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Take an injective (continuous) map $T:\mathbb{S}_1\to\mathbb{S}_1$. It's an obvious fact (though I can only prove it with nontrivial facts about homotopy) that $T$ is automatically surjective.

I have two questions:

  • Is this the case for all 'endomorphisms' of spheres? (I think so, but have not proven it)
  • Does there exist a compact manifold without boundary which embeds nontrivially into itself?

I suspect the answer to the second question is 'no', but I have little evidence for the conjecture.

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    Thanks for the reference - very cool :)2012-10-16

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