Vol. 22, No. 2, 1967

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Some topological properties of piercing points

Daniel Russell McMillan, Jr.

Vol. 22 (1967), No. 2, 313–322

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 The corollary to Theorem 2 gives an alternate proof of an as yet unpublished fact, first proven by R. H. Bing: a topological 2-sphere in S8 is arcwise accessible at each point by a tame arc from at least one of its complementary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.

Mathematical Subject Classification
Primary: 54.78
Received: 25 July 1966
Published: 1 August 1967
Daniel Russell McMillan, Jr.