Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Atomic maps and
the Chogoshvili-Pontrjagin claim

Authors: M. Levin and Y. Sternfeld
Journal: Trans. Amer. Math. Soc. 350 (1998), 4623-4632
MSC (1991): Primary 54F45
MathSciNet review: 1433123
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • 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$f(X) \geq k$, 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 $2$-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 $m$-dimensioalen Menge in $R_k$, Ergebnisse Math. Kolloq., 4(1933), 24-25.
  • 15. S. Mardesic, Compact subsets of ${\mathbb R}^n$ 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.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 54F45

Retrieve articles in all journals with MSC (1991): 54F45

Additional Information

M. Levin
Affiliation: Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel

Y. Sternfeld
Affiliation: Department of Mathematics, Haifa University, Mount Carmel, Haifa 31905, Israel

Received by editor(s): January 17, 1996
Received by editor(s) in revised form: December 5, 1996
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society