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The Alexander horned ball construction gives a closed embedding from the ball into the sphere, $D^3 \hookrightarrow S^3$. Its complement has zero homology but has a non-trivial $\pi_1$. Since the homology is zero, this should mean that $\pi_1(S^3 - im(D^3))$ has trivial abelianization. Is there a nice characterization of this group?
More generally, one can construct a Kan complex from maps from simplices into this space and ask for its homotopy type.

Furthermore, one can further embed $S^3$ into $S^4$ and ask about the complement $S^4 - im(D^3)$. This seems to be weakly contractible. Is there an actual contraction of this space to a point?

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    $\begingroup$ I think it is plausible this question would get a better reception on MSE. I do not want to write details right now, but your questions have quick answers: (1) A presentation for the fundamental group is obtained here. (2) The complement is a smooth manifold, hence a CW complex, hence equivalent to your Kan complex. (3) If $C$ is any compact connected set in $S^3$, the manifold $S^3 - C$ is aspherical by the sphere theorem in 3-manifold topology + Hurewicz on the universal cover. (4) A weakly contractible manifold is contractible. $\endgroup$
    – mme
    Commented Oct 22, 2022 at 0:42
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    $\begingroup$ Thanks! That was helpful. $\endgroup$
    – Charles Wang
    Commented Oct 22, 2022 at 3:56

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