Wild cells in 𝐸⁴ in which every arc is tame

Daverman, Robert J.

doi:10.1090/S0002-9939-1972-0295312-1

Wild cells in $E^{4}$ in which every arc is tame
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by Robert J. Daverman

Proc. Amer. Math. Soc. 34 (1972), 270-272

DOI: https://doi.org/10.1090/S0002-9939-1972-0295312-1

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Abstract:

Seebeck has proved that if an m-cell C in Euclidean n-space ${E^n}$ factors k-times, $m \leqq n - 2$, and $n \geqq 5$, then every embedding of a compact k-dimensional polyhedron in C is tame relative to ${E^n}$. We prove the analogous result for $n = 4$ and $m \leqq 3$.

References

R. H. Bing, Pushing a 2-sphere into its complement, Michigan Math. J. 11 (1964), 33–45. MR 160194
J. L. Bryant, On embeddings with locally nice cross-sections, Trans. Amer. Math. Soc. 155 (1971), 327–332. MR 276983, DOI 10.1090/S0002-9947-1971-0276983-6

Characterizations of tame subsets of

spheres in three-dimensional space

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R. J. Daverman and W. T. Eaton, An equivalence for the embeddings of cells in a $3$-manifold, Trans. Amer. Math. Soc. 145 (1969), 369–381. MR 250280, DOI 10.1090/S0002-9947-1969-0250280-8
Charles L. Seebeck III, Tame arcs on wild cells, Proc. Amer. Math. Soc. 29 (1971), 197–201. MR 281177, DOI 10.1090/S0002-9939-1971-0281177-X
R. B. Sher, Tame polyhedra in wild cells and spheres, Proc. Amer. Math. Soc. 30 (1971), 169–174. MR 281178, DOI 10.1090/S0002-9939-1971-0281178-1