A converse of the Jordan-Brouwer separation theorem in three dimensions

Abstract

That the closed (n-1)-manifold immersed in euclidean space, En, of n dimensions (n>2), separates En into just two domains of which it is the common boundary, was shown by Brouwer in 1912.t That the points of the manifold are accessible from each of its complementary domains Brouwer proved in an accompanying paper,: and in the latter connection he gave an example to show that a bounded and closed point set which separates En into just two domains and every point of which is accessible from each of these domains, is not necessarily homeomorphic with a closed manifold. The above results of Brouwer, in so far as the connectivity of the set residual to the manifold in En is concerned, received considerable extension at the hands of J. W. Alexander, who not only demonstrated, using modulo 2 Betti numbers, that the residual set is just two connected domains, but that a certain duality exists between the connectivity numbers of the set and those of its complement.? If we denote the ith connectivity number of a set F by Ri(F), then, for the particular case where Ci is a set in En homeomorphic with an i-sphere, Alexander showed that