Mapping spaces and fibrewise homotopy theory

Moreira, Manuel F. (2007) Mapping spaces and fibrewise homotopy theory. Masters thesis, Memorial University of Newfoundland.

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Abstract

If q : Y → B is a fibration and Z is a space, then the free range mapping space Y!Z has a collection of partial maps from Y to Z as underlying set, namely those maps whose domains are individual fibres of q. -- It is shown in [B3] that these maps have applications to several topics in homotopy theory. Three results [B3, Ths. 5.1, 6.1 and 7.1], concerning identifications, cofibrations and sectioned fibrations, are given in complete detail. The necessary topological foundations for two more complicated applications, to the cohomology of fibrations and the classification of Moore-Postnikov systems, are also given. The applications themselves are outlined in Chapters 8 and 9 of [B3]. -- The argument of [B3] is in the context of the usual category of all topological spaces, and this necessarily introduces some limitations. Whenever we work with exponential laws for mapping spaces in that category, we usually find that we are forced to assume that some of the spaces are locally compact and Hausdorff. These conditions detract considerably from the generality of the results obtained. -- In this thesis we develop the aforementioned topological foundations in the category of compactly generated or CG-spaces, which is free of the inconvenient assumptions mentioned above. Furthermore we do not require the Hausdorff condition for CG-ification as in [S]. Thus we obtain the CG-space versions of the applications to identifications, cofibrations and sectioned fibrations, a theorem on infinite CW-complexes, and establish improved foundations for the CG-versions of the other two applications, i.e. the cohomology of fibrations and the classification theorem for Moore-Postnikov factorizations.

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