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Explicit normal-framed collapse of an equatorial sphere

Example

For $S^{n} = {(x, 0)} \subset S^{n + 1} \subset R^{n + 1} \times R$ , use its upward unit normal framing and $0 < a < 1$ . An explicit collapse to $S_{+}^{n} \land S^{1}$ is $c (\sqrt{1 - t^{2}} x, t) = [x, t / a] (∣ t ∣ < a), c = * (∣ t ∣ \geq a) .$ Here $S^{1} = [- 1, 1] / {- 1, 1}$ . Projecting the first smash factor by $S_{+}^{n} \to S^{0}$ gives the framed collapse to $S^{1}$ .

Facts & Assumptions

Given: $n \geq 0$ , the standard equator and the specified upward normal framing.

[F1]

Pontryagin–Thom collapse with specified normal data fixes the compatible tube and collapse.

[F2]

Trivial Thom spaces as suspension smash products identifies its trivial line target.

Verification

1.1F1given

The chart $Φ (x, t) = (\sqrt{1 - t^{2}} x, t)$ is a diffeomorphism onto the band $∣ t ∣ < 1$ . Its derivative in the normal direction at $t = 0$ is $(0, 1)$ , so it respects exactly the specified framing. The metric disk of radius $a$ is its closed band, and [F1] gives the displayed formula.

2.1F2step 1.1∎

The two band boundaries go to the circle basepoint, so the outside constant formula pastes continuously. By [F2] its target is $S_{+}^{n} \land S^{1}$ . The continuous based map $S_{+}^{n} \to S^{0}$ sends the whole equator to the nonbasepoint; smashing it with the identity produces the claimed sphere-valued framed collapse. For $n = 0$ the same formula has two band components and remains valid.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stanford Math 215B notes, Lectures 14–15, Theorems 138–139 (standard reference, not scraped)
May, A Concise Course in Algebraic Topology, Chapter 23 §5 (standard reference, not scraped)