Sharif Digital Repository

Topological combinatorics is an almost novel branch in mathematics whose dawn began to shine originally from Lovász studies at late 70's. The main purpose of the pursued subjects in this branch is to apply topological methods in order to conclude various combinatorial results based on the famous Borsuk-Ulam's theorem which itself is a very deep statement in algebraic topology and has many equivalent versions. For instance, one of these versions states that there is no continuous map f:S^n⟶S^(n-1) which behaves antipodally. In this thesis, we intend to take a considerably long journey in homology theory to pave the way toward a complete proof of this theorem and some of its generalizations....