I saw the following surprising statement in Wikipedia:

When $D\subseteq\Bbb C$ is a simply connected compact set, then its complement $E=D^c$ is a simply connected domain in the Riemann sphere that contains $\infty$, ...

This property sounds like something specific to $S^2$. Is it true that the complement of a compact simply connected subset of $S^n$ is simply connected? Is this true if $D$ is not necessarily compact/closed? How do you prove this?

A related question (which may need to be moved elsewhere): is there a subset $D\subseteq S^2$ such that there are no nontrivial paths in either $D$ or $D^c$?

  • $\begingroup$ what do you mean by $D$ not necessarily closed? in a Hausdorff space (like $S^n$) compact sets are closed $\endgroup$
    – Riccardo
    Jun 19 '16 at 23:10
  • $\begingroup$ @Riccardo I mean not necessarily compact or closed, just simply connected. $\endgroup$ Jun 19 '16 at 23:10
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    $\begingroup$ re the related question, I think the answer is yes. There are continuum many paths in $S^2$, say $\{P_\alpha:\alpha<\frak c\}$ so by transfinite recursion one may pick two points $a_\alpha,b_\alpha$ in $P_\alpha$ with $\{a_\alpha,b_\alpha\}\cap\{a_\beta,b_\beta\}=\emptyset$ for all $\beta<\alpha$, and let $D=\{a_\alpha:\alpha<\frak c\}$. $\endgroup$
    – Mirko
    Dec 21 '17 at 2:29

I consider path-connectedness to be part of "simply connected". As a counterexample to your question when the set is not closed, take the Warsaw circle (a closed up topologist's sine curve), delete a point on the "reasonable" part of the curve, and take the complement of the result. The punctured Warsaw circle is not path-connected, but its complement is simply connected.

The complement of the obvious $S^2$ in $S^n$ is homotopy equivalent to $S^{n - 3}$, so no (take $n=3,4$). You can do arbitrarily badly; many many groups appear as the fundamental group of knotted 2-spheres in $S^4$, and the same for n-spheres in $S^{n+2}$ etc.

Indeed, fix $n \geq 3$; Kervaire proved that a finitely presented group $G$ is the fundamental group of the complement of some smoothly embedded $S^n \hookrightarrow S^{n+2}$ iff the abelianization of $G$ is $\Bbb Z$, $H_2(G;\Bbb Z) = 0$, and $G$ is normally generated by some element $m$. These are not too violent to ask.

  • $\begingroup$ Yikes! Most weird and wonderful! $\endgroup$ Jun 20 '16 at 1:59
  • $\begingroup$ Edit: "Topologist's some curve" changed to "topologist's sine curve". A typo that I do,often, is "sunset" for "subset". $\endgroup$ Jun 20 '16 at 14:47
  • $\begingroup$ @user254665 Thanks for the correction. $\endgroup$
    – user98602
    Jun 20 '16 at 14:48

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