Cantor sets and homotopy connectedness of manifolds

Wright, David G.

doi:10.1090/S0002-9939-1975-0370594-9

Cantor sets and homotopy connectedness of manifolds
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Proc. Amer. Math. Soc. 50 (1975), 463-470 Request permission

Abstract:

We prove that a topological manifold $M$ of dimension $n$ is $(n - 2)$-connected if each Cantor set in $M$ is contained in an open $n$-ball of $M$. An immediate consequence is that a compact manifold $N$ of dimension $n(n \geq 5)$ is homeomorphic to the $n$-sphere if and only if every Cantor set of $N$ is contained in an open $n$-ball of $N$. This consequence generalizes a $3$-dimensional theorem of Doyle and Hocking.

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Cantor sets and homotopy connectedness of manifolds