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Milnor sphere bundles are simply connected

Statement

Assume the Axiom of Choice as inherited from the bundle-to-fibration supplier. For all integers $h, j$ , the Milnor sphere bundle $M_{h, j}$ of The Milnor sphere and disk bundles $M_{h, j}$ and $W_{h, j}$ is path-connected and simply connected; in particular this holds for the bundles with Euler number $\pm 1$ used to construct homotopy seven-spheres.

Facts & Assumptions

Given: Integers $h, j$ and the smooth $S^{3}$ -bundle $S^{3} \to M_{h, j} \to S^{4}$ of The Milnor sphere and disk bundles $M_{h, j}$ and $W_{h, j}$ .

[A1]

The Axiom of Choice is assumed (The Axiom of Choice).

[L1]

The two product charts of this bundle admit a support-subordinate finite partition on $S^{4}$ (choose radial chart cutoffs and normalize their positive sum), so it is numerable. Under AC it is a Hurewicz, hence Serre, fibration (Numerable fiber bundles are hurewicz fibrations), so there is a long exact sequence of homotopy groups $\dots \to π_{1} (S^{3}) \to π_{1} (M_{h, j}) \to π_{1} (S^{4}) \to π_{0} (S^{3}) \to π_{0} (M_{h, j}) \to π_{0} (S^{4}) \to 0$ (Long exact sequence of homotopy groups of a fibration).

[L2]

For $n \geq 2$ the sphere $S^{n}$ is path-connected (For $n \geq 2$ , the sphere $S^{n - 1}$ is path-connected and connected) and simply connected ( $S^{n}$ is simply connected for every $n \geq 2$ ); in particular $π_{1} (S^{3}) = π_{1} (S^{4}) = 0$ and $π_{0} (S^{3}) = π_{0} (S^{4}) = 0$ , the latter meaning a single path component.

Proof

technique · direct

1.1L1L2A1given

By [L1] the fibration $S^{3} \to M_{h, j} \to S^{4}$ gives the exact segment $π_{1} (S^{3}) \to π_{1} (M_{h, j}) \to π_{1} (S^{4})$ ; both outer groups vanish by [L2], so exactness forces $π_{1} (M_{h, j}) = 0$ .

2.1step 1.1L1L2

Exactness of the pointed-set component sequence shows that every component of $M_{h, j}$ mapping to the base component lies in the image of $π_{0} (S^{3})$ . The base has only that component, and the fibre has only one component, so $M_{h, j}$ has only one path component.

3.1step 2.1∎

A path-connected space with trivial fundamental group is simply connected, so $M_{h, j}$ is simply connected; nothing in the argument depends on $h + j$ , so it applies in particular when the Euler number is $\pm 1$ .

Depends on

Used by

Euler number ±1 makes the Milnor sphere bundle a homotopy seven-sphere Corollary

Dependency tree · two levels

35 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

John Milnor, On Manifolds Homeomorphic to the 7-Sphere, Annals of Mathematics 64 (1956), 399-405 (standard reference, not scraped)
Allen Hatcher, Algebraic Topology, Cambridge University Press 2002 (complete book) (standard reference, not scraped)