Colourful Cohomology

Abstract

The thesis starts out by explaining connections between graph theory, category theory and homology. Thereafter, the very abstract is translated into geometrical concepts, simplicial cohomology is especially derived. Enlightening theory is included along with relevant examples. Studying the four colour problem with simplicial cohomology gives a reformulation in terms of equations involving the coboundary operator. The equations give a direct connection of historical discoveries by P. G. Tait and O. Veblen. Solutions by Hamiltonian cycles as well as connectedness of triangulations are discussed. In the end, a short proof of the weaker five colour problem is presented.