nLab
Hurewicz connection

Context

Topology

topology (point-set topology, point-free topology)

see also algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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]Bu:[0,1]\to 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 January 9, 2017 15:23:51 by Potato? (109.67.250.91)