Shrinking certain sliced decompositions of
Authors:
Robert J. Daverman and D. Kriss Preston
Journal:
Proc. Amer. Math. Soc. 79 (1980), 477483
MSC:
Primary 54B15; Secondary 54B10
MathSciNet review:
567997
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Abstract: We set forth a connection, based on relatively elementary techniques, between the shrinkability of product decompositions of and that of sliced decompositions. In particular, if G is a decomposition of such that each decomposition element g is contained in some horizontal slice and if the decomposition of , consisting of those subsets g of for which , expands to a shrinkable decomposition of , we show then that G itself is shrinkable.
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 [A]
 S. Armentrout, Cellular decompositions of 3manifolds that yield 3manifolds, Mem. Amer. Math. Soc. No. 107 (1971). MR 0413104 (54:1225)
 [B1]
 R. H. Bing, A homeomorphism between the 3sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354362. MR 0049549 (14:192d)
 [B2]
 , Upper semicontinuous decompositions of , Ann. of Math. (2) 65 (1957), 363374. MR 0092960 (19:1187f)
 [C]
 J. W. Cannon, Shrinking celllike 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), 299302. MR 0151944 (27:1925)
 [D1]
 R. J. Daverman, Every crumpled ncube is a closed ncellcomplement, Michigan Math. J. 24 (1977), 225241. MR 0488066 (58:7637)
 [D2]
 , Detecting the disjoint discs property 254 (1979), 217236.
 [D3]
 , Applications of local contractibility of manifold homeomorphism groups (to appear).
 [DP]
 R. J. Daverman and D. K. Preston, Celllike 1dimensional decompositions of are 4manifold factors (to appear).
 [DR]
 R. J. Daverman and W. H. Row, Celllike 0dimensional decompositions of are 4manifold factors, Trans. Amer. Math. Soc. 254 (1979), 217236. MR 539916 (82g:54011)
 [DH]
 E. Dyer and M. E. Hamstrom, Completely regular mappings, Fund. Math. 45 (1958), 103118. MR 0092959 (19:1187e)
 [Ed]
 R. D. Edwards, Approximating certain celllike maps by homeomorphisms, manuscript. See Notices Amer. Math. Soc. 24 (1977), p. A649, Abstract 751G5.
 [EK]
 R. D. Edwards and R. C. Kirby, Deformations of spaces of embeddings, Ann. of Math. (2) 93 (1971), 6388. MR 0283802 (44:1032)
 [Ev]
 D. L. Everett, Embedding theorems for decomposition spaces, Houston J. Math. 3 (1977), 351368. 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), 501507. MR 0388319 (52:9156)
 [M]
 R. L. Moore, Concerning upper semicontinuous collections of continua, Trans. Amer. Math. Soc. 27 (1925), 416428. MR 1501320
 [S]
 L. C. Siebenmann, Approximating cellular maps by homeomorphisms, Topology 11 (1972), 271294. MR 0295365 (45:4431)
 [W]
 E. P. Woodruff, Decomposition spaces having arbitrarily small neighborhoods with 2sphere boundaries, Trans. Amer. Math. Soc. 232 (1977), 195204. MR 0442944 (56:1319)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005679978
PII:
S 00029939(1980)05679978
Keywords:
Upper semicontinuous decomposition,
celllike,
shrinkable decomposition,
shrinkability criterion,
sliced decomposition,
embedding dimension
Article copyright:
© Copyright 1980
American Mathematical Society
