Combinatorial Proofs of Some Theorems in Algebraic Topology

Two important theorems in algebraic topology are the Brouwer Fixed Point theorem and the Borsuk-Ulam theorem. The theorems require the development of homology in their standard proofs. However, each theorem has an equivalent combinatorial result involving triangulating the relevant surface and coloring the vertices of the triangulation. Then by taking the limit of a sequence of finer triangulations, it is possible to prove results about continuous functions. In this paper, I will prove Sperner’s lemma and Tucker’s lemma and then use them to prove the Brouwer Fixed Point theorem and Borsuk-Ulam theorem, respectively.