Setup and definition
Given a continuous map of topological spaces, one constructs the mapping cocylinder as the pullback
where is the path space in Top, the space of continuous paths in , and where is the map sending a path to its value . The cocylinder can be realized as a subspace of consisting of pairs where and are such that .
A Hurewicz connection is any continuous section
of the map given by .
Characterization of Hurewicz fibrations
A map is a Hurewicz fibration iff there exists at least one Hurewicz connection for .
To see that consider the following diagram
where is the restriction of the projection to the factor and the map is the evaluation for . The right-hand square is commutative and this square defines a homotopy lifting problem. If is a cofibration this universal homotopy lifting problem has a solution, say . By the hom-mapping space adjunction (exponential law) this map corresponds to some map . One can easily check that this map is a section of .
Conversely, let a Hurewicz connection exist, and fill the right-hand square of the diagram with diagonal obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: , with ; let furthermore be the map obtained from by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping such that and . Now notice that the by composing the horizontal lines we obtain upstairs and downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by . Simple checking finishes the proof.
Of course there are many other equivalent characterizations of Hurewicz fibrations.
Special cases and properties
If is a covering space where is Hausdorff, then is a homeomorphism; thus in that case the Hurewicz connection is unique.
If is a smooth principal bundle equipped with a distribution of horizontal spaces forming an Ehresmann connection, then one can define a corresponding “smooth” Hurewicz connection in the sense that the Ehresmann connection provides a continuous choice of smooth path lifting, with prescribed initial point, of a smooth path in the base. This can be expressed in terms as a continuous section of where the subspaces of smooth paths are used.
The original article is
- Witold Hurewicz, On the concept of fiber space, Proc. Nat. Acad. Sci. USA 41 (1955) 956–961; MR0073987 (17,519e) PNAS,pdf.
A review is for instance in
- James Eells, Jr., Fibring spaces of maps, in Richard Anderson (ed.) Symposium on infinite-dimensional topology