Extensions of the Jordan-Brouwer separation theorem and its converse

Abstract

In Wilder's colloquium [2] a Jordan-Brouwer type separation theorem is proved for generalized manifolds (see Definition 2). It states that if K is a connected orientable (n-1)-gcm (generalized closed manifold) in a connected orientable n-gcm S such that pn-I(S)=O (the (n-1)-Betti number), then S-K is the union of disjoint domains A and B that have K as their common boundary. It is shown in this paper that under the conditions of the above theorem, the sets A =AkUK and B = BUK are each generalized manifolds with boundary K (see Definition 2). Furthermore if we are interested in proving only that Ai and B are manifolds with boundary, this can be done without the hypothesis pn-'(S) =0 and without assuming K is connected or orientable. The result then becomes the following: If S-K=AUJB separate, where K is an (n-1)-gcm in the connected orientable n-gcm S, then Af and B are generalized manifolds with boundaries consisting of some of the components of K. More generally it can be said that the closure of each component of S-K is a gm with boundary formed by some of the components of K (if S-K is disconnected). Wilder also considers converses of the Jordan-Brouwer theorem and other related theorems which are all concerned with the case where the boundary of a ulcr (uniformly locally-s-connected s < r) open subset of a connected orientable n-gcm is a connected orientable (n 1)-gcm. We answer the more general question as to when the closure of such a set is a gm with boundary. As before the hypothesis pn-l(S) =0 can be eliminated if connectedness is not required in the conclusion. Thus we show among other things that the closure of an open ulcn-' subset of a connected orientable n-gcm whose boundary is (n 1)-dimensional is an n-gm with boundary. In order to eliminate the assumption pn-l(S) = 0 it is necessary to prove an extension of the Alexander type duality theorem which does not include that hypothesis. The extended result states that if K is a closed subset of the connected orientable n-gcm S such that S-K has m components (m may be infinite), then mr-1 pn-l'(K) <(m-1)+pn-l(S). Throughout the paper we shall assume that the space S is a compact Hausdorff space. The homology theory used will be that of