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Multifibrations. A class of shape fibrations with the path lifting property. (English) Zbl 1079.55503

Summary: In this paper we introduce a class of maps possessing a multivalued homotopy lifting property with respect to every topological space. We call these maps multifibrations and they represent a formally stronger concept than that of shape fibration. Multifibrations have the interesting property of being characterized in a completely intrinsic way by a path lifting property involving only the total and the base space of the fibration. We also show that multifibrations (and also, with some restrictions, shape fibrations) have a lifting property for homotopies of fine multivalued maps. This implies, when the spaces considered are metric compacta, that the possibility of lifting a fine multivalued map is a property of the corresponding strong shape morphism and not of the particular map considered.

MSC:

55P55 Shape theory
55R05 Fiber spaces in algebraic topology
54C56 Shape theory in general topology
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References:

[1] K. Borsuk: On movable compacta. Fund. Math. 66 (1969), 137-146. · Zbl 0189.53802
[2] K. Borsuk: Theory of Shape (Monografie Matematyczne 59). Polish Scientific Publishers, Warszawa, 1975.
[3] F. W. Cathey: Shape fibrations and strong shape theory. Topology Appl. 14 (1982), 13-30. · Zbl 0505.55020 · doi:10.1016/0166-8641(82)90044-X
[4] Z. Čerin: Shape theory intrinsically. Publ. Mat. 37 (1993), 317-334. · Zbl 0808.54014 · doi:10.5565/PUBLMAT_37293_06
[5] Z. Čerin: Proximate topology and shape theory. Proc. Roy. Soc. Edinburgh 125 (1995), 595-615. · Zbl 0837.55006 · doi:10.1017/S0308210500032704
[6] Z. Čerin: Approximate fibrations. To appear. · Zbl 0872.55015
[7] D. Coram and P. F. Duvall, Jr.: Approximate fibrations. Rocky Mountain J. Math. 7 (1977), 275-288. · Zbl 0367.55019 · doi:10.1216/RMJ-1977-7-2-275
[8] J. M. Cordier and T. Porter: Shape Theory. Categorical Methods of Approximation (Ellis Horwood Series: Mathematics and its Applications). Ellis Horwood Ltd., Chichester, 1989.
[9] J. Dydak and J. Segal: Shape Theory: An Introduction (Lecture Notes in Math. 688). Springer-Verlag, Berlin, 1978. · Zbl 0401.54028
[10] J. Dydak and J. Segal: A list of open problems in shape theory. J. Van Mill and G. M. Reed: Open problems in Topology, North Holland, Amsterdam, 1990, pp. 457-467.
[11] J. E. Felt: \(\epsilon \)-continuity and shape. Proc. Amer. Math. Soc. 46 (1974), 426-430. · Zbl 0292.55013
[12] A. Giraldo: Shape fibrations, multivalued maps and shape groups. Canad. J. Math 50 (1998), 342-355. · Zbl 0904.54010 · doi:10.4153/CJM-1998-018-7
[13] A. Giraldo and J. M. R. Sanjurjo: Strong multihomotopy and Steenrod loop spaces. J. Math. Soc. Japan. 47 (1995), 475-489. · Zbl 0842.55006 · doi:10.2969/jmsj/04730475
[14] R. W. Kieboom: An intrinsic characterization of the shape of paracompacta by means of non-continuous single-valued maps. Bull. Belg. Math. Soc. 1 (1994), 701-711. · Zbl 0814.54013
[15] K. Kuratowski: Topology I. Academic Press, New York, 1966. · Zbl 0158.40901
[16] S. Mardešić: Approximate fibrations and shape fibrations. Proc. of the International Conference on Geometric Topology, PWN, Polish Scientific Publishers, 1980, pp. 313-322.
[17] S. Mardešić and T. B. Rushing: Shape fibrations. General Topol. Appl. 9 (1978), 193-215. · Zbl 0398.55011 · doi:10.1016/0016-660X(78)90023-5
[18] S. Mardešić and T. B. Rushing: Shape fibrations II. Rocky Mountain J. Math. 9 (1979), 283-298. · Zbl 0448.55006 · doi:10.1216/RMJ-1979-9-2-283
[19] S. Mardešić and J. Segal: Shape Theory. North Holland, Amsterdam, 1982.
[20] E. Michael: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152-182. · Zbl 0043.37902 · doi:10.2307/1990864
[21] M. A. Morón and F. R. Ruiz del Portal: Multivalued maps and shape for paracompacta. Math. Japon. 39 (1994), 489-500. · Zbl 0808.54015
[22] J. M. R. Sanjurjo: A non-continuous description of the shape category of compacta. Quart. J. Math. Oxford (2) 40 (1989), 351-359. · Zbl 0697.55012 · doi:10.1093/qmath/40.3.351
[23] J. M. R. Sanjurjo: Multihomotopy sets and transformations induced by shape. Quart. J. Math. Oxford (2) 42 (1991), 489-499. · Zbl 0760.54012 · doi:10.1093/qmath/42.1.489
[24] J. M. R. Sanjurjo: An intrinsic description of shape. Trans. Amer. Math. Soc. 329 (1992), 625-636. · Zbl 0748.54005 · doi:10.2307/2153955
[25] J. M. R. Sanjurjo: Multihomotopy, Čech spaces of loops and shape groups. Proc. London Math. Soc. (3) 69 (1994), 330-344. · Zbl 0826.55004 · doi:10.1112/plms/s3-69.2.330
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