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