There are two major relations betweeen topology and
computer:

applications of topology to computer science, and

the use of computer for computations in topology.
The former is new and many interesting ideas have
been discovered. For example, Gaucher introduced the
notion of
model category
in the study of parallel processing. It seems that
Voevodsky began to apply homotopy theory to logic and
computer science.
There are, of course, attempts for computing homology
and homotopy groups by using computer. One of the most
popular projects is the computation of the stable
homotopy groups of spheres by using the
Adams spectral sequence
.
Nowadays, it is not hard to manipulate topological
objects such as
simplicial complexes
and
surfaces
on computer displays. Probably visualizations by
computer will be useful when teaching undergraduate
students.
