Its abelianisation can be identified with the first homology group of the space.

Thus the intersection form can also be thought of as a pairing on the 2nd homology group.

In particular, if X is a connected contractible space, then all its homology groups are 0, except .

The homology groups of P over the integers can be calculated.

A superperfect group is one whose first two integral homology groups vanish.

Each yields three types of homology groups, which fit into an exact triangle.

As for the other homology groups, computations are easier.

This phenomenon is studied in a systematic way using (co)homology groups.

In general, this does not allow (co)homology groups of a space to be completely computed.

The map from the sum to the homology group of the product is called the cross product.