When are two topological spaces homeomorphic? We can look for instance
at the fundamental group, which is a topological invariant of the
space. But S^3 and S^4 have the same fundamental group, and
nevertheless are not homeomorphic. We can associate to a space a number
of groups, called homology groups, which are roughly speaking the
generalisation of the fundamental group to higher dimensions. This
gives us more possibilities to distinguish between spaces.

Homology groups also turn out to be useful if you look at fixpoints
of maps between spaces. An important result is Lefschetz's fixed point
theorem, from wich we can deduce Brouwers fixed point theorem.