## Embedding phenomena based upon decomposition theory: locally spherical but wild codimension one spheres

HTML articles powered by AMS MathViewer

- by Robert J. Daverman PDF
- Proc. Amer. Math. Soc.
**90**(1984), 139-144 Request permission

## Abstract:

For $n \geqslant 7$ we describe an $(n - 1)$-sphere $\Sigma$ wildly embedded in the $n$-sphere yet every point of $\Sigma$ has arbitrarily small neighborhoods bounded by flat $(n - 1)$-spheres, each intersecting $\Sigma$ in an $(n - 2)$-sphere. Not only do these examples for large $n$ run counter to what can occur when $n = 3$, they also illustrate the sharpness of high-dimensional taming theorems developed by Cannon and Harrold and Seebeck. Furthermore, despite their wildness, they have mapping cylinder neighborhoods, which both run counter to what is possible when $n = 3$ and also partially illustrate the sharpness of another high-dimensional taming theorem due to Bryant and Lacher.## References

- Edward G. Begle,
*The Vietoris mapping theorem for bicompact spaces*, Ann. of Math. (2)**51**(1950), 534β543. MR**35015**, DOI 10.2307/1969366 - J. L. Bryant and R. C. Lacher,
*Embeddings with mapping cylinder neighborhoods*, Topology**14**(1975), 191β201. MR**394680**, DOI 10.1016/0040-9383(75)90027-0 - C. E. Burgess,
*Characterizations of tame surfaces in $E^{3}$*, Trans. Amer. Math. Soc.**114**(1965), 80β97. MR**176456**, DOI 10.1090/S0002-9947-1965-0176456-2 - J. W. Cannon,
*Characterization of taming sets on $2$-spheres*, Trans. Amer. Math. Soc.**147**(1970), 289β299. MR**257996**, DOI 10.1090/S0002-9947-1970-0257996-6 - J. W. Cannon,
*$\textrm {ULC}$ properties in neighbourhoods of embedded surfaces and curves in $E^{3}$*, Canadian J. Math.**25**(1973), 31β73. MR**314037**, DOI 10.4153/CJM-1973-004-1 - J. W. Cannon,
*Shrinking cell-like decompositions of manifolds. Codimension three*, Ann. of Math. (2)**110**(1979), no.Β 1, 83β112. MR**541330**, DOI 10.2307/1971245 - A. V. ΔernavskiΔ,
*The identity of local flatness and local simple connectedness for imbeddings of $(n-1)$-dimensional into $n$-dimensional manifolds when $n>4$*, Mat. Sb. (N.S.)**91(133)**(1973), 279β286, 288 (Russian). MR**0334222** - Robert J. Daverman,
*Locally nice codimension one manifolds are locally flat*, Bull. Amer. Math. Soc.**79**(1973), 410β413. MR**321095**, DOI 10.1090/S0002-9904-1973-13190-8 - Robert J. Daverman,
*Embedding phenomena based upon decomposition theory: wild Cantor sets satisfying strong homogeneity properties*, Proc. Amer. Math. Soc.**75**(1979), no.Β 1, 177β182. MR**529237**, DOI 10.1090/S0002-9939-1979-0529237-7 - W. T. Eaton,
*A note about locally spherical spheres*, Canadian J. Math.**21**(1969), 1001β1003. MR**244969**, DOI 10.4153/CJM-1969-110-8 - Robert D. Edwards,
*The topology of manifolds and cell-like maps*, Proceedings of the International Congress of Mathematicians (Helsinki, 1978) Acad. Sci. Fennica, Helsinki, 1980, pp.Β 111β127. MR**562601** - O. G. Harrold Jr.,
*Locally peripherally unknotted surfaces in $E^{3}$*, Ann. of Math. (2)**69**(1959), 276β290. MR**105660**, DOI 10.2307/1970182 - O. G. Harrold and C. L. Seebeck,
*Locally weakly flat spaces*, Trans. Amer. Math. Soc.**138**(1969), 407β414. MR**239597**, DOI 10.1090/S0002-9947-1969-0239597-0 - V. L. Klee Jr.,
*Some topological properties of convex sets*, Trans. Amer. Math. Soc.**78**(1955), 30β45. MR**69388**, DOI 10.1090/S0002-9947-1955-0069388-5 - Chris Lacher and Alden Wright,
*Mapping cylinders and $4$-manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp.Β 424β427. MR**0271951** - L. D. Loveland,
*Tame surfaces and tame subsets of spheres in $E^{3}$*, Trans. Amer. Math. Soc.**123**(1966), 355β368. MR**199850**, DOI 10.1090/S0002-9947-1966-0199850-3 - M. H. A. Newman,
*The engulfing theorem for topological manifolds*, Ann. of Math. (2)**84**(1966), 555β571. MR**203708**, DOI 10.2307/1970460 - Victor Nicholson,
*Mapping cylinder neighborhoods*, Trans. Amer. Math. Soc.**143**(1969), 259β268. MR**248788**, DOI 10.1090/S0002-9947-1969-0248788-4 - T. M. Price and C. L. Seebeck III,
*Somewhere locally flat codimension one manifolds with $1-\textrm {ULC}$ complements are locally flat*, Trans. Amer. Math. Soc.**193**(1974), 111β122. MR**346796**, DOI 10.1090/S0002-9947-1974-0346796-8 - Edwin H. Spanier,
*Algebraic topology*, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR**0210112**

## Additional Information

- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**90**(1984), 139-144 - MSC: Primary 57N50; Secondary 54B15, 57M30, 57N15, 57N45
- DOI: https://doi.org/10.1090/S0002-9939-1984-0722432-5
- MathSciNet review: 722432