Editing Homotopy groups of spheres (section)

==={{math|π<sub>2</sub>(''S''<sup>2</sup>) {{=}} Z}}===

[[Image:Sphere wrapped round itself.png|200px|thumb|right|Illustration of how a 2-sphere can be wrapped twice around another 2-sphere. Edges should be identified.]]

Mappings from a 2-sphere to a 2-sphere can be visualized as wrapping a plastic bag around a ball and then sealing it. The sealed bag is topologically equivalent to a 2-sphere, as is the surface of the ball. The bag can be wrapped more than once by twisting it and wrapping it back over the ball. (There is no requirement for the continuous map to be [[injective]] and so the bag is allowed to pass through itself.) The twist can be in one of two directions and opposite twists can cancel out by deformation. The total number of twists after cancellation is an integer, called the ''[[degree of a map|degree]]'' of the mapping. As in the case mappings from the circle to the circle, this degree identifies the homotopy group with the group of integers, {{math|Z}}.{{citation needed|date=July 2024}}

These two results generalize: for all {{math|''n'' > 0}}, {{math|π<sub>''n''</sub>(''S''<sup>''n''</sup>) {{=}} Z}} (see [[#General theory|below]]).