In this section we will associate S with a structure that looks unwieldy - an infinite sequence of free abelian groups, where each group has an uncountable rank. This would be of no practical utility if it weren't for the boundary operator, which maps one group into the next. The resulting quotient groups are often manageable, and they form the homology of S. The free abelian groups are derived from the simplex, hence this branch of mathematics is sometimes called simplicial homology. However, simplicial homology sometimes means something else, so it is more accurately referred to as singular homology. The two concepts coincide (we'll prove this later), so the distinction is rarely significant.
Before we proceed, you need to be familiar with simplexes, and homology theory. The former is straightforward; so you may want to review the theorems in that topic now. The latter is rather abstract, and can become confusing when studied out of context. In fact, homology and cohomology were developed in concert with algebraic topology, to support the structures described in these pages. So you may want to step through both topics in parallel. If you try to learn all the theorems in homology first, then return to this page, you will probably get confused, because you won't see how and why each theorem is used. If homology theory is new to you, read the introduction, and I'll try to reference other theorems as they are needed.
When you refer to the theorems in homology, remember that the boundary operator goes up, from G3 into G4, for example. In the world of algebraic topology the boundary operator goes down, from G4 into G3. So you need to turn the arrows around. No big deal - I just didn't want you to get confused.
Why then is homology pulled out, as a separate topic? Why not include those theorems here as we go along? Wouldn't that be easier?
Perhaps, but homology has taken on a life of its own as a separate algebraic discipline, beyond the context of topology. In fact, homology has been generalized from abelian groups to modules, where it serves as a foundation for many definitions and theorems. The module operators tor and ext, for instance, are based on module homology. But I digress.
If you can get through the next few theorems, you'll find some lovely results. For instance, each euclidean space Rn is distinct. There is no function f that equates the line and the plane, and preserves the topology. Two and three dimensions really are different worlds. Another application of homology is Brouwer's fixed point theorem. Take a closed ball in n dimensions and map it (continuously) into itself. There will always be an x such that f(x) = x. Beyond these immediate applications, singular homology leads to cohomology, cellular homology, and much more. So take a deep breath and turn the page.