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)



On classes of maps which preserve finitisticness

Authors: Akira Koyama and Manuel A. Moron
Journal: Proc. Amer. Math. Soc. 130 (2002), 3091-3096
MSC (2000): Primary 55M10, 54F45
Published electronically: March 13, 2002
MathSciNet review: 1908934
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We shall prove the following: $(1)$ Let $r:X \to Y$ be a refinable map between paracompact spaces. Then $X$ is finitistic if and only if $Y$ is finitistic. $(2)$ Let $f:X \to Y$ be a hereditary shape equivalence between metric spaces. Then if $X$ is finitistic, $Y$ is finitistic.

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

  • [A] F. Ancel, Proper hereditary shape equivalence property C, Topology and its Appl., 19(1985), 71-74. MR 86g:54026
  • [B] G. E. Bredon, Introduction to compact transformation groups, Academic Press, 1972, New York. MR 54:1265
  • [C-V] A. Chigogidze and V. Valov, Extension dimension and refinable maps, preprint(1999).
  • [De-P] S. Deo and A. R. Pears, A completely finitistic space is finite-dimensional, Bull. London Math. Soc., 17(1985), 49-51. MR 85k:54041
  • [De-Si$_1$] S. Deo and T. B. Singh, On the converse of some theorems about orbit spaces, J. London Math. Soc., 25(1982), 162-170. MR 83k:54039
  • [De-Si$_2$] -, On certain constructions in finitistic spaces, Internat. J. Math. Math. Sci., 6(1983), 477-482. MR 85c:54058
  • [De-Si-S] S. Deo, T. B. Singh, and R. A. Shukra, On an extension of localization theorem and generalized Conner conjecture, Trans. Amer. Math. Soc., 269(1982), 395-402. MR 83a:57051
  • [De-T] S. Deo and H. S. Tripathi, Compact Lie group actions on finitistic spaces, Topology, 21(1982), 393-399. MR 83k:54042
  • [Di] J. J. Dijkstra, A dimension raising hereditary shape equivalence, Fund. Math., 149(1996), 265-274. MR 97f:54040
  • [Di-Mo] J. J. Dijkstra and J. Mogilski, Countable dimensionality and dimension raising cell-like maps, 80(1997), 73-79. MR 98i:54017
  • [Dr] A. N. Dranishnikov, Cohomological dimension is not preserved under Stone-Cech compactification, Comptes Rendus Bulgarian Acad. of Sci., 41(1988), 9-10. MR 90e:55002
  • [Dy-M-S] J. Dydak, S. N. Mishra and R. A. Shukla, On finitistic spaces, Topology and its Appl., 97(1999), 217-229. MR 2000i:55003
  • [Dy-W] J. Dydak and J. Walsh, Spaces without cohomological dimension preserving compactifications, Proc. Amer. Math. Soc., 113(1991), 1155-1162. MR 92c:54039
  • [F-R] Jo Ford and J. Rogers, Jr., Refinable maps, Colloq. Math., 39(1978), 263-269. MR 80d:54009
  • [G-Ro] D. Garity and D. Rohm, Property C, refinable maps and dimension raising maps, Proc. Amer. Math. Soc., 98(1986), 336-340. MR 87i:54077
  • [H1] Y. Hattori, A note on finitistic spaces, Questions and Answers Gen. Topology, 3(1985), 47-55. MR 86m:54025
  • [H2] -, Finitistic spaces and dimension, Houston J. Math., 25(1999), 687-696.
  • [Ka] H. Kato, A note on infinite dimension under refinable maps, Proc. Amer. Math. Soc., 88(1983), 177-180. MR 84c:54064
  • [Ka-Ko] H. Kato and A. Koyama, A role of refinable maps - a survey, Topology Proc., 11(1986), 317-348. MR 89f:54030
  • [Ko1] A. Koyama, Refinable maps in dimension theory, Topology and its Appl., 17(1984), 247-255. MR 86d:54056
  • [Ko2] -, Refinable maps in dimension theory II, Bull. Pol. Math. Soc., 42(1994), 255-261. CMP 2001:07
  • [Ko-Sh] A. Koyama and R. Sher, Approximable dimension and acyclic dimension, Fund. Math., 152(1997), 43-53.
  • [Ku] V. I. Kuz'minov, Homological dimension theory, Russian Math. Surveys, 23(1968), 1-45. MR 39:2158
  • [L] M. Levin, Some examples in cohomological dimension theory, preprint(1999).
  • [Mi] R. Millspaugh, Proper hereditary shape equivalences preserve small weak infinite dimensionality, Proc. Amer. Math. Soc., 90(1984), 1055-1061. MR 91d:54041
  • [Ru-Sc$_1$] L. Rubin and P. Schapiro, Cell-like maps onto non-compact spaces of finite cohomological dimension, Topology and its Appl., 27(1987), 221-244. MR 89b:55002
  • [Ru-Sc$_2$] -, Compactifications which preserve cohomological dimension, Glasnik Mat., 28(48)(1993), 155-165. MR 95g:54029
  • [Sw] R. G. Swan, A new method in fixed point theory, Comment. Math. Helv., 34(1960), 1-16. MR 22:5978

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 55M10, 54F45

Retrieve articles in all journals with MSC (2000): 55M10, 54F45

Additional Information

Akira Koyama
Affiliation: Division of Mathematical Sciences, Osaka Kyoiku University, Kashiwara, Osaka 582-8582, Japan

Manuel A. Moron
Affiliation: Unidad Dovente de Matematicas, E. T. S. I. Montes, Universidad Polittecnica, 28040, Madrid, Spain

Keywords: Finitistic spaces, refinable maps, c-refinable maps, hereditary shape equivalences, extension dimension, cohomological dimension
Received by editor(s): December 12, 2000
Received by editor(s) in revised form: April 24, 2001
Published electronically: March 13, 2002
Additional Notes: This work was started when the first author visited Departmento de Geometria y Topologia, Facultad de Ciencias Matematicas, Universidad Complutense de Madrid.
Communicated by: Alan Dow
Article copyright: © Copyright 2002 American Mathematical Society