A set is a 3 cell if its cartesian product with an arc is a 4 cell

A set is a $3$ cell if its cartesian product with an arc is a $4$ cell
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by R. H. Bing

Proc. Amer. Math. Soc. 12 (1961), 13-19

DOI: https://doi.org/10.1090/S0002-9939-1961-0123303-2

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References

R. H. Bing, The Kline sphere characterization problem, Bull. Amer. Math. Soc. 52 (1946), 644–653. MR 16645, DOI 10.1090/S0002-9904-1946-08614-0
R. H. Bing, A decomposition of $E^3$ into points and tame arcs such that the decomposition space is topologically different from $E^3$, Ann. of Math. (2) 65 (1957), 484–500. MR 92961, DOI 10.2307/1970058
R. H. Bing, Necessary and sufficient conditions that a $3$-manifold be $S^{3}$, Ann. of Math. (2) 68 (1958), 17–37. MR 95471, DOI 10.2307/1970041
R. H. Bing, The cartesian product of a certain non-manifold and a line is $E_{4}$, Bull. Amer. Math. Soc. 64 (1958), 82–84. MR 97034, DOI 10.1090/S0002-9904-1958-10160-3
R. H. Bing, The cartesian product of a certain nonmanifold and a line is $E^{4}$, Ann. of Math. (2) 70 (1959), 399–412. MR 107228, DOI 10.2307/1970322

A surface $S$ is tame in ${E^3}$ if ${E^3}- S$ is locally simply connected at each point of $S$

Corollary to a proof due to Bing

Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979–990. MR 27512, DOI 10.2307/1969408
James Glimm, Two Cartesian products which are Euclidean spaces, Bull. Soc. Math. France 88 (1960), 131–135. MR 125573
Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
A. Kirkor, A remark about Cartesian division by a segment, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 6 (1958), 379–381. MR 0097033

Les décompositions de l’hypercube en produit topologique

Frank Raymond, Separation and union theorems for generalized manifolds with boundary, Michigan Math. J. 7 (1960), 7–21. MR 120638

is the cartesian product of a totally non-Euclidean space and

M. Szumbarski, Sur la décomposition des eléments euclidiens en produits cartésiens, Rec. Math. [Mat. Sbornik] N.S. 16(58) (1945), 39–42 (French, with Russian summary). MR 0013296

A certain open manifold whose group is unity

R. L. Wilder, A converse of the Jordan-Brouwer separation theorem in three dimensions, Trans. Amer. Math. Soc. 32 (1930), no. 4, 632–657. MR 1501556, DOI 10.1090/S0002-9947-1930-1501556-8
R. L. Wilder, On the properties of domains and their boundaries in $E^n$, Math. Ann. 109 (1934), no. 1, 273–306. MR 1512894, DOI 10.1007/BF01449139
Gail S. Young Jr., On the factors and fiberings of manifolds, Proc. Amer. Math. Soc. 1 (1950), 215–223. MR 34582, DOI 10.1090/S0002-9939-1950-0034582-7