Bachelor thesis in mathematics
Jens Jakob Kjær · 2011
This thesis deals with the homology and cohomology groups of simplicial complexes, and especially with the two duality theorems which willdemonstrate links between the two notions. For this we will need some results, and we will therefore prove the long exact sequence for the reduced relative homology group; and to link the homology and cohomolgy to our geometric notion of a simplicial complex we will see that these are connected to the Euler characteristic of the complex. Then by rewriting the relative homology group we will be able to prove the Alexander duality which is the major theorem of this thesis. We will use the result that if realizations of two different simplicial complexes are homotopic then their homology and cohomology groups are isomorphic in order to prove another duality, by showing that there is a different complex constructed by the nonfaces of our complex which is homotopic to the Alexander dual of the complex. For this we shall need some homotopy theorems which will be duly proven.