9

Let $U,V$ be disjoint non-empty connected open subsets of the sphere $S^2$ such that $\partial U=\partial V$ and $\operatorname{cl}(U\cup V)=S^2$. Must $U$ and $V$ be simply connected?

This seems intuitively obvious, but I'm not sure how to best show it. I have a painstaking proof where one basically "rasterizes" the problem and reduces it to some discrete statement about any finite grid, but this feels like a poor way to go about it. Is there a better way to show this statement?

Milo Brandt
  • 61,938
  • In $S^3$ it is not true. Consider the Alexander horned sphere (see https://en.wikipedia.org/wiki/Alexander_horned_sphere). It separates $S^3$ in two non-empty components $U, V$ as in your question, but one of these is not simply connected. – Paul Frost Jan 08 '19 at 17:22
  • I wonder if it is possible to show that the common boundary of $U, V$ is connected? – Paul Frost Jan 08 '19 at 17:29
  • @PaulFrost It's certainly true that the boundary is connected; the discrete method I know about for this problem also establishes that the boundary is connected, but it's not a very satisfying proof. – Milo Brandt Jan 08 '19 at 17:35
  • I asked because one can show that the complement of compact connected $B \subset S^2$ splits into the disjoint union of at most countably many open disks. – Paul Frost Jan 08 '19 at 17:42
  • @PaulFrost Ah, that's interesting, I didn't know that. That seems like it would be most of the way to a proof - though I don't have a quick way to see that the boundary is indeed connected. (But that might be easier; it might even be doable via homology) – Milo Brandt Jan 08 '19 at 20:35
  • See my answer to https://math.stackexchange.com/q/2897660 (especially the last paragraph). These are highly non-trivial results. – Paul Frost Jan 08 '19 at 20:42

1 Answers1

2

This follows from the fact that a connected open subset of $S^2$ is simply connected iff its complement is connected. This fact itself is proved in many complex analysis textbooks, see also Complement is connected iff Connected components are Simply Connected

Lukas Geyer
  • 18,654