---
tag: 3VQ2V5HH3R
kind: concept
name: Sphere bundle
msc: [55R25, 55R10]
specializes: [CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Sphere_bundle
same_as:
  wikidata: Q51544038
  wikipedia: Sphere bundle
---

## Summary

A sphere bundle is a [[fiber bundle|CX4M8K61AB]] whose fibers are spheres of a fixed dimension.
The main source of examples is the unit sphere bundle of a real vector bundle with an inner product.
Sphere bundles carry the [[Euler class|BJX6QS0ANT]] and the [[Gysin sequence|QJSPWM9F9P]], and the [[Hopf fibration|M5AKNYC4QQ]] is a circle bundle over the $2$-sphere.
Milnor's exotic $7$-spheres were found as total spaces of $S^3$-bundles over $S^4$.

## Explanations

### L1 human

Over each point of a base space, place a sphere, and let these spheres vary continuously.
The torus is a circle bundle over the circle: above each point of one circle sits another circle.
The Klein bottle is also a circle bundle over the circle, but with a reflection as one goes around.
For a surface, the unit tangent vectors at each point form a circle, and together they make a circle bundle over the surface.