Wild spheres in $E^{n}$ that are locally flat modulo tame Cantor sets

Abstract:

Kirby has given an elementary geometric proof showing that if an $(n - 1)$-sphere $\Sigma$ in Euclidean $n$-space ${E^n}$ is locally flat modulo a Cantor set that is tame relative to both $\Sigma$ and ${E^n}$, then $\Sigma$ is locally flat. In this paper we illustrate the sharpness of the result by describing a wild $(n - 1)$-sphere $\Sigma$ in ${E^n}$ such that $\Sigma$ is locally flat modulo a Cantor set $C$ and $C$ is tame relative to ${E^n}$. These examples then are used to contrast certain properties of embedded spheres in higher dimensions with related properties of spheres in ${E^3}$. Rather obviously, as Kirby points out in [11], his result cannot be weak-ened by dismissing the restriction that the Cantor set be tame relative to ${E^n}$. It is well known that a sphere in ${E^n}$ containing a wild (relative to ${E^n}$) Cantor set must be wild. Consequently the only variation on his work that merits consideration is the one mentioned above. The phenomenon we intend to describe also occurs in $3$-space. Alexander’s horned sphere [1] is wild but is locally flat modulo a tame Cantor set. In fact, at one spot methods used here parallel those used to construct that example. However, other properties of $3$-space are strikingly dissimilar to what can be derived from the higher dimensional examples constructed here, for, as discussed in §2, natural analogues to some important results concerning locally flat embeddings in ${E^3}$ are false.