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Classification of exterior and proper fibrations. (English) Zbl 1448.55014
This technical paper fits in the fields of topological categories and homotopy theory in algebraic topology. Specifically, it concerns some questions in standard proper homotopy theory that are however studied in a much more general setup, that is the category of “exterior” spaces where problems of “exterior” homotopy theory can be solved with the help of methods of standard homotopy theory.
For instance, the category of topological spaces with proper maps can actually be embedded into a larger category which has the advantage of being both complete and co-complete: the category of “exterior spaces”.
Let us now give some definitions. An exterior space is a topological space \(X\) endowed with a nonempty family (subfamily of the topology of \(X\)) of so-called exterior sets which can be thought of as a neighborhood system at infinity. Given two exterior spaces \(X\) and \(Y\), an exterior map is a continuous map from \(X\) to \(Y\) for which the pre-image of any exterior set of \(Y\) is an exterior set of \(X\). A proper fibration is a map in the proper category which is an exterior fibration when considered in the exterior category through the embedding of the category of topological spaces with proper maps into that of exterior spaces.
Finally, there are also the tricky notions of (relative) exterior and based exterior CW-complexes: for what we need now, let us just say that a finite exterior CW-complex is in general a finite dimensional infinite classical CW-complex, while any classical CW-complex is an exterior CW-complex, as well as open manifolds and PL-manifolds (whenever they admit a locally finite countable triangulation).
The aim of the present paper is to classify the set of equivalence classes of exterior fiber sequences over a given exterior path-connected based CW-complex \(X\) with fiber another exterior path-connected based CW-complex \(F\).
This results allows the authors to describe and classify all based proper fibrations between countable, locally finite relative CW-complexes, with fiber \(F\) regarded as an exterior based space.
MSC:
55P57 Proper homotopy theory
55R05 Fiber spaces in algebraic topology
55R15 Classification of fiber spaces or bundles in algebraic topology
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