Cell-like closed-$0$-dimensional decompositions of $R^{3}$ are $R^{4}$ factors
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- by Robert D. Edwards and Richard T. Miller
- Trans. Amer. Math. Soc. 215 (1976), 191-203
- DOI: https://doi.org/10.1090/S0002-9947-1976-0383411-3
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Abstract:
It is proved that the product of a cell-like closed-0-dimensional upper semicontinuous decomposition of ${R^3}$ with a line is ${R^4}$. This establishes at once this feature for all the various dogbone-inspired decompositions of ${R^3}$. The proof makes use of an observation of L. Rubin that the universal cover of a wedge of circles admits a 1-1 immersion into the wedge crossed with ${R^1}$.References
- W. R. Alford and R. B. Sher, Defining sequences for compact $0$-dimensional decompositions of $E^{n}$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 17 (1969), 209–212 (English, with Russian summary). MR 254824
- J. J. Andrews and M. L. Curtis, $n$-space modulo an arc, Ann. of Math. (2) 75 (1962), 1–7. MR 139153, DOI 10.2307/1970414
- Steve Armentrout, Monotone decompositions of $E^{3}$, Topology Seminar (Wisconsin, 1965) Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N.J., 1966, pp. 1–25. MR 0222865
- Steve Armentrout, Local properties of decomposition spaces, Proc. First Conf. on Monotone Mappings and Open Mappings (SUNY at Binghamton, Binghamton, N.Y., 1970) State Univ. of New York at Binghamton, Binghamton, N.Y., 1971, pp. 98–111. MR 0276942 —, A survey of results on decompositions, in: D. Day et al., eds., Proc. University of Oklahoma Topology Conference, University of Oklahoma, 1972, pp. 1-12.
- R. H. Bing, The cartesian product of a certain nonmanifold and a line is $E^{4}$, Ann. of Math. (2) 70 (1959), 399–412. MR 107228, DOI 10.2307/1970322
- Robert D. Edwards and Leslie C. Glaser, A method for shrinking decompositions of certain manifolds, Trans. Amer. Math. Soc. 165 (1972), 45–56. MR 295357, DOI 10.1090/S0002-9947-1972-0295357-6
- Leslie C. Glaser, On double suspensions of arbitrary nonsimply connected homology $n$-spheres, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, Ga., 1969) Markham, Chicago, Ill., 1970, pp. 5–17. MR 0328953
- R. C. Lacher, Cell-like mappings. I, Pacific J. Math. 30 (1969), 717–731. MR 251714, DOI 10.2140/pjm.1969.30.717
- H. W. Lambert and R. B. Sher, Point-like $0$-dimensional decompositions of $S^{3}$, Pacific J. Math. 24 (1968), 511–518. MR 225308, DOI 10.2140/pjm.1968.24.511
- D. R. McMillan Jr., A criterion for cellularity in a manifold. II, Trans. Amer. Math. Soc. 126 (1967), 217–224. MR 208583, DOI 10.1090/S0002-9947-1967-0208583-7
- D. R. McMillan Jr. and Harry Row, Tangled embeddings of one-dimensional continua, Proc. Amer. Math. Soc. 22 (1969), 378–385. MR 246267, DOI 10.1090/S0002-9939-1969-0246267-7
- Leonard R. Rubin, A general class of factors of $E^{4}$, Trans. Amer. Math. Soc. 166 (1972), 215–224. MR 295314, DOI 10.1090/S0002-9947-1972-0295314-X
- R. B. Sher and W. R. Alford, A note on $0$-dimensional decompositions of $E^{3}$, Amer. Math. Monthly 75 (1968), 377–378. MR 226615, DOI 10.2307/2313418
- A. Marin and Y. M. Visetti, A general proof of Bing’s shrinkability criterion, Proc. Amer. Math. Soc. 53 (1975), no. 2, 501–507. MR 388319, DOI 10.1090/S0002-9939-1975-0388319-X
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 215 (1976), 191-203
- MSC: Primary 57A10
- DOI: https://doi.org/10.1090/S0002-9947-1976-0383411-3
- MathSciNet review: 0383411