# Is the Alexander horned sphere a cofibration?

The Alexander horned sphere is a closed embedding of $$S^2$$ into $$S^3$$ which is not flat because otherwise the Schoenflies Theorem would be true for it. However, not being flat is not the same as not being able to find a neighborhood deformation retract. I suspect the answer to the question is no, but I have no valid argument. To be precise, let $$\iota:S^2\hookrightarrow S^3$$ be the embedding whose image is the Alexander horned sphere. Is $$\iota$$ a cofibration?

We'll make use of the following.

If $$X$$ is an ANR and $$j:A\subseteq X$$ is a closed subspace, then $$A$$ is an ANR if and only if the inclusion $$j$$ is a cofibration.

I don't know a good reference for this, although I suspect it may be in Borsuk's book in some form or another. A reference I do have which will cover the case at hand is found in chapter 3 of Daverman's book Decompositions of Manifolds, where Theorem 6 states that

If $$A$$ is a closed subset of a metric space $$X$$, then $$A$$ has HEP in $$X$$ with respect to every ANR $$Y$$.

The point is that if $$A$$ is itself an ANR, then so is $$A\times I$$, and in particular also the mapping cyclinder $$M_j=A\times I\cup X\times 0$$. In this case we see that $$j:A\subseteq X$$ is a cofibration by noticing that it satisfies the HEP with respect to $$M_j$$ (i.e. we show that $$M_j$$ is a retract of $$X\times I$$).

The point of course is that $$S^3$$ is an ANR, as is any closed subspace $$A\subseteq S^3$$ which is homeomorphic to $$S^2$$.

• I learnt from this answer that a reference for your first statement (with a very similar proof as what you describe) is Proposition A.6.7 of Cellular structures in topology, by Fritsch and Piccinini. – Pierre PC Oct 15 '20 at 19:36
• I find this amazing. Thank you very much for the clear and nice answer! I presume on the other hand that at least the Alexander horned sphere cannot have a topological regular neighborhood as defined by Edwards here arxiv.org/pdf/0904.4665.pdf ? – daniel Oct 16 '20 at 1:53