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

X f E i 0 π X×I ϕ B\array{ X & \overset{f}{\to} & E \\ i_0 \downarrow & & \downarrow \pi \\ X \times I & \underset{\phi}{\to} & B }

where i 0i_0 is the composite inclusion XX×{0}X×IX \cong X \times \{0\} \hookrightarrow X \times I. We are trying to show that ϕ\phi lifts through π\pi.

As I recall, the trick proceeds by considering the bundle

(ϕ 0) *E×(,0]ϕ *E(ϕ 1) *E×[1,) (X×(,0])(X×I)(X×[1,)\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×X \times \mathbb{R} and ϕ t\phi_t is the restriction of ϕ\phi along X×{t}X×IX \times \{t\} \hookrightarrow X \times I, and then one constructs a bundle lifting of the homeomorphism

X×X×:(x,t)(x,t+1)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×[1,0]X \times [-1, 0] (see below), so it has a section, and one transports this section along the slide to give a section σ\sigma over the part

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

and then the composition

X×Iσϕ *EEX \times I \overset{\sigma}{\to} \phi^* E \to E

gives the desired homotopy lifting.

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

XfEπBX \overset{f}{\to} E \overset{\pi}{\to} B

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

π *EE× BEE×G\pi^* E \cong E \times_B E \to E \times G

over EE.

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 ρ α\rho_\alpha subordinate to the pulled back (locally finite) numerable cover U αU_\alpha of X×IX \times I. One is supposed to well-order the α\alpha, and then transfinitely compose a bunch of mini-slides over (x,t)(x,t+ρ α(x))(x, t) \mapsto (x, t + \rho_{\alpha}(x)). The transfinite composition is well-defined on the fiber over any xx because the arrows in the composite are non-identity only for those U αU_\alpha which contain xx, and there are only finitely many of these by local finiteness.

To be continued.

Last revised on April 4, 2021 at 05:40:38. See the history of this page for a list of all contributions to it.