Algebraic Topology, An Introduction


As the name suggests, algebraic topology is a marriage of algebra and topology. These branches of mathematics seem unrelated, yet they can be connected in several ways. For instance, a topological space S might be assigned a group G, based on the propertyies of S, in such a way that any space homeomorphic to S will be given the same group G. Therefore, two spaces that exhibit different groups cannot be homeomorphic.

Proving a negative is always difficult; this is often the only way we can prove two spaces are different from each other. Consider a simple example, the line and the plane. These spaces seem different enough, yet they have the same cardinality. There are invertible functions that map the plane onto the line and back again. Perhaps one of these functions preserves open sets. Perhaps one dimension is really the same as two, when viewed from the right perspective.

This isn't true of course; the plane is different from the line. But we need algebraic topology to prove it. The groups associated with these spaces (actually their compactifications) are different, and that closes the case.

The first group that we will assign to a space S is its homotopy group, but before we can do that we need to know what a homotopy is.