Previous |  Up |  Next


shape fibration; multivalued map; path lifting property; strong shape
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.
[B1] K. Borsuk: On movable compacta. Fund. Math. 66 (1969), 137–146. MR 0251698 | Zbl 0189.53802
[B2] K. Borsuk: Theory of Shape (Monografie Matematyczne 59). Polish Scientific Publishers, Warszawa, 1975. MR 0418088
[C] F. W. Cathey: Shape fibrations and strong shape theory. Topology Appl. 14 (1982), 13–30. DOI 10.1016/0166-8641(82)90044-X | MR 0662809 | Zbl 0505.55020
[Č1] Z. Čerin: Shape theory intrinsically. Publ. Mat. 37 (1993), 317–334. DOI 10.5565/PUBLMAT_37293_06 | MR 1249234
[Č2] Z. Čerin: Proximate topology and shape theory. Proc. Roy. Soc. Edinburgh 125 (1995), 595–615. DOI 10.1017/S0308210500032704 | MR 1359494
[Č3] Z. Čerin: Approximate fibrations. To appear.
[CD] D. Coram and P. F. Duvall, Jr.: Approximate fibrations. Rocky Mountain J. Math. 7 (1977), 275–288. DOI 10.1216/RMJ-1977-7-2-275 | MR 0442921
[CP] J. M. Cordier and T. Porter: Shape Theory. Categorical Methods of Approximation (Ellis Horwood Series: Mathematics and its Applications). Ellis Horwood Ltd., Chichester, 1989. MR 1000348
[DS1] J. Dydak and J. Segal: Shape Theory: An Introduction (Lecture Notes in Math. 688). Springer-Verlag, Berlin, 1978. MR 0520227
[DS2] 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. MR 1078663
[F] J. E. Felt: $\epsilon $-continuity and shape. Proc. Amer. Math. Soc. 46 (1974), 426–430. MR 0362206
[G] A. Giraldo: Shape fibrations, multivalued maps and shape groups. Canad. J. Math 50 (1998), 342–355. DOI 10.4153/CJM-1998-018-7 | MR 1618314 | Zbl 0904.54010
[GS] A. Giraldo and J. M. R. Sanjurjo: Strong multihomotopy and Steenrod loop spaces. J. Math. Soc. Japan. 47 (1995), 475–489. DOI 10.2969/jmsj/04730475 | MR 1331325
[K] 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. MR 1315365 | Zbl 0814.54013
[Ku] K. Kuratowski: Topology I. Academic Press, New York, 1966. MR 0217751
[M] S. Mardešić: Approximate fibrations and shape fibrations. Proc. of the International Conference on Geometric Topology, PWN, Polish Scientific Publishers, 1980, pp. 313–322. MR 0656763
[MR1] S. Mardešić and T. B. Rushing: Shape fibrations. General Topol. Appl. 9 (1978), 193–215. DOI 10.1016/0016-660X(78)90023-5 | MR 0510901
[MR2] S. Mardešić and T. B. Rushing: Shape fibrations II. Rocky Mountain J. Math. 9 (1979), 283–298. DOI 10.1216/RMJ-1979-9-2-283 | MR 0519943
[MS] S. Mardešić and J. Segal: Shape Theory. North Holland, Amsterdam, 1982. MR 0676973
[Mi] E. Michael: Topologies on spaces of subsets. Trans. Amer. Math. Soc. 71 (1951), 152–182. DOI 10.1090/S0002-9947-1951-0042109-4 | MR 0042109 | Zbl 0043.37902
[MoR] M. A. Morón and F. R. Ruiz del Portal: Multivalued maps and shape for paracompacta. Math. Japon. 39 (1994), 489–500. MR 1278864
[S1] J. M. R. Sanjurjo: A non-continuous description of the shape category of compacta. Quart. J. Math. Oxford (2) 40 (1989), 351–359. MR 1010825 | Zbl 0697.55012
[S2] J. M. R. Sanjurjo: Multihomotopy sets and transformations induced by shape. Quart. J. Math. Oxford (2) 42 (1991), 489–499. MR 1135307 | Zbl 0760.54012
[S3] J. M. R. Sanjurjo: An intrinsic description of shape. Trans. Amer. Math. Soc. 329 (1992), 625–636. DOI 10.1090/S0002-9947-1992-1028311-X | MR 1028311 | Zbl 0748.54005
[S4] J. M. R. Sanjurjo: Multihomotopy, Čech spaces of loops and shape groups. Proc. London Math. Soc. (3) 69 (1994), 330–344. MR 1281968 | Zbl 0826.55004
Partner of
EuDML logo