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Title:	Fibration theory of polish spaces category
Authors:	Assakta Khalil Bashier Mohammed (P76001)
Supervisor:	Abd Ghafur Ahmad, Prof. Dr.
Keywords:	Fibration Dimension theory (Topology)
Issue Date:	10-Jan-2017
Description:	In fibration theory, it is appropriate to deal with a class of spaces which satisfies the standard notions of the theory and large enough, to contain all the spaces which are considered interesting. This has motivated the researchers to examine the fibration theory for distinct types of spaces. It is acknowledged that the theory of fibration varies from one space to another depending on the properties of the space being examined. In this thesis, the case where general Hurewicz fibrations of topological spaces are restricted to Polish spaces is explored. Polish spaces form a large class of topological spaces, having various desirable properties and are quite manageable. The fibrations of Polish spaces (Polish fibrations) are firstly defined in the study and their main properties determined. The main result identifies that category Pol of Polish spaces preserves the structure of a fibration category based on Baues, provided that the fibration means Polish fibration and that weak equivalence means homotopy equivalence in Pol. The category Bor, whose objects are the pairs (X;a) consisting of a Polish space X and a Borel equivalence relation a, is constructed. Homomorphisms in Bor are continuous maps compatible with the equivalence relations and the concepts of B-homotopy and B-fibrations are defined. The notion of B-fibration exhibits many of the usual properties of the standard theory of fibration and are closed under composition. The concept of Polish fibrations is then applied in order to prove various results related to actions on Polish spaces which thereby generalize known results on fibrations related to H-spaces. The problem of classifying Polish fibrations is further considered by defining and examining a new homotopy invariant subgroup of the fundamental groups using the space of probability measures on Polish spaces. The relationship between regular maps and locally trivial fibrations of Polish spaces is examined. Finally, the special case of fibrations for Polish semigroups (P-fibrations) is considered. The concept of attaching fibrations, using fiber homotopy equivalences for P-fibrations and Polish fibrations are investigated. A generalization of the theorem by Tulley on strong fiber homotopy equivalence is obtained. The results clearly show that the notions of fiber homotopy coincide with that of strong fiber homotopy in the category Pol. Furthermore, the findings of the study indicate both the feasibility and effectiveness of the theory of fibration in the category of Polish spaces. This category is much closer to bundle theory and a convenient category for performing fibrations.,Certification of Master's/Doctoral Thesis" is not available,Ph.D.
Pages:	132
Call Number:	QA3.M843 2017 tesis
Publisher:	UKM, Bangi
Appears in Collections:	Faculty of Science and Technology / Fakulti Sains dan Teknologi

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