Mapping spaces and homotopy theory

ISSN： 1607-3606

Source： Quaestiones Mathematicae, Vol.27, Iss.4, 2004-12, pp. : 415-430

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Abstract

Let q : Y → B be a map (= continuous function) and Z be a space. We will use Y ! Z to denote a set of partial maps from Y to Z, i.e.those such maps whose domains are individual fibres of q. If B is a T₁ -space, we will equip Y ! Z with a suitable version of the compact-open topology, thereby defining the freerange mapping space Y ! Z. Then there is an obvious associated freerange projection q ! Z : Y ! Z → B. If B is a Hausdorff space, we will show that maps from Y to Z determine sections (= right inverses) to q ! Z. Further, if Y is a compactly generated space, then this correspondence will be bijective. These properties are actually just particular cases of our main result, i.e.an exponential law for freerange mapping spaces. We demonstrate the versatility and utility of these concepts by giving applications to identifications, cofibrations, fibrations with cofibration cross sections, the cohomology of fibrations, and Moore-Postnikov Factorizations.