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  "body": "\n## Summary\n\nA sphere bundle is a [[fiber bundle|CX4M8K61AB]] whose fibers are spheres of a fixed dimension.\nThe main source of examples is the unit sphere bundle of a real vector bundle with an inner product.\nSphere bundles carry the [[Euler class|BJX6QS0ANT]] and the [[Gysin sequence|QJSPWM9F9P]], and the [[Hopf fibration|M5AKNYC4QQ]] is a circle bundle over the $2$-sphere.\nMilnor's exotic $7$-spheres were found as total spaces of $S^3$-bundles over $S^4$.\n\n## Explanations\n\n### L1 human\n\nOver each point of a base space, place a sphere, and let these spheres vary continuously.\nThe torus is a circle bundle over the circle: above each point of one circle sits another circle.\nThe Klein bottle is also a circle bundle over the circle, but with a reflection as one goes around.\nFor a surface, the unit tangent vectors at each point form a circle, and together they make a circle bundle over the surface.\n",
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