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Question 2.B.6 in Allen Hatcher's Algebraic Topology page 176:

Modify the construction of the Alexander horned sphere to produce an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ for which neither component of $\mathbb{R}^3 − S^2$ is simply-connected.

Using the hint given by the comments, I want to build a modified version of the Alexander horned sphere. In my new sphere, there is another set of horns formed on the inside of the sphere. I want to show this is homeomorphic to $S^2$. I then embed it in $\mathbb{R}^3$.

The argument for the unbounded component of $\mathbb{R}^3 − S^2$ not being simply connected should be the same as the case of the Alexander horned sphere. Intuitively, the bounded component of $\mathbb{R}^3 − S^2$ shouldn't be simply connected by the same argument on why the unbounded component isn't.

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    $\begingroup$ Hint: add another set of horns inside the sphere. $\endgroup$ – Wojowu Mar 2 at 21:49
  • $\begingroup$ It seems that you want to answer Question 2.B.6. But why do you add Proposition 2B.1? This only makes clear that homology cannot distinguish between different embeddings. $\endgroup$ – Paul Frost Mar 3 at 9:36

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