Local homology and cohomology

Abstract

The idea of associating local homology and cohomology groups to each point of a space is now over fifty years old, and it has proved useful in the theory of transformation groups (as in the local Smith theorems), in dimension theory and elsewhere. Nevertheless there is no uniformity as to definition or notation, and a good deal of the theory is written in a notation now superseded and often impenetrable to the modern reader. In this paper all the important definitions are collected in a single framework, and the relations between the various groups are explored. Particular attention is paid to the case of a cohomologically locally connected space, for which relatively complete results are obtained.