Homology Groups of Relations

Abstract

Any relation between the elements of a set X and the elements of a set Y is associated with two simplicial complexes K and L. A simplex of K is a finite set of elements of X related to a common element of Y; a simplex of L is a finite set of elements of Y related to a common element of X. In particular, the relation of being an element of a set of a covering is a relation between the points of a space and the sets of the covering the space. One of the complexes associated with this relation is the nerve of the covering; its homology and cohomology groups are used in defining the Cech homology and cohomology groups of the space. The other associated complex, which has points of the space as its vertices, is used in defining the Vietoris homology groups and the Alexander cohomology groups. The two complexes associated with a relation will be shown (Theorem 1) to have isomorphic homology and cohomology groups; if the complexes are geometrically realized, they even have the same homotopy type. In particular, the nerve and the Vietoris complex of any covering have isomorphic homology and cohomology groups. It follows that, when the Cech [5] and Vietoris [12] homology groups are based on the same family of coverings, these groups are isomorphic for arbitrary spaces.' It also follows that, when the Alexander [2] and Cech cohomology groups are based on the same family of coverings, they are isomorphic for arbitrary spaces.2 The Alexander cohomology theory based on all open coverings is found to satisfy the seven Eilenberg-Steenrod axioms.3 The proof consists of showing that this cohomology theory is isomorphic with the Cech cohomology theory which is known [6] to satisfy the axioms.