Editing Homotopy groups of spheres (section)

==={{math|π<sub>1</sub>(''S''<sup>2</sup>) {{=}} 0}}===
[[Image:P1S2all.jpg|400px|thumb|A homotopy from a circle around a sphere down to a single point]]
Any continuous mapping from a circle to an ordinary sphere can be continuously deformed to a one-point mapping, and so its homotopy class is trivial. One way to visualize this is to imagine a rubber-band wrapped around a frictionless ball: the band can always be slid off the ball. The homotopy group is therefore a [[trivial group]], with only one element, the identity element, and so it can be identified with the [[subgroup]] of {{math|Z}} consisting only of the number zero. This group is often denoted by 0.  Showing this rigorously requires more care, however, due to the existence of
[[space-filling curve]]s.{{sfn|Hatcher|2002|p=348}}

This result generalizes to higher dimensions. All mappings from a lower-dimensional sphere into a sphere of higher dimension are similarly trivial: if {{math|''i'' < ''n''}}, then  {{math|π<sub>''i''</sub>(''S''<sup>''n''</sup>) {{=}} 0}}.  This can be shown as a consequence of the [[cellular approximation theorem]].{{sfn|Hatcher|2002|p=349}}