HOMOLOGY THEORY AND THE EXACTNESS AXIOM

Abstract

In this article we give a description of a certain new homology theory on fairly wide categories of topological spaces, as well as a survey of papers concerning this construction. In contrast to the Aleksandrov-Cech homology theory this theory satisfies all the Eilenberg-Steenrod axioms, including exactness. In the end it turns out to be equivalent to the Steenrod homology theory and very close to the Borel-Moore homology theory, being isomorphic to it when the coefficient module is finitely generated (without this condition the Borel-Moore theory is not well-defined). We show that many results of the Borel-Moore theory take their most definitive and natural form in the homology theory under discussion. The methods of sheaf theory, which are used in the Borel-Moore homology theory, can be applied just as effectively in our case.