Homogeneous continua in Euclidean $(n+1)$-space which contain an $n$-cube are $n$-manifolds

Abstract:

Let $X$ be a homogeneous continuum and let ${E^n}$ be Euclidean $n$-space. We prove that if $X$ is properly contained in a connected $(n + 1)$-manifold, then $X$ contains no $n$-dimensional umbrella (i.e. a set homeomorphic to the set $\{ ({x_1}, \ldots ,{x_{n + 1}}) \in {E^{n + 1}}:x_1^2 + \cdots + x_{n + 1}^2 \leq 1$ and ${x_{n + 1}} \leq 0$ and either ${x_1} = \cdots = {x_n} = 0$ or ${x_{n + 1}} = 0\}$). Combining this fact with an earlier result of the author we conclude that if $X$ lies in ${E^{n + 1}}$ and topologically contains ${E^n}$, then $X$ is an $n$-manifold.