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

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