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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“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)