Smale of the Poincaré conjecture in higher dimensions changed matters somewhat, in that it made dimensions three and four seem the hardest; and indeed they required new methods, while the freedom of higher dimensions meant that questions could be reduced to computational methods available in surgery theory.
A number of advances, on the Poincaré conjecture and the Thurston conjecture programme, and on knot invariants such as new knot polynomials, had the effect of changing geometric topology. They certainly broke down the barriers with other parts of topology and geometry, making the subject more part of the mainstream.
See also: list of geometric topology topics
