On the Connectivity Ring of an Abstract Space
James W. Alexander · Annals of Mathematics · 1936
(integrands of multiple integrals) that are independent, in the large, modulo the derived forms. The geometrical method of approach has been extended to compact metric spaces by Vietoris' and to still more general spaces by tech.2 Moreover, this branch of the theory has been very greatly perfected by the introduction of Pontrjagin's cycles with real coefficients reduced modulo 1. Now, if we use Pontrjagin's cycles, the kth connectivity group of a compact, metric space becomes a compact, metric group. Moreover, by a theorem of Pontrjagin,3 every such group may be identified with the character group of a countable, discrete group. This immediately suggests the advisability of regarding the discrete group, rather than its equivalent (though more complicated) metric character group, as the kth invariant of the space, and of looking for a revised theoretical treatment leading simply and directly to this group. We give such a treatment below, based on a suitable combinatory adaptation of the second, or analytic, method of approach. One decided advantage of taking the discrete groups rather than their metric character groups as the fundamental connectivity groups of the space is that we can then define the product4 (as distinguished from the sum) of two elements of the same or of different groups. The combined groups of all dimensionalities (or, more precisely, their direct sum) will thus become a connectitvity ring, as distinguished from a set of isolated connectivity groups.