nLab
Hurewicz connection

Contents

Setup and definition

Given a continuous map π:EB\pi : E\to B of topological spaces, one constructs the mapping cocylinder Cocyl(π)Cocyl(\pi) as the pullback

Cocyl(π) pr 𝒫(B) 𝒫(B) pr E E π B \array{ Cocyl(\pi) &\overset{pr_{\mathcal{P}(B)}}\to & \mathcal{P}(B) \\ \pr_E\downarrow && \downarrow \\ E & \stackrel{\pi}\to & B }

where 𝒫(B)\mathcal{P}(B) is the path space in Top, the space of continuous paths u:[0,1]Bu:[0,1]\to B in BB, and where 𝒫(B)B\mathcal{P}(B)\to B is the map sending a path uu to its value u(0)u(0). The cocylinder can be realized as a subspace of E×𝒫(B)E\times \mathcal{P}(B) consisting of pairs (e,u)(e,u) where eEe\in E and u:[0,1]𝒫(B)u:[0,1]\to \mathcal{P}(B) are such that π(e)=u(0)\pi(e)=u(0).

Definition

A Hurewicz connection is any continuous section

s:Cocyl(π)𝒫(E)s:Cocyl(\pi)\to \mathcal{P}(E)

of the map π !:𝒫(E)Cocyl(π)\pi_!:\mathcal{P}(E)\to Cocyl(\pi) given by π !(u)=(u(0),πu)\pi_!(u)=(u(0),\pi\circ u).

Characterization of Hurewicz fibrations

Theorem

A map π:EB\pi:E\to B is a Hurewicz fibration iff there exists at least one Hurewicz connection for π !\pi_!.

Proof

To see that consider the following diagram

Y θ Cocyl(π) pr E E σ 0 σ 0 π Y×I θ×I Cocyl(π)×I ev B\begin{matrix} Y& \stackrel{\theta}\to & Cocyl(\pi) &\overset{pr_E}\to & E\\ \sigma_0\downarrow&&\sigma_0\downarrow&&\downarrow \pi\\ Y\times I&\stackrel{\theta\times I}\underset{}{\to}& Cocyl(\pi)\times I& \underset{ev}\to &B \end{matrix}

where pr E:Cocyl(π)Epr_E: Cocyl(\pi)\to E is the restriction of the projection E×B IEE\times B^I\to E to the factor EE and the map Cocyl(π)×IBCocyl(\pi)\times I\to B is the evaluation (e,u,t)u(t)(e,u,t)\mapsto u(t) for (e,u)Cocyl(π)(e,u)\in Cocyl(\pi). The right-hand square is commutative and this square defines a homotopy lifting problem. If π\pi is a cofibration this universal homotopy lifting problem has a solution, say s˜:Cocyl(p)×IE\tilde{s}:Cocyl(p)\times I\to E. By the hom-mapping space adjunction (exponential law) this map corresponds to some map s:Cocyl(π)𝒫(E)s:Cocyl(\pi)\to \mathcal{P}(E). One can easily check that this map is a section of π !\pi_!.

Conversely, let a Hurewicz connection ss exist, and fill the right-hand square of the diagram with diagonal s˜\tilde{s} obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: f˜:YE\tilde{f}:Y\to E, F:Y×IBF:Y\times I\to B with F 0=pf˜:YEF_0 = p\circ \tilde{f}:Y\to E; let furthermore F:Y𝒫(B)F':Y\to \mathcal{P}(B) be the map obtained from FF by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping θ:YCocyl(π)\theta: Y\to Cocyl(\pi) such that pr 𝒫(B)θ=F:Y𝒫(B)pr_{\mathcal{P}(B)}\circ\theta=F':Y\to \mathcal{P}(B) and pr Eθ=f˜:YEpr_E\circ\theta =\tilde{f}:Y\to E. Now notice that the by composing the horizontal lines we obtain f˜\tilde{f} upstairs and FF downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by s˜(θ×id I):Y×IE\tilde{s}\circ (\theta\times id_I):Y\times I\to E. Simple checking finishes the proof.

Of course there are many other equivalent characterizations of Hurewicz fibrations.

Special cases and properties

If π:EB\pi:E\to B is a covering space where BB is Hausdorff, then π !\pi_! is a homeomorphism; thus in that case the Hurewicz connection is unique.

If π\pi 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 π ! smooth:𝒫 smooth(E)Cocyl smooth(π)\pi_!^{smooth}:\mathcal{P}^{smooth}(E)\to Cocyl^{smooth}(\pi) where the subspaces of smooth paths are used.

References

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

Revised on December 8, 2013 02:10:19 by Urs Schreiber (89.204.135.58)