Hopf fibration
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In differential topology, the Hopf fibration describes a 3-sphere in terms of circles and an ordinary sphere. Discovered by Heinz Hopf in 1931, it is an influential early example of a fiber bundle. Te
Pairwise linked keyrings mimic part of the Hopf fibration.
Homotopy groups of spheres
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In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, whic
Hopf fibration
…2 and S 1 although locally it is indistinguishable from it. This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. It also provides a basic example of a principal bundle, by identifying the fiber with the circle group. Stereographic…
A homotopy from a circle around a sphere down to a single point