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sphere fiber bundle in nLab

Contents

Context

Bundles

bundles

Spheres

n-sphere

low dimensional n-spheres

Contents

Idea

A sphere fiber bundle is a fiber bundle whose fibers are spheres S nS^n of some dimension nn.

Often, but not always, this is considered in homotopy theory or even in stable homotopy theory, hence for fibers which have the (stable) homotopy type of a sphere, in which case one speaks of spherical fibrations. See there for more.

Properties

Vertical tangent bundles of sphere bundles

The following appears in FSS 20, Sec. 3 (somewhat implicit in v1, explicitly in v2):

Proof

By the pasting law we find that the homotopy fiber of the homotopy fiber inclusion, and hence (by the discssion at principal infinity-bundle) the total space of the bundle it classifies, is ΩBO(n+1)≃O(n+1)\Omega B O(n+1) \simeq O(n+1):

Moreover, we have an evident isomorphism

given by acting with O(n+1)O(n+1) on the canonical orthonormal basis (v 0,v 1,⋯,v n)(v_0, v_1, \cdots, v_n) of ℝ n+1\mathbb{R}^{n+1}, regarded as a point v 0v_0 on S n=S(ℝ n+1)S^n = S(\mathbb{R}^{n+1}) equipped with a frame (v 1,⋯,v n)(v_1, \cdots, v_n) of its tangent space T v 0S(ℝ n+1)T_{v_0} S(\mathbb{R}^{n+1}).

This isomorphism is manifestly O(n)O(n)-equivariant, and its quotient on both sides is manifestly S nS^n, so that this is actually an isomorphism of O(n)O(n)-principal bundles.

In parametrized generalization of this situation, it follows that:

Tangent bundles of sphere bundles

The following generalizes Cor. to the full tangent bundle of sphere-fiber bundles, now assuming that the base is a smooth manifold and giving a more traditional differential-geometric proof (the following statement appears, without proof, as Crowley-Escher 03, Fact. 3.1, apparently reading between the lines in Milnor 56, p. 403):

Proof

Consider first the actual tangent bundle but to the open ball/disk-fiber bundle D(𝒱)D(\mathcal{V}) that fills the given sphere-fiber bundle: By the standard splitting (this Prop.) this is the direct sum

T(D(𝒱))≃(D(p) *TM)⊕ MT pD(𝒱), T \big( D(\mathcal{V}) \big) \;\simeq\; \big( D(p)^\ast T M \big) \oplus_M T_p D(\mathcal{V}) \,,

where T pD(𝒱)T_p D(\mathcal{V}) is the vertical tangent bundle of the disk bundle. But, by definition of disk bundles, this is the restriction of the vertical tangent bundle of the vector bundle 𝒱\mathcal{V} itself, and that is just the pullback of that vector bundle along itself (by this Example):

⋯ ≃(D(p) *TM)⊕ M(D(p) *𝒱) ≃D(p) *(TM⊕ M𝒱). \begin{aligned} \cdots & \simeq\; \big( D(p)^\ast T M \big) \oplus_M \big( D(p)^\ast \mathcal{V} \big) \\ & \simeq\; D(p)^\ast \big( T M \oplus_M \mathcal{V} \big) \,. \end{aligned}

To conclude, it just remains to observe that the normal bundle of the n-sphere-boundary inside the (n+1)(n+1)-ball is manifestly trivial, so that the restriction of the tangent bundle of D(𝒱)D(\mathcal{V}) to S(𝒱)S(\mathcal{V}) is the stable tangent bundle of S(𝒱)S(\mathcal{V}).

Examples

Unit sphere bundles

These appear in the discussion of Thom spaces and hence of Thom spectra, as well as in the discussion of wave front sets.

References

Early discussion of sphere bundles (and introducing the general notion of fiber bundles):

With focus on 3-sphere-fiber bundles over the 4-sphere and the construction of exotic 7-spheres:

On sphere fiber bundles as Kaluza-Klein compactifications in supergravity and string theory:

On sphere bundles with formal Sullivan models:

Last revised on June 21, 2023 at 16:29:15. See the history of this page for a list of all contributions to it.