Shrinking certain sliced decompositions of

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 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.

**[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*, 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**are 4-manifold factors*(to appear).**[DR]**R. J. Daverman and W. H. Row,*Cell-like 0-dimensional decompositions of**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)**

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