Suppose C is a Cantor set in the Euclidean plane, or even in R^3. Suppose h is a homeomorphism of C onto itself. Can h be extended to a homeomorphism of the whole space? What about if h preserves the ´end points´ of the Cantor set? Seems that the latter is a necessary condition but I don´t know how to prove it.


In $\mathbb{R}^2$, Schonflies theorem ensures that any two embeddings of Cantor sets are equivalently embedded, or equivalently that any homeomorphism $h$ between the two Cantor sets (it can be proven that any two Cantor sets have a homeomorphism between them - I think this is proven in Edwin E. Moise's book Geometric Topology in Dimensions 2 and 3) can be extended to a homeomorphism $H \colon \mathbb{R}^2 \to \mathbb{R}^2$.

In $\mathbb{R}^3$, we have many examples of rigid Cantor sets; a personal favourite is the following paper which generalises Skora's construction of a wildly embedded Cantor set in $S^3$ with a simply connected complement, to produce a rigid wildly embedded Cantor set in $\mathbb{R}^3$, with a simply connected complement.

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    $\begingroup$ Clarification: The proof that there is a homeomorphism between any two Cantor sets was given by Brouwer in 1909 ("On the structure of perfect sets of points", found at the publisher's site: dwc.knaw.nl/DL/publications/PU00013496.pdf). Moise's text proves that if both are planar then the homeomorphism extends to the plane. $\endgroup$ Dec 15 '16 at 2:59
  • $\begingroup$ Do you know if exist any generalization of the Moise result to closed surface instead of $\mathbb{R}^2$? $\endgroup$
    – jon jones
    Jan 5 '20 at 19:15

This depends on what you want to call a Cantor set. The usual middle-thirds Cantor set is one embedding of the cantor set into the plane/$3$-space, however there are many other ways of embedding the Cantor set into a space and there appears to be a rich theory behind all the possible ways (I don't work in this area so I can't give too many details).

The paper Homogeneity groups of ends of open $3$-manifolds, Garity & Repovs, offers a brief glimpse of some of the questions that can arise, and you can get a feel for how non-trivial some of these embeddings can be by reading the introduction.

The standard middle thirds Cantor set embedding in the plane is known to be strongly homeogenously embedded which means any automorphism extends to the whole of the plane. However, on the other end of things there exist embeddings of the Cantor set in $\mathbb{R}^n$ such that no non-trivial automorphism of the Cantor set can be extended - these are called rigidly embedded.

There are many open problems related to these terms involving tame and wild embeddings in Eucldiean space.

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    $\begingroup$ Very interesting reference. Thank you! $\endgroup$ Mar 29 '14 at 6:22

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