ABSTRACT
A fundamental problem of topology is that of determining, for two spaces, whether or not they are homeomorphic. To show two spaces are homeomorphic, one needs to construct a continuous bijective map, with continuous inverse, mapping one space to the other. To show two spaces are not homeomorphic involves showing that such a map does not exist. Classifying all compact surfaces up to homeomorphism, for instance, demands more sophisticated topological invariants than these. Algebraic topology originated in the attempts by such mathematicians as Poincare and Betti to construct such topological invariants. Poincare introduced a certain group, called the fundamental group of a topological space; it is by its definition a topological invariant. There are several different ways of defining homology groups, all of which lead to the same results for spaces that are sufficiently 'nice'. A space that is the polytope of a simplicial complex will be called a polyhedron.
