Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Shrinking certain sliced decompositions of $ E\sp{n+1}$


Authors: Robert J. Daverman and D. Kriss Preston
Journal: Proc. Amer. Math. Soc. 79 (1980), 477-483
MSC: Primary 54B15; Secondary 54B10
DOI: https://doi.org/10.1090/S0002-9939-1980-0567997-8
MathSciNet review: 567997
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We set forth a connection, based on relatively elementary techniques, between the shrinkability of product decompositions of $ {E^{n + 1}}$ and that of sliced decompositions. In particular, if G is a decomposition of $ {E^{n + 1}}$ such that each decomposition element g is contained in some horizontal slice $ {E^n} \times \{ s\} $ and if the decomposition $ {G^s}$ of $ {E^n}$, consisting of those subsets g of $ {E^n}$ for which $ g \times \{ s\} \in G$ , expands to a shrinkable decomposition $ {G^s} \times {E^1}$ of $ {E^n} \times {E^1}$, we show then that G itself is shrinkable.


References [Enhancements On Off] (What's this?)

  • [A] S. Armentrout, Cellular decompositions of 3-manifolds that yield 3-manifolds, Mem. Amer. Math. Soc. No. 107 (1971). MR 0413104 (54:1225)
  • [B1] R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354-362. MR 0049549 (14:192d)
  • [B2] -, Upper semicontinuous decompositions of $ {E^3}$, Ann. of Math. (2) 65 (1957), 363-374. MR 0092960 (19:1187f)
  • [C] J. W. Cannon, Shrinking cell-like decompositions of manifolds. Codimension three, Ann. of Math. (to appear). MR 541330 (80j:57013)
  • [CD] J. W. Cannon and R. J. Daverman, A totally wild flow (to appear). MR 611226 (82m:57006)
  • [CM] M. L. Curtis and D. R. McMillan, Cellularity of sets in products, Michigan Math. J. 9 (1962), 299-302. MR 0151944 (27:1925)
  • [D1] R. J. Daverman, Every crumpled n-cube is a closed n-cell-complement, Michigan Math. J. 24 (1977), 225-241. MR 0488066 (58:7637)
  • [D2] -, Detecting the disjoint discs property 254 (1979), 217-236.
  • [D3] -, Applications of local contractibility of manifold homeomorphism groups (to appear).
  • [DP] R. J. Daverman and D. K. Preston, Cell-like 1-dimensional decompositions of $ {S^3}$ are 4-manifold factors (to appear).
  • [DR] R. J. Daverman and W. H. Row, Cell-like 0-dimensional decompositions of $ {S^3}$ are 4-manifold factors, Trans. Amer. Math. Soc. 254 (1979), 217-236. MR 539916 (82g:54011)
  • [DH] E. Dyer and M. E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103-118. MR 0092959 (19:1187e)
  • [Ed] R. D. Edwards, Approximating certain cell-like maps by homeomorphisms, manuscript. See Notices Amer. Math. Soc. 24 (1977), p. A-649, Abstract 751-G5.
  • [EK] R. D. Edwards and R. C. Kirby, Deformations of spaces of embeddings, Ann. of Math. (2) 93 (1971), 63-88. MR 0283802 (44:1032)
  • [Ev] D. L. Everett, Embedding theorems for decomposition spaces, Houston J. Math. 3 (1977), 351-368. MR 0464241 (57:4175)
  • [HW] W. Hurewicz and H. Wallman, Dimension theory, Princeton Univ. Press, Princeton, N. J., 1948. MR 0006493 (3:312b)
  • [K] G. Kozlowski, Images of ANR's, Trans. Amer. Math. Soc. (to appear).
  • [MV] A. Marin and Y. M. Visetti, A general proof of Bing's Shrinkability Criterion, Proc. Amer. Math. Soc. 53 (1975), 501-507. MR 0388319 (52:9156)
  • [M] R. L. Moore, Concerning upper semi-continuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416-428. MR 1501320
  • [S] L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271-294. MR 0295365 (45:4431)
  • [W] E. P. Woodruff, Decomposition spaces having arbitrarily small neighborhoods with 2-sphere boundaries, Trans. Amer. Math. Soc. 232 (1977), 195-204. MR 0442944 (56:1319)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54B15, 54B10

Retrieve articles in all journals with MSC: 54B15, 54B10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0567997-8
Keywords: Upper semicontinuous decomposition, cell-like, shrinkable decomposition, shrinkability criterion, sliced decomposition, embedding dimension
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society