Homology Theory - General Idea

General Idea

To any topological space and any natural number, one can associate a set, whose elements are called (-dimensional) homology classes. There is a well-defined way to add and subtract homology classes, which makes into an abelian group, called the th homology group of . In heuristic terms, the size and structure of gives information about the number of -dimensional holes in . For example, if is a figure eight, then it has two holes, which in this context count as being one-dimensional. The corresponding homology group can be identified with the group of pairs of integers, with one copy of for each hole. While it seems very straightforward to say that has two holes, it is surprisingly hard to formulate this in a mathematically rigorous way; this is a central purpose of homology theory.

For a more intricate example, if is a Klein bottle then can be identified with . This is not just a sum of copies of, so it gives more subtle information than just a count of holes.

The formal definition of can be sketched as follows. The elements of are one-dimensional cycles, except that two cycles are considered to represent the same element if they are homologous. The simplest kind of one-dimensional cycles are just closed curves in, which could consist of one or more loops. If a closed curve can be deformed continuously within to another closed curve, then and are homologous and so determine the same element of . This captures the main geometric idea, but the full definition is somewhat more complex. For details, see singular homology. There is also a version (called simplicial homology) that works when is presented as a simplicial complex; this is smaller and easier to understand, but technically less flexible.

For example, let be a torus, as shown on the right. Let be the pink curve, and let be the red one. For integers and, we have another closed curve that goes times around and then times around ; this is denoted by . It can be shown that any closed curve in is homologous to for some and, and thus that is again isomorphic to .