I am bit puzzled by two facts from two Hatcher's books, they seem contradictory to me:
1) In Notes on 3-Manifolds [3MF], he proves that smooth embeddings $S^2\to\mathbb{R}^3$ bound a ball. Before the proof, he notes, that this doesn't hold for merely topological embeddings since the Alexander horned sphere is a counterexample. Yet
2) in Algebraic Topology [AT], he proves that the horned sphere IS a boundary of embedded ball.
What am I missing? Am I reading something incorrectly?
[AT] Example 2B.2 on page 170,
[3MF] Theorem 1.1 on page 1.
Best Answer
Suppose you embed Alexander's horned sphere into $S^3$ instead of $\mathbb{R}^3$. The complement of this horned sphere in $S^3$ is now made of two parts, one part that is ball-shaped and one part that is not ball-shaped. If you puncture $S^3$ in the not-ball-shaped region, you get the usual embedding of the horned sphere into $\mathbb{R}^3$. But if you puncture $S^3$ in the ball-shaped region, you get a different embedding into $\mathbb{R}^3$ where the horns are on the inside. This embedding is the counterexample mentioned in [3MF].