On classes of maps which preserve finitisticness

Koyama, Akira; Moron, Manuel

doi:10.1090/S0002-9939-02-06402-X

On classes of maps which preserve finitisticness
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by Akira Koyama and Manuel A. Moron PDF

Proc. Amer. Math. Soc. 130 (2002), 3091-3096 Request permission

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

Fredric D. Ancel, Proper hereditary shape equivalences preserve property $\textrm {C}$, Topology Appl. 19 (1985), no. 1, 71–74. MR 786082, DOI 10.1016/0166-8641(85)90086-0
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
A. Chigogidze and V. Valov, Extension dimension and refinable maps, preprint(1999).
Satya Deo and Alan R. Pears, A completely finitistic space is finite-dimensional, Bull. London Math. Soc. 17 (1985), no. 1, 49–51. MR 766447, DOI 10.1112/blms/17.1.49
Satya Deo and Tej Bahadur Singh, On the converse of some theorems about orbit spaces, J. London Math. Soc. (2) 25 (1982), no. 1, 162–170. MR 645873, DOI 10.1112/jlms/s2-25.1.162
Satya Deo and Mohan Singh, On certain constructions in finitistic spaces, Internat. J. Math. Math. Sci. 6 (1983), no. 3, 477–482. MR 712566, DOI 10.1155/S0161171283000423
Satya Deo, Tej Bahadur Singh, and Ram Anugrah Shukla, On an extension of localization theorem and generalized Conner conjecture, Trans. Amer. Math. Soc. 269 (1982), no. 2, 395–402. MR 637697, DOI 10.1090/S0002-9947-1982-0637697-0
Satya Deo and Hari Shankar Tripathi, Compact Lie group actions on finitistic spaces, Topology 21 (1982), no. 4, 393–399. MR 670743, DOI 10.1016/0040-9383(82)90019-2
Jan J. Dijkstra, A dimension raising hereditary shape equivalence, Fund. Math. 149 (1996), no. 3, 265–274. MR 1383210, DOI 10.4064/fm-149-3-265-274
Jan J. Dijkstra and Jerzy Mogilski, Countable dimensionality and dimension raising cell-like maps, Topology Appl. 80 (1997), no. 1-2, 73–79. MR 1469468, DOI 10.1016/S0166-8641(97)00006-0
A. N. Dranishnikov, The cohomological dimension is not preserved under the Stone-Čech compactification, C. R. Acad. Bulgare Sci. 41 (1988), no. 12, 9–10 (Russian). MR 985881
J. Dydak, S. N. Mishra, and R. A. Shukla, On finitistic spaces, Topology Appl. 97 (1999), no. 3, 217–229. MR 1711355, DOI 10.1016/S0166-8641(99)00064-4
Jerzy Dydak and John J. Walsh, Spaces without cohomological dimension preserving compactifications, Proc. Amer. Math. Soc. 113 (1991), no. 4, 1155–1162. MR 1081695, DOI 10.1090/S0002-9939-1991-1081695-3
Jo Ford and J. W. Rogers Jr., Refinable maps, Colloq. Math. 39 (1978), no. 2, 263–269. MR 522365, DOI 10.4064/cm-39-2-263-269
Dennis J. Garity and Dale M. Rohm, Property C, refinable maps and dimension raising maps, Proc. Amer. Math. Soc. 98 (1986), no. 2, 336–340. MR 854043, DOI 10.1090/S0002-9939-1986-0854043-7
Yasunao Hattori, A note on finitistic spaces, Questions Answers Gen. Topology 3 (1985), no. 1, 47–55. MR 798585
—, Finitistic spaces and dimension, Houston J. Math., 25(1999), 687–696.
Hisao Kato, A note on infinite-dimension under refinable maps, Proc. Amer. Math. Soc. 88 (1983), no. 1, 177–180. MR 691304, DOI 10.1090/S0002-9939-1983-0691304-6
Hisao Kato and Akira Koyama, The role of refinable maps—a survey, Proceedings of the 1986 Topology Conference (Lafayette, LA, 1986), 1986, pp. 317–348. MR 945507
Akira Koyama, Refinable maps in dimension theory, Topology Appl. 17 (1984), no. 3, 247–255. MR 752274, DOI 10.1016/0166-8641(84)90045-2
—, Refinable maps in dimension theory II, Bull. Pol. Math. Soc., 42(1994), 255–261.
A. Koyama and R. Sher, Approximable dimension and acyclic dimension, Fund. Math., 152(1997), 43–53.
V. I. Kuz′minov, Homological dimension theory, Uspehi Mat. Nauk 23 (1968), no. 5 (143), 3–49 (Russian). MR 0240813
M. Levin, Some examples in cohomological dimension theory, preprint(1999).
Richard P. Millspaugh, Proper hereditary shape equivalences preserve small weak infinite-dimensionality, Proc. Amer. Math. Soc. 110 (1990), no. 4, 1055–1061. MR 1037215, DOI 10.1090/S0002-9939-1990-1037215-1
Leonard R. Rubin and Philip J. Schapiro, Cell-like maps onto noncompact spaces of finite cohomological dimension, Topology Appl. 27 (1987), no. 3, 221–244. MR 918533, DOI 10.1016/0166-8641(87)90088-5
Leonard R. Rubin and Philip J. Schapiro, Compactifications which preserve cohomological dimension, Glas. Mat. Ser. III 28(48) (1993), no. 1, 155–165 (English, with English and Serbo-Croatian summaries). MR 1283905
Richard G. Swan, A new method in fixed point theory, Comment. Math. Helv. 34 (1960), 1–16. MR 115176, DOI 10.1007/BF02565923