Low-dimensional Examples
The low-dimensional examples of homotopy groups of spheres provide a sense of the subject, because these special cases can be visualized in ordinary 3-dimensional space (Hatcher 2002). However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.
Read more about this topic: Homotopy Groups Of Spheres
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“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)