What is Topology? 

      Algebraic topology occupies a very important position in modern mathematics. In many ways it is a microcosm of 20th century mathematics, illustrating features such as the increasing emphasis on global questions, the importance of functoriality, and the use of rather general and abstract machinery to solve quite specific problems. Since many of these developments (equally visible in subjects such as algebraic geometry and number theory) actually have their roots in algebraic topology, it can be very useful even for those not directly interested in its subject matter to have some familiarity with its methods, especially as it is probably the most accessible of the subjects which require "heavy machinery". 

      Of course its importance is not merely conceptual - it has very many applications. For example:  low-dimensional topology (topics such as knot theory and three-manifold theory); classification of high-dimensional manifolds and study of their smooth structures; study of vector fields on manifolds and other hard questions of linear algebra; study of spaces of functions and of solutions of differential equations (index theory); all manner of uses in modern geometry and physics; and so on. 

      Although algebraic topology is formulated primarily for studying the properties of arbitrary topological spaces up to homotopy equivalence (a rather general class of spaces with a fairly brutal form of deformation), it can also be used to study finer structures, such as manifolds up to homeomorphism.
                                             -- Justin Roberts, UC San Diego Math Professsor 

Last Updated December 17, 2007