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Atomic maps and the Chogoshvili-Pontrjagin claim
Author(s):
M.
Levin;
Y.
Sternfeld
Journal:
Trans. Amer. Math. Soc.
350
(1998),
4623-4632.
MSC (1991):
Primary 54F45
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Abstract:
It is proved that all spaces of dimension three or more disobey the Chogoshvili-Pontrjagin claim. This is of particular interest in view of the recent proof (in Certain 2-stable embeddings, by Dobrowolski, Levin, and Rubin, Topology Appl. 80 (1997), 81-90) that two-dimensional ANRs obey the claim. The construction utilizes the properties of atomic maps which are maps whose fibers ( point inverses) are atoms ( hereditarily indecomposable continua). A construction of M. Brown is applied to prove that every finite dimensional compact space admits an atomic map with a one-dimensional range.
References:
- 1.
- Y. Sternfeld, Stability an dimension-a counter example to a conjecture of Chogoshvili, Transactions A.M.S, 340(1993), 243-251. MR 94a:54087
- 2.
- M. Levin and Y. Sternfeld, Mappings which are stable with respect to the property dim
 , Top. and Appl., 52(1993), 241-265. MR 95c:54059 - 3.
- M. Levin and Y. Sternfeld, Monotone basic embeddings of hereditarily indecomposable continua, Top. and Appl. 68 (1996), no. 3, 241-249. MR 96m:54061
- 4.
- R.H. Bing, Higher dimensional hereditarily indecomposable continua, Transactions AMS, 71(1951), 267-273. MR 13:265c
- 5.
- A.N. Dranisknikov, On Chogoshvili's conjecture, preprint.
- 6.
- M. Brown, Continuous collections of higher dimensional hereditarily indecomposable continua, Thesis, University of Wisconsin, 1958.
- 7.
- K. Kuratowski, Topology II, Academic Press and PWN, 1968. MR 41:4467
- 8.
- W. Lewis, The Pseudo-Arc, Marcel-Dekker, in preparation.
- 9.
- T. Dobrowolski, M. Levin and L.R. Rubin, Certain
 -stable embeddings, Top. and Appl. 80 (1997), 81-90. CMP 98:01 - 10.
- G. Chogoshvili, On a theorem in the theory of dimensionality, Compositio Math., 5(1938), 292-298.
- 11.
- P. Alexandroff, Zum allgeminen Dimensions problem, Gott. nachrichten, 37(1928).
- 12.
- F.D. Ancel and T. Dobrowolski, A variant of Sternfeld's counter example to a conjecture of Chogoshvili-Pontrjagin, preprint.
- 13.
- K. Sitnikov, An example of a 2-dimensional set in 3-dimensional Euclidean space allowing arbitrarily small deformation into a 1-dimensional polyhedron and a certain new characterization of dimension of sets in Euclidean spaces, Dokl. Akad. Nauk SSSR, 88(1953), 21-24.
- 14.
- G. Nobeling, Die Projektioner einer kompakten
 -dimensioalen Menge in  , Ergebnisse Math. Kolloq., 4(1933), 24-25. - 15.
- S. Mardesic, Compact subsets of
 and dimension of their projections, Proceedings A.M.S., 41(1973), 631-633. MR 48:12480 - 16.
- D. Kiguradze, Some properties of metric dimension (Russian), Soobsch. Akad. Nauk. Gruz. SSR, 132(3)(1988), 485-488. MR 90k:54047
- 17.
- Y. Sternfeld, Uniformly separating families of functions, Israel J. Math., 29(1978), 61-91. MR 58:7570
- 18.
- Y. Sternfeld, Hilbert's 13th problem and dimension, in Geometrical aspects of functional analysis, (J. Lindenstrauss and V. Milman, editors) Notes in Math. 1376, Springer-Verlag, Berlin, 1987-8, 1-48. MR 90j:54035
- 19.
- A. Dranishnikov, D. Repovs and E. Schepin, On intervention of compacta in Euclidean spaces: The metastable case., Tsukuba J. Math., 17(2)(1993), 549-564. MR 95a:55002
- 20.
- R. Pol, A two-dimensional compactum in the product of two 1-dimensional compacta which does not contain any rectangle, Topology Proceedings, 16(1991), 133-135. MR 94b:54101
- 21.
- Y. Sternfeld, On finite dimensional maps and other maps with ``small'' fibers, Fundamenta Math., 147(1995), 127-133. MR 96e:54030
- 22.
- M. Levin, A short construction of hereditarily infinite dimensional compacta, Top. and Appl., 65(1995), 97-99. MR 97b:54044
- 23.
- M Levin and Y. Sternfeld, The space of subcontinua of a 2-dimensional continuum is infinite dimensional, Proceedings A.M.S., 125 (1997), 2771-275. MR 97j:54012
- 24.
- M Levin and Y. Sternfeld, Hyperspaces of two-dimensional continua, Fundamenta Math., 150 (1996), no. 1, 17-24. MR 97d:54015
- 25.
- M. Levin, Bing maps and finite-dimensional maps, Fundamenta Math., 151 (1996), no. 1, 47-52. MR 97e:54031
- 26.
- B. Knaster, Un continu dont tout sous-continu est indecomposable, Fundamenta Math., 3(1922), 247-286.
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Additional Information:
M.
Levin
Affiliation:
Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel
Email:
levin@mathcs2.haifa.ac.il
Y.
Sternfeld
Affiliation:
Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel
Email:
yaki@mathcs2.haifa.ac.il
DOI:
10.1090/S0002-9947-98-01995-3
PII:
S 0002-9947(98)01995-3
Received by editor(s):
January 17, 1996
Received by editor(s) in revised form:
December 5, 1996
Copyright of article:
Copyright
1998,
American Mathematical Society
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