Let K be the closure of one of
the complementary domains of a 2-sphere S topologically embedded in the 3-sphere,
S3. We give first (Theorem 1) a characterization of those points p ∈ S with the
following property: there exists a homeomorphism h : K → S8 such that h(S) can be
pierced with a tame arc at h(p). The topological property of K which distinguishes
such a “piercing point” p is this: K − p is 1-ULC. Using this result, we find
(Theorems 2 and 3) that p is a piercing point of K if and only if S is arcwise
accessible at p by a tame arc from lS3− K (note: perhaps S cannot be
pierced with a tame arc at p, even if p is a piercing point of K). Thus, the
“tamely arcwise accessible” property is independent of the embedding of
K in S3. The corollary to Theorem 2 gives an alternate proof of an as yet
unpublished fact, first proven by R. H. Bing: a topological 2-sphere in S8 is arcwise
accessible at each point by a tame arc from at least one of its complementary
domains.
In the last section of the paper, we give two applications of the above theorems.
First, we show in Theorem 4 that S can be pierced with a tame arc at p if and only if
p is a piercing point of both K and the closure of S8− K. Finally, we remark
in Theorem 5 that S can be pierced with a tame arc at each of its points
if it is “free” in the sense that for each 𝜖 > 0,S can be mapped into each
of its complementary domains by a mapping which moves each point less
than 𝜖. It is not known whether each 2-sphere S with this last property is
tame.
