# nLab Hurewicz connection

### Context

#### Topology

topology

algebraic topology

# Contents

## Setup and definition

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

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

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

###### Definition

A Hurewicz connection is any continuous section

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

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

## 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

$\begin{array}{ccccc}Y& \stackrel{\theta }{\to }& \mathrm{Cocyl}\left(\pi \right)& \stackrel{{\mathrm{pr}}_{E}}{\to }& E\\ {\sigma }_{0}↓& & {\sigma }_{0}↓& & ↓\pi \\ Y×I& \stackrel{\theta ×I}{\underset{}{\to }}& \mathrm{Cocyl}\left(\pi \right)×I& \underset{\mathrm{ev}}{\to }& B\end{array}$\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 ${\mathrm{pr}}_{E}:\mathrm{Cocyl}\left(\pi \right)\to E$ is the restriction of the projection $E×{B}^{I}\to E$ to the factor $E$ and the map $\mathrm{Cocyl}\left(\pi \right)×I\to B$ is the evaluation $\left(e,u,t\right)↦u\left(t\right)$ for $\left(e,u\right)\in \mathrm{Cocyl}\left(\pi \right)$. 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 $\stackrel{˜}{s}:\mathrm{Cocyl}\left(p\right)×I\to E$. By the hom-mapping space adjunction (exponential law) this map corresponds to some map $s:\mathrm{Cocyl}\left(\pi \right)\to 𝒫\left(E\right)$. 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 $\stackrel{˜}{s}$ obtained by hom-mapping space adjunction. Let the data for the general homotopy lifting problem be given: $\stackrel{˜}{f}:Y\to E$, $F:Y×I\to B$ with ${F}_{0}=p\circ \stackrel{˜}{f}:Y\to E$; let furthermore $F\prime :Y\to 𝒫\left(B\right)$ 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 $\theta :Y\to \mathrm{Cocyl}\left(\pi \right)$ such that ${\mathrm{pr}}_{𝒫\left(B\right)}\circ \theta =F\prime :Y\to 𝒫\left(B\right)$ and ${\mathrm{pr}}_{E}\circ \theta =\stackrel{˜}{f}:Y\to E$. Now notice that the by composing the horizontal lines we obtain $\stackrel{˜}{f}$ 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 $\stackrel{˜}{s}\circ \left(\theta ×{\mathrm{id}}_{I}\right):Y×I\to E$. 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 ${\pi }_{!}^{\mathrm{smooth}}:{𝒫}^{\mathrm{smooth}}\left(E\right)\to {\mathrm{Cocyl}}^{\mathrm{smooth}}\left(\pi \right)$ where the subspaces of smooth paths are used.

Revised on March 9, 2012 10:31:37 by Andrew Stacey (129.241.15.200)