Topology

Several areas of research in modern mathematics have developed as a result of interaction between two or more specialized areas. For example, the subject of algebraic topology associates with topological spaces various algebraic structures and uses their properties to answer topological questions. An elegant proof of the theorem that R m and R n with their respective standard topologies, are not homeomorphic for m≠n is provided by computing the homology of the one point compactification of these spaces. Indeed, the problem of classifying topological spaces up to homeomorphism was fundamental in the creation of algebraic topology. In general, however, the knowledge of these algebraic structures is not enough to decide whether two topological spaces are homeomorphic. The equivalence of algebraic structures follows from a weaker relation among topological spaces, namely, that of homotopy equivalence. In fact, homotopy equivalent spaces have isomorphic homotopy and homology structures. Equivalence of algebraic structures associated to two topological spaces is a necessary but not sufficient condition for their homeomorphism. Thus, one may think of homotopy and homology as providing obstructions to the existence of homeomorphisms. As we impose further structure on a topological space such as piecewise linear, differentiable, or analytic structures other obstructions may arise.