A loop condition for embedded compacts
HTML articles powered by AMS MathViewer
- by D. Coram, R. J. Daverman and P. F. Duvall PDF
- Proc. Amer. Math. Soc. 53 (1975), 205-212 Request permission
Abstract:
In this paper we consider a version $({\text {SLC}})$ of the cellularity criterion which applies to embeddings of arbitrary compacta. It is shown that if the continuum $X \subset {E^n},\;\dim X \leqslant n - 3$, has the shape of a finite $k$-complex $K,\;2k + 1 \leqslant n$, and satisfies the ${\text {SLC}}$, then $X$ has arbitrarily small neighborhoods which are regular neighborhoods of a copy of $K$.References
- Steve Armentrout, $\textrm {UV}$ properties of compact sets, Trans. Amer. Math. Soc. 143 (1969), 487β498. MR 273573, DOI 10.1090/S0002-9947-1969-0273573-7
- Karol Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223β254. MR 229237, DOI 10.4064/fm-62-3-223-254
- Morton Brown, Wild cells and spheres in higher dimensions, Michigan Math. J. 14 (1967), 219β224. MR 221481
- J. L. Bryant, Approximating embeddings of polyhedra in codimension three, Trans. Amer. Math. Soc. 170 (1972), 85β95. MR 307245, DOI 10.1090/S0002-9947-1972-0307245-7 D. Coram, Semicellularity of compact subsets of manifolds, Ph. D. Thesis, University of Wisconsin, Madison, Wis., 1970.
- Donald Coram, Semicellularity, decompositions and mappings in manifolds, Trans. Amer. Math. Soc. 191 (1974), 227β244. MR 356068, DOI 10.1090/S0002-9947-1974-0356068-3
- M. L. Curtis and D. R. McMillan, Cellularity of sets in products, Michigan Math. J. 9 (1962), 299β302. MR 151944, DOI 10.1307/mmj/1028998767
- Robert J. Daverman, On weakly flat $1$-spheres, Proc. Amer. Math. Soc. 38 (1973), 207β210. MR 310894, DOI 10.1090/S0002-9939-1973-0310894-X
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606 P. F. Duvall, Jr., Neighborhoods of $1$-connected ${\text {ANR}}$βs in high dimensional piecewise linear manifolds, Ph. D. Thesis, University of Georgia, Athens, Ga., 1967.
- Paul F. Duvall Jr., Weakly flat spheres, Michigan Math. J. 16 (1969), 117β124. MR 246300
- Samuel Eilenberg and R. L. Wilder, Uniform local connectedness and contractibility, Amer. J. Math. 64 (1942), 613β622. MR 7100, DOI 10.2307/2371708
- J. P. Hempel and D. R. McMillan Jr., Locally nice embeddings of manifolds, Amer. J. Math. 88 (1966), 1β19. MR 205257, DOI 10.2307/2373043
- Herman Gluck, Embeddings in the trivial range, Ann. of Math. (2) 81 (1965), 195β210. MR 173243, DOI 10.2307/1970614
- Witold Hurewicz and Henry Wallman, Dimension Theory, Princeton Mathematical Series, vol. 4, Princeton University Press, Princeton, N. J., 1941. MR 0006493
- Sibe MardeΕ‘iΔ and Jack Segal, Shapes of compacta and ANR-systems, Fund. Math. 72 (1971), no.Β 1, 41β59. MR 298634, DOI 10.4064/fm-72-1-41-59
- D. R. McMillan Jr., A criterion for cellularity in a manifold, Ann. of Math. (2) 79 (1964), 327β337. MR 161320, DOI 10.2307/1970548
- D. R. McMillan Jr., Piercing a disk along a cellular set, Proc. Amer. Math. Soc. 19 (1968), 153β157. MR 220266, DOI 10.1090/S0002-9939-1968-0220266-2
- D. R. McMillan and E. C. Zeeman, On contractible open manifolds, Proc. Cambridge Philos. Soc. 58 (1962), 221β224. MR 139154, DOI 10.1017/s0305004100036434
- R. B. Sher, Defining subsets of manifolds by cells, Bull. Acad. Polon. Sci. SΓ©r. Sci. Math. Astronom. Phys. 17 (1969), 363β365 (English, with Russian summary). MR 256403
- R. B. Sher, Determining the cellularity of a $i$-complex by properties of its arcs, Proc. Amer. Math. Soc. 26 (1970), 491β498. MR 270353, DOI 10.1090/S0002-9939-1970-0270353-7
- Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112
- John Stallings, The piecewise-linear structure of Euclidean space, Proc. Cambridge Philos. Soc. 58 (1962), 481β488. MR 149457, DOI 10.1017/S0305004100036756
- John Stallings, On topologically unknotted spheres, Ann. of Math. (2) 77 (1963), 490β503. MR 149458, DOI 10.2307/1970127
- D. G. Stewart, On unions of cellular sets, Topology Conference (Arizona State Univ., Tempe, Ariz., 1967) Arizona State Univ., Tempe, Ariz., 1968, pp.Β 262β264. MR 0238290 E. C. Zeeman, Notes on combinatorial topology, Inst. Hautes Γtudes Sci., Paris, 1964-1966 (mimeographed notes).
Additional Information
- © Copyright 1975 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 53 (1975), 205-212
- MSC: Primary 57A60; Secondary 57A40
- DOI: https://doi.org/10.1090/S0002-9939-1975-0377902-3
- MathSciNet review: 0377902