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.