Strong homology groups of systems

Abstract

In this section we define strong homology groups \({\bar H_m}(X;G),\) of inverse systems of spaces X. They are homology groups of the total complex of a certain bicomplex, whose boundary operators are the boundary operators â of singular complexes and the Nöbeling — Roos operators b of abelian pro-groups (defined in 11.5). We also define a sequence of homology groups \(\bar H_m^{(r)}(X;G),r = 0,1, \ldots ,\) of height r. They approximate the strong homology group \({\bar H_m}(X;G),\) which can be viewed as homology of height oo. Homology of height 0 coincides with Cech homology and gives the coarsest approximation. The precise relation between the homology groups \({\bar H_m}(X;G),\) for different heights r, as well as the relation between their limit and the strong group \({\bar H_m}(X;G),\) are given by certain exact sequences (the Miminoshvili sequences). These sequences enable us to show that mappings f: X → Y, which induce isomorphisms of homology pro-groups also induce isomorphisms of strong homology groups.