Singular regular neighborhoods and local flatness in codimension one
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- by Robert J. Daverman PDF
- Proc. Amer. Math. Soc. 57 (1976), 357-362 Request permission
Abstract:
For an $(n - 1)$-manifold $S$ topologically embedded as a closed subset of an $n$-manifold $N$, we define what it means for $S$ to have a singular regular neighborhood in $N$. The principal result demonstrates that $S$ has a singular regular neighborhood in $N$ if and only if the homotopy theoretic condition holds that $N - S$ is locally simply connected ($1$-LC) at each point of $S$. Consequently, $S$ has a singular regular neighborhood in $N$ if and only if $S$ is locally flatly embedded $(n \ne 4)$.References
- R. H. Bing, A surface is tame if its complement is $1$-ULC, Trans. Amer. Math. Soc. 101 (1961), 294β305. MR 131265, DOI 10.1090/S0002-9947-1961-0131265-1
- 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 A. V. ΔernavskiΔ, The equivalence of local flatness and local $1$-connectedness for imbeddings of $(n - 1)$-dimensional manifolds in $n$-dimensional manifolds, Mat. Sb. 91 (133) (1973), 276-286 = Math. USSR Sbornik 20 (1973), 297-304.
- 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
- Samuel Eilenberg and Norman Steenrod, Foundations of algebraic topology, Princeton University Press, Princeton, N.J., 1952. MR 0050886
- D. B. A. Epstein, The degree of a map, Proc. London Math. Soc. (3) 16 (1966), 369β383. MR 192475, DOI 10.1112/plms/s3-16.1.369
- John Hempel, A surface in $S^{3}$ is tame if it can be deformed into each complementary domain, Trans. Amer. Math. Soc. 111 (1964), 273β287. MR 160195, DOI 10.1090/S0002-9947-1964-0160195-7 N. Hosay, The sum of a real cube and a crumpled cube is ${S^3}$, Notices Amer. Math. Soc. 10 (1963), 666; Errata, ibid. 11 (1964), 152. Abstract #607-17.
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- Lloyd L. Lininger, Some results on crumpled cubes, Trans. Amer. Math. Soc. 118 (1965), 534β549. MR 178460, DOI 10.1090/S0002-9947-1965-0178460-7
- M. H. A. Newman, Local connection in locally compact spaces, Proc. Amer. Math. Soc. 1 (1950), 44β53. MR 33530, DOI 10.1090/S0002-9939-1950-0033530-3
- Paul Olum, Mappings of manifolds and the notion of degree, Ann. of Math. (2) 58 (1953), 458β480. MR 58212, DOI 10.2307/1969748
- 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
- Warren White, Some tameness conditions involving singular disks, Trans. Amer. Math. Soc. 143 (1969), 223β234. MR 248790, DOI 10.1090/S0002-9947-1969-0248790-2
- 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
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 57 (1976), 357-362
- MSC: Primary 57A45
- DOI: https://doi.org/10.1090/S0002-9939-1976-0420630-7
- MathSciNet review: 0420630