nLab
Hurewicz connection

Setup and definition

Given a continuous map $\pi :E\to B$ of topological spaces, one constructs the cocylinder $\mathrm{Cocyl}(\pi )$ as the pullback

$$\begin{array}{ccc}\mathrm{Cocyl}(\pi )& \stackrel{{\mathrm{pr}}_{𝒫(B)}}{\to }& 𝒫(B)\\ {pr}_E\downarrow & & \downarrow \\ E& \stackrel{\pi }{\to }& B\end{array}$$\array{ Cocyl(\pi) &\overset{pr_{\mathcal{P}(B)}}\to & \mathcal{P}(B) \\ \pr_E\downarrow && \downarrow \\ E & \stackrel{\pi}\to & B }

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

Definition

A Hurewicz connection is any continuous section

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

of the map ${\pi }_{!}:𝒫(E)\to \mathrm{Cocyl}(\pi )$ given by ${\pi }_{!}(u)=(u(0),\pi \circ u)$.

Characterization of Hurewicz fibrations

Theorem

A map $\pi :E\to B$ is a Hurewicz fibration iff there exists at least one Hurewicz connection for ${\pi }_{!}$.

Proof

To see that consider the following diagram

$$\begin{array}{ccccc}Y& \stackrel{\theta }{\to }& \mathrm{Cocyl}(\pi )& \stackrel{{\mathrm{pr}}_E}{\to }& E\\ {\sigma }_0\downarrow & & {\sigma }_0\downarrow & & \downarrow \pi \\ Y\times I& \stackrel{\theta \times I}{\underset{}{\to }}& \mathrm{Cocyl}(\pi )\times I& \underset{\mathrm{ev}}{\to }& B\end{array}$$\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 ${\mathrm{pr}}_E:\mathrm{Cocyl}(\pi )\to E$ is the restriction of the projection $E\times B^I\to E$ to the factor $E$ and the map $\mathrm{Cocyl}(\pi )\times I\to B$ is the evaluation $(e,u,t)\mapsto u(t)$ for $(e,u)\in \mathrm{Cocyl}(\pi )$. The right-hand square is commutative and this square defines a homotopy lifting problem. If $\pi$ is a cofibration this universal homotopy lifting problem has a solution, say $\tilde{s}:\mathrm{Cocyl}(p)\times I\to E$. By the hom-mapping space adjunction (exponential law) this map corresponds to some map $s:\mathrm{Cocyl}(\pi )\to 𝒫(E)$. One can easily check that this map is a section of ${\pi }_{!}$.

Conversely, let a Hurewicz connection $s$ exist, and fill the right-hand square of the diagram with diagonal $\tilde{s}$ obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: $\tilde{f}:Y\to E$, $F:Y\times I\to B$ with $F_0=p\circ \tilde{f}:Y\to E$; let furthermore $F\prime :Y\to 𝒫(B)$ be the map obtained from $F$ by the hom-mapping space adjunction. By the universal property of the cocylinder (as a pullback), there is a unique mapping $\theta :Y\to \mathrm{Cocyl}(\pi )$ such that ${\mathrm{pr}}_{𝒫(B)}\circ \theta =F\prime :Y\to 𝒫(B)$ and ${\mathrm{pr}}_E\circ \theta =\tilde{f}:Y\to E$. Now notice that the by composing the horizontal lines we obtain $\tilde{f}$ upstairs and $F$ downstairs, hence the external square is the square giving the homotopy lifting problem for this pair. The lifting is then given by $\tilde{s}\circ (\theta \times {\mathrm{id}}_I):Y\times I\to E$. Simple checking finishes the proof.

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

Special cases and properties

If $\pi :E\to B$ is a covering space where $B$ is Hausdorff, then ${\pi }_{!}$ is a homeomorphism; thus in that case the Hurewicz connection is unique.

If $\pi$ 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 ${\pi }_{!}^{\mathrm{smooth}}:{𝒫}^{\mathrm{smooth}}(E)\to {\mathrm{Cocyl}}^{\mathrm{smooth}}(\pi )$ where the subspaces of smooth paths are used.