Topología algebraica: homología singular

Abstract

In this paper we introduce the singular homology theory. First, we develop its main characteristics and properties. Then, we mainly focus on using its power to deduce some important results in topology such as the Brouwer fixed point theorem in arbitrary dimension, the Invariance of Dimension and Domain, or the Jordan-Brouwer separation theorem. We also give a rough idea about simplicial homology and compare it with the singular one. After computing the homology of the spheres, we briefly introduce the degree theory, presenting its main properties and using them to prove the non existence of a continuous tangent vector field on S n and the uniqueness of Z2 as a nontrivial group acting freely on S n, both when n is even. As a final point, we present a generalization of the Jordan-Brouwer theorem that explains the difference between homology and homotopy: the Alexander horned sphere shows that Schönflies theorem
fails in dimension ≥ 3, but the failure is homotopic, not homologic.