Bundle of spheres

A torus is a product of two circles and thus a bundle of circles over the circle

In mathematics , bundles of spheres are spaces that look locally like a product space , one factor of which is a sphere . These include, in particular, bundles of circles .

definition

A bundle of spheres is a bundle of fibers whose fiber is a sphere . ${\displaystyle S^{n}}$

For one speaks of a bundle of circles . ${\displaystyle n=1}$

Examples

• The unit tangent bundle of a differentiable manifold is a bundle of spheres.
• A product manifold is a (trivial) bundle of spheres. ${\displaystyle M\times S^{n}}$
• The torus and the Klein bottle are bundles of circles above the circle .
• The nontriviality of a bundle of spheres is measured by its Euler class , which in turn is used in the Gysin sequence .

literature

• Raoul Bott , Loring Tu : Differential forms in algebraic topology. Graduate Texts in Mathematics 82. Springer-Verlag, New York-Berlin, 1982. ISBN 0-387-90613-4