Homotopy and Homotopy Type

So far we have been discussing topological spaces under the assumption that two spaces may be identified if their topological type is the same, namely, when they are homeomorphic. The topological type of a space however is a very strong invariant of the space in the sense that for many applications two spaces may be thought of as the same although they are not homeomorphic. For these applications it is many times useful to think about two spaces as identical if one can continuously deformed into the other. For example, any Euclidean space can be deformed continuously into a single point. However the circle or any sphere or the torus cannot be deformed to a point. It is also not hard to see that the circle cannot be deformed into a 2-sphere and generally an n-sphere cannot be deformed into a k-sphere is n $\neq$ k. This motivates the idea of homotopy and homotopy type. A brief description follows.

Suppose f, g : X$\rTo$Y are two maps. We say that f and g are homotopic if in some precise sense f can be deformed continuously into g. It is not hard to see even from this intuitive definition that this relation on maps from X to Y is an equivalence relation. Two spaces X and Y are said to be homotopy equivalent if there are maps f : X$\rTo$Y and g : Y$\rTo$X such that fog is homotopic to the identity of Y and gof is homotopic to the identity of X. Once more, it is not hard to see that homotopy equivalence is indeed an equivalence relation on topological spaces. The homotopy type of a space X is the class of X under this equivalence relation. The study of homotopy types and related features of topological spaces forms a rich, well developed mathematical discipline called homotopy theory. In this section we describe some of the fundamental concepts of homotopy theory with an eye towards geometric applications.