The ends of product manifolds

Abstract

In this paper we provide simple point set properties which characterize the m-spheres Sn, open m-cells Em, closed cells Im, and annuli Am = [0, 1) x Sm-1. It is important to notice that the Poincare conjecture is not used in dimension 3 or 4. In this paper we provide simple point set properties which characterize m-spheres Sm and open m-cells Em among m-manifolds. Accepting these two main theorems, we establish corollaries which characterize closed cells I'll and annuli A= [0, 1) x Sm-. Then we prove the two main theorems. It is important to note that the Poincare conjecture is not used in dimension 3 or 4. In the following M will mean an ni-manifold, for mr>2, and P is a point in M. We will always be asking for geometric consequences of supposing that a certain subset of M (such as M-P) is a product A x B of topological spaces where neither A nor B is a single point. Recall that such factors A, B are necessarily generalized manifolds in the sense of Wilder ([1], [2]). We will repeatedly use the fact that A or B is therefore a manifold if its dimension is <2 [1]. 1. The proofs of the first two theorems are deferred until the next section. THEOREM 1. M=Em is the only connected noncompact m-manifold such that M-P is a product space. THEOREM 2. M=Sm is the only connected compact m-manifold such that M-P is a product space. COROLLARY 3. It is not necessary in Theorem 2 to assume that M is connected. PROOF. Let N be the component of M with P in N. By Theorem 2, N=Stm. Now let M-P=Xx Yand XO, YO be the components of Xand Y Received by the editors August 22, 1971. AMS 1970 subject classifications. Primary 54F65, 55A40, 57A15, 57B99; Secondary 54E45, 57A05, 57A10, 57C99, 57D99.