The Fundamental Group

A standard problem in topology is the classification of spaces and continuous functions up to homeomorphisms and there are limited tools, in general, topology to deal with it. For example, one would like to know whether or not Euclidean spaces of different dimensions are homeomorphic; alternatively, one may be interested in finding a topological property which can be used to distinguish between a circular disc and an annulus (i.e., a disc with a hole). Notice that topological properties we have studied thus far are not helpful in answering these questions. A solution to such a problem is generally obtained by converting the problem into algebraic questions in such a way that the process always assigns isomorphic groups, rings, etc., to homeomorphic spaces, that is, the associated algebraic structure is a topological invariant. Such an invariant, called the Fundamental Group or Poincare Group of the space, was first defined by the great French mathematician Henri Poincare in 1895. It will be treated in the present chapter.