# Modify the Alexander horned sphere for an embedding $S^2 \hookrightarrow ֓\mathbb{R}^3$ s.t. neither component of $\mathbb{R}^3 − S^2$ is 1-connected.

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.

• Hint: add another set of horns inside the sphere. – Wojowu Mar 2 at 21:49
• 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. – Paul Frost Mar 3 at 9:36