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Principal-bundle classification can fail without numerability

Claim

Let $L$ be the smooth long line and let $P = F (T L)$ be the frame bundle of its tangent line bundle. Then

$P ⟶ L$

is a locally trivial principal ${GL}_{1} (R)$ -bundle which is not numerable. Hence it is not the pullback of Milnor's universal numerable bundle along any map $L \to B {GL}_{1} (R)$ . The space $L$ is locally compact Hausdorff, hence CGWH, but it is outside the library's second-countable manifold convention.

Facts & Assumptions

[F1]

Nyikos's long line is a connected Hausdorff differentiable $1$ -manifold and is nonmetrizable; each bounded closed order interval is metrizable.

[F2]

Smooth coordinate changes make $F (T L)$ a locally trivial principal ${GL}_{1} (R)$ -bundle in the sense of Principal g bundle and associated fiber bundle.

[F3]

A numeration is a locally finite partition of unity whose supports lie in assigned trivializing opens (Locally trivial fiber bundle).

[F4]

Pullback of a support-subordinate numeration is again a support-subordinate numeration (Numerable principal bundles are classified by maps to BG).

[F5]

A locally finite sum of continuous functions is continuous (A locally finite family of continuous nonnegative functions has a continuous pointwise sum).

Verification

Given: The smooth long line $L$ and its tangent frame bundle $P$ .

1.1

The derivative of a change of one-dimensional chart is a continuous nonzero scalar, so the frame-coordinate changes take values in ${GL}_{1} (R)$ and act freely and transitively on each frame fiber. Thus [F2] gives the asserted locally trivial principal bundle.

F1F2

1.2

Suppose for contradiction that it is numerable. Let $(U_{i}, φ_{i})$ be the [F3] data. In the frame over $U_{i}$ , declare the selected frame to have squared norm $1$ ; this defines a continuous positive quadratic form $g_{i}$ on $T L ∣_{U_{i}}$ . Extend $φ_{i} g_{i}$ by zero away from $U_{i}$ . Support containment makes the extension continuous, and local finiteness together with [F5] makes

$g = \sum_{i} φ_{i} g_{i}$

a continuous quadratic form on $T L$ . Since $\sum_{i} φ_{i} = 1$ and every $g_{i}$ is positive on nonzero tangent vectors, $g$ is positive definite. [F3, F5]

2.1

This $g$ metrizes $L$ , as follows. Define $d_{g} (p, q)$ as the infimum of the $g$ -lengths of piecewise smooth paths from $p$ to $q$ . In a connected smooth manifold, the points reachable from a fixed point by such paths form a nonempty open-and-closed set, so every two points are joined and $d_{g} (p, q) < \infty$ . Positivity gives $d_{g} (p, q) > 0$ when $p \neq q$ : choose a coordinate interval $V$ about $p$ whose smaller closed subinterval $K$ contains $p$ in its interior. On $K$ , the coefficient of $g$ has a positive lower bound, so every path leaving $K$ has a fixed positive length, and within $K$ coordinate displacement has the corresponding lower bound. An upper bound for the coefficient on a still smaller interval shows short coordinate segments have arbitrarily small $g$ -length. Therefore sufficiently small $d_{g}$ -balls lie in $V$ , while a sufficiently small coordinate interval lies in any prescribed $d_{g}$ -ball. The metric topology is exactly the original topology.

F1step 1.2

3.1

Step 2.1 contradicts the nonmetrizability in [F1], so $P$ is not numerable. Every pullback of Milnor's bundle is numerable by [F4], applied to its join-coordinate numeration. Therefore no map $L \to B {GL}_{1} (R)$ pulls Milnor's bundle back to $P$ . This does not contradict the classification theorem, whose right side contains only numerable bundles. $□$

F1F4step 2.1

Depends on

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Sources

Peter J. Nyikos, The Topological Structure of the Tangent and Cotangent Bundles on the Long Line (standard reference, not scraped)
Dale Husemoller, Fibre Bundles, Third Edition (standard reference, not scraped)