Continuous Connectivity Groups in Terms of Limit Groups

Abstract

It is known that the concept of a group spectrum and its limit group yields a particularly elegant method for defining the Cech homology and cohomology groups of arbitrary spaces. The object of this paper is to show that these fundamental group-theoretic notions can also be applied to define the connectivity groups, in the sense of homology theory, based on continuous mappings of n-cells into the space. With this approach, the homology groups appear as direct limits and the cohomology groups are inverse limits, in exact contrast with the Cech theory. This approach avoids the cumbersome use of singular chains and has the additional advantage that in many cases theorems established for polytopes can be immediately extended to arbitrary spaces. It will be remarked that the Vietoris homology theory can also be developed within the framework of the limit group concept, and it turns out that the spectrum which has the Vietoris group as its inverse limit group yields, for its direct limit group, the comology group of a space as originally defined by Alexander in terms of antisymmetric functions of sets of points. It will moreover be shown that the Alexander and Cech cohomology groups are isomorphic for a wide class of spaces.