nLab sphere fiber bundle

Idea

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

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):

Proposition

Let denote the universal $n$-spherical fibration over the classifying space of the orthogonal group, where

is the canonical inclusion.

Then its homotopy fiber inclusion is the classifying map $\vdash Fr(S^n)$ of the orthonormal frame bundle of the n-sphere:

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 $\Omega B O(n+1) \simeq O(n+1)$:

Moreover, we have an evident isomorphism

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

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

In parametrized generalization of this situation, it follows that:

Corollary

(once-stabilized vertical tangent bundles to sphere-fiber bundles are pulled back from base) Let $S(p) \colon S(\mathcal{V}) \to X$ be an $n$-spherical fibration, associated to an oriented real vector bundle $p \colon \mathcal{V} \to X$, hence fitting into a homotopy pullback-diagram as shown here: Then:

1. the top map shown classifies the vertical tangent bundle $T_{S(p)} S(\mathcal{V})$;

2. hence the homotopy-commutativity of the diagram says that the once-stabilized vertical tangent bundle is the pullback of the original bundle on the base:

   $$T_{S(p)} S(\mathcal{V}) \times \mathbb{R} \;\simeq_{{}_{X}}\; S(p)^\ast \big( \mathcal{V} \big) \,.$$

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):

Proposition

(stable tangent bundle of unit sphere bundle)
The once-stabilized tangent bundle of a unit sphere bundle $S(\mathcal{V})$ in a real vector bundle $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example ) over a smooth manifold $M$ is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

$$T S(\mathcal{V}) \times \mathbb{R} \; \simeq \; S(p)^\ast \big( T M \oplus_M \mathcal{V} \big) \,.$$

Proof

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

$$T \big( D(\mathcal{V}) \big) \;\simeq\; \big( D(p)^\ast T M \big) \oplus_M T_p D(\mathcal{V}) \,,$$

where $T_p D(\mathcal{V})$ 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 $\mathcal{V}$ itself, and that is just the pullback of that vector bundle along itself (by this Example):

$$\begin{aligned} \cdots & \simeq\; \big( D(p)^\ast T M \big) \oplus_M \big( D(p)^\ast \mathcal{V} \big) \\ & \simeq\; D(p)^\ast \big( T M \oplus_M \mathcal{V} \big) \,. \end{aligned}$$

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

Remark

Prop. implies that every stable characteristic class of the tangent bundle of an orthogonal sphere-fiber bundle – i.e all polynomials in its Pontryagin classes – are basic, i.e. pulled back from the base space.

Examples

Unit sphere bundles

Example

A key example of sphere fiber bundles are the unit sphere bundles inside of real vector bundles that are equipped with orthogonal structure: the bundles whose fibers are the unit spheres in the corresponding fiber of the given real vector bundle.

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

Related concepts

• Sullivan model of a spherical fibration

• twisted Cohomotopy

References

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

• Hassler Whitney, On the theory of sphere-bundles, Proceedings of the National Academy of Sciences 26 2 (1940) 148-153 [doi:10.1073/pnas.26.2.148]

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

• John Milnor, On manifolds homeomorphic to the 7-sphere, Annals of Mathematics 64 (2): 399–405 (1956) (pdf, doi:10.1142/9789812836878_0001)

• Diarmuid Crowley, Christine Escher, A classification of $S^3$-bundles over $S^4$, Differential Geometry and its Applications Volume 18, Issue 3, May 2003, Pages 363-380 (doi:10.1016/S0926-2245(03)00012-3))

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

• Federico Bonetti, Ruben Minasian, Valentí Vall Camell, Peter Weck, Consistent truncations from the geometry of sphere bundles [arXiv:2212.08068]

On sphere bundles with formal Sullivan models:

• Jiawei Zhou, Formality of Sphere Bundles [arXiv:2304.09594]