Tame polyhedra in wild cells and spheres

Sher, R. B.

doi:10.1090/S0002-9939-1971-0281178-1

Tame polyhedra in wild cells and spheres
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Proc. Amer. Math. Soc. 30 (1971), 169-174 Request permission

Abstract:

It is shown that each arc on a disk D in ${E^4}$ can be homeomorphically approximated by an arc in D which is tame in ${E^4}$. Some applications of this are given. Also, we construct an everywhere wild $(n - 1)$-sphere in ${E^n},n \geqq 3$, each of whose arcs is tame in ${E^n}$.

References

Steve Armentrout, Homotopy properties of decomposition spaces, Trans. Amer. Math. Soc. 143 (1969), 499–507. MR 273574, DOI 10.1090/S0002-9947-1969-0273574-9
R. H. Bing, A wild surface each of whose arcs is tame, Duke Math. J. 28 (1961), 1–15. MR 123302
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
J. C. Cantrell, $n$-frames in euclidean $k$-space, Proc. Amer. Math. Soc. 15 (1964), 574–578. MR 164330, DOI 10.1090/S0002-9939-1964-0164330-1
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

Each disk in

can be squeezed to an arc

Herman Gluck, Embeddings in the trivial range, Ann. of Math. (2) 81 (1965), 195–210. MR 173243, DOI 10.2307/1970614
John Hollingsworth and R. B. Sher, Triangulating neighborhoods in topological manifolds, General Topology and Appl. 1 (1971), 345–348. MR 296952
C. Lacher, Locally flat strings and half-strings, Proc. Amer. Math. Soc. 18 (1967), 299–304. MR 212805, DOI 10.1090/S0002-9939-1967-0212805-1
T. B. Rushing, Everywhere wild cells and spheres, Rocky Mountain J. Math. 2 (1972), no. 2, 249–258. MR 301741, DOI 10.1216/RMJ-1972-2-2-249

Tame arcs on wild cells

R. B. Sher, Determining the cellularity of a $i$-complex by properties of its arcs, Proc. Amer. Math. Soc. 26 (1970), 491–498. MR 270353, DOI 10.1090/S0002-9939-1970-0270353-7