  Higher dimensional graphs can be used to colour two-dimensional geometric
graphs. If G the boundary of a three dimensional graph H for example, we can
refine the interior until it is colourable with 4 colours. The later goal is
achieved if all interior edge degrees are even. Using a refinement process
which cuts the interior along surfaces we can adapt the degrees along the
boundary of that surface. More efficient is a self-cobordism of G with itself
with a host graph discretizing the product of G with an interval. It follows
from the fact that Euler curvature is zero everywhere for three dimensional
geometric graphs, that the odd degree edge set O is a cycle and so a boundary
if H is simply connected. A reduction to minimal colouring would imply the four
colour theorem. The method is expected to give a reason "why 4 colours suffice"
and suggests that every two dimensional geometric graph of arbitrary degree and
orientation can be coloured by 5 colours: since the projective plane can not be
a boundary of a 3-dimensional graph and because for higher genus surfaces, the
interior H is not simply connected, we need in general to embed a surface into
a 4-dimensional simply connected graph in order to colour it. This explains the
appearance of the chromatic number 5 for higher degree or non-orientable
situations, a number we believe to be the upper limit. For every surface type,
we construct examples with chromatic number 3,4 or 5, where the construction of
surfaces with chromatic number 5 is based on a method of Fisk. We have
implemented and illustrated all the topological aspects described in this paper
on a computer. So far we still need human guidance or simulated annealing to do
the refinements in the higher dimensional host graph.
