1

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential Topology by Morris Hirsch that said ``if a subset of $\mathbb{R}^2$ is homeomorphic to $S^1$ then it is a $C^0$ submanifold of $\mathbb{R}^2$'' which requires Schoenflies' Theorem to prove (given as a hint). As this special case requires that much machinery, I think in general it is not true.

Paladin
  • 119
  • 5
    I imagine that the counterexamples to Schoenflies problem, ie. the wild spheres $S^2 \subseteq \mathbb{R}^3$ could give what you're looking for. If such a wild sphere was a submanifold, it would be locally flat and by a theorem of Morton Brown (see http://en.wikipedia.org/wiki/Schoenflies_problem) thus not wild at all, giving a contradiction. – Piotr Pstrągowski Apr 04 '14 at 20:47
  • 3
    In fact someone already gave this example and, as I understand it, answered this question here http://mathoverflow.net/questions/79431/locally-flat-submanifold?rq=1 – Piotr Pstrągowski Apr 04 '14 at 21:02
  • 1
    Could you include a definition of a $C^0$ submanifold? – Alexandre Eremenko Apr 05 '14 at 14:01
  • wiki "local flatness" – Paladin Apr 05 '14 at 14:57

0 Answers0