Homology:

The k-th homology group is made up of classes of k-cycles, where two k-cycles are in the same class if together the form the boundary of some (k+1)-dimensional object. So for example, two points are in the same class provided there is a path that connects them (so the 0-dimensional points are the boundary of some 1-dimensional curve). Similarly, two curves represent the same homology class if together they form the boundary of some region. For instance, on a torus, two curves that go around the hole together bound a region, so they are in the same homology class.

The identity in the k-th homology group is the class made up of the k-cycles that bound regions all by themselves. Such cycles are called trivial cycles. For example, on the sphere every curve bounds a region, so every curve is trivial and there is only one homology class; thus the first homology group of the sphere is the trivial group.

For closed, connected surfaces, since there is a path connecting every pair of points, the 0-th homology group is trivial. Since there is only one 2-dimensional region without a boundary (namely the entire surface itself), the 2nd homology group is also trivial. So the most important homology group for surfaces is the 1st homology group.

We have seen that for the sphere, the 1st homology group is trivial. For the torus, it has two generators (a cycle that goes around the hole, and a cycle that goes around the tube). The real projective plane has a single generator for its 1st homology group.

The homology groups are a very powerful tool in the study of surfaces.