A surface in 𝐸³ is tame if it has round tangent balls

Loveland, L. D.

doi:10.1090/S0002-9947-1970-0270381-6

A surface in $E^{3}$ is tame if it has round tangent balls
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Trans. Amer. Math. Soc. 152 (1970), 389-397 Request permission

Abstract:

R. H. Bing has asked if a $2$-sphere $S$ in ${E^3}$ is tame when it is known that for each point $p$ in $S$ there exist two round balls which are tangent to each other at $p$ and which lie, except for $p$, in opposite complementary domains of $S$. The main result in this paper is that Bing’s question has an affirmative answer.

References

R. H. Bing, Approximating surfaces with polyhedral ones, Ann. of Math. (2) 65 (1957), 465–483. MR 87090
R. H. Bing, A surface is tame if its complement is $1$-ULC, Trans. Amer. Math. Soc. 101 (1961), 294–305. MR 131265, DOI 10.1090/S0002-9947-1961-0131265-1
R. H. Bing, Spheres in $E^{3}$, Amer. Math. Monthly 71 (1964), 353–364. MR 165507, DOI 10.2307/2313236

Additional questions on $3$-manifolds

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Embeddings of surfaces in

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