A characterization of manifold boundaries in 𝐸_{𝑛} dependent only on lower dimensional connectivities of the complement

Abstract

In my recent paper Generalized closed manifolds in n-space it was shown} that a compact point set B in Eny common boundary of (at least) two domains and D which are respectively u.li-c.§ for OSi^j and O^i^n—j—3 (where » 2 > j ^ ( n 3 ) / 2 ) , and such that the Betti numbers p'(D)t p*(D), • • • , p~{D) are finite, is a g.c.(» —l)-m. This constituted a generalization of a former result|| to the effect that when n = 3, and Dx are u.l.O-c, and p (D) is finite, B is a closed 2-manifold. In the present note I propose to show, as principal result, that the above conditions on the numbers p'+(D), • • • , p~{D) are irrelevant, and furthermore that it is immaterial whether we place the restriction as to finiteness on pi(D) or on p~~{D^, I t turns out that the only essential requirements are that the upper limits on the dimensions for which and D are u.l.i-c. must total at least w —3, and that one of the domains have a finite Betti number as just stated. For the sake of brevity we make the following definitions. We shall understand without explicit statement hereafter that the imbedding space is En(n^3) (euclidean space of n dimensions).