nLab Milnor slide trick

This page meant to recall the proof of the local recognition of Hurewicz fibrations (see there). But it didn’t and doesn’t.

Let $\pi: E \to B$ be a principal $G$-fiber bundle which has a numerable cover (this condition obtains if for example $B$ is paracompact). Suppose given a commutative diagram in Top:

$$\array{ X & \overset{f}{\to} & E \\ i_0 \downarrow & & \downarrow \pi \\ X \times I & \underset{\phi}{\to} & B }$$

where $i_0$ is the composite inclusion $X \cong X \times \{0\} \hookrightarrow X \times I$. We are trying to show that $\phi$ lifts through $\pi$.

As I recall, the trick proceeds by considering the bundle

$$\array{ (\phi_0)^*E \times (-\infty, 0] \cup \phi^* E \cup (\phi_1)^* E \times [1, \infty) \\ \downarrow \\ (X \times (-\infty, 0]) \cup (X \times I) \cup (X \times [1, \infty) }$$

where the base is $X \times \mathbb{R}$ and $\phi_t$ is the restriction of $\phi$ along $X \times \{t\} \hookrightarrow X \times I$, and then one constructs a bundle lifting of the homeomorphism

$$X \times \mathbb{R} \to X \times \mathbb{R}: (x, t) \mapsto (x, t + 1)$$

This bundle lifting is “the slide”. Now the bundle is trivial over $X \times [-1, 0]$ (see below), so it has a section, and one transports this section along the slide to give a section $\sigma$ over the part

$$\array{ \phi^* E\\ \downarrow \\ X \times [0, 1] }$$

and then the composition

$$X \times I \overset{\sigma}{\to} \phi^* E \to E$$

gives the desired homotopy lifting.

To see that the bundle restricted over $X \times (-\infty, 0]$ is trivial, we just need to check that $(\phi_0)^* E$ is trivial over $X$. However, by the original commutative square, $\phi_0$ equals the composite

$$X \overset{f}{\to} E \overset{\pi}{\to} B$$

and already $\pi^* E$ is trivial (over $E$) essentially because $\pi$ is a $G$-torsor: there is a bundle isomorphism

$$\pi^* E \cong E \times_B E \to E \times G$$

over $E$.

The construction of the slide is where transfinite composition comes in. The details are at this moment a little hazy, but the rough idea is to construct a partition of unity $\rho_\alpha$ subordinate to the pulled back (locally finite) numerable cover $U_\alpha$ of $X \times I$. One is supposed to well-order the $\alpha$, and then transfinitely compose a bunch of mini-slides over $(x, t) \mapsto (x, t + \rho_{\alpha}(x))$. The transfinite composition is well-defined on the fiber over any $x$ because the arrows in the composite are non-identity only for those $U_\alpha$ which contain $x$, and there are only finitely many of these by local finiteness.

To be continued.