K-6 - Homology

Abstract

Homologies were introduced by H. Poincare on the eve of the 20th century that followed the discovery of cohomologies. The singular homology theory was developed in the works of many as an extension of the simplicial homology to general spaces. It is topologically invariant by definition and coincides with the simplicial homology for simplicial complexes. Any two homology theories with isomorphic coefficient groups on the category of compact polyhedral pairs are isomorphic. This is extended to the category of finite CW-complexes. The Eilenberg–Steenrod theorem introduces the possibility of an axiomatic approach to algebraic topology. Thus, if the Dimension Axiom is dropped, the notion of extraordinary homology theory can be obtained. The Eilenberg–Steenrod theorem holds for cohomologies as well. Singular homology and cohomology are defined for arbitrary topological spaces, but the natural area of their applications is the category of finite complexes. It can be automatically enlarged to spaces homotopically equivalent to finite complexes—such as compact absolute neighborhood retract (ANR). More general spaces require a different construction of homologies. One of the areas of application of cohomology in general topology is dimension theory.