Editing Homotopy groups of spheres (section)

==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. However, such visualizations are not mathematical proofs, and do not capture the possible complexity of maps between spheres.

==={{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}===
[[Image:Fundamental group of the circle.svg|300px|thumb|Elements of {{math|π<sub>1</sub>(''S''<sup>1</sup>)}}]]

The simplest case concerns the ways that a circle (1-sphere) can be wrapped around another circle. This can be visualized by wrapping a [[rubber band]] around one's finger: it can be wrapped once, twice, three times and so on.  The wrapping can be in either of two directions, and wrappings in opposite directions will cancel out after a deformation. The homotopy group {{math|π<sub>1</sub>(''S''<sup>1</sup>)}} is therefore an [[infinite cyclic group]], and is [[isomorphic]] to the group {{math|Z}} of [[integer]]s under addition: a homotopy class is identified with an integer by counting the number of times a mapping in the homotopy class wraps around the circle. This integer can also be thought of as the [[winding number]] of a loop around the [[origin (mathematics)|origin]] in the [[plane (mathematics)|plane]].{{sfn|Hatcher|2002|p=29}}

The identification (a [[group isomorphism]]) of the homotopy group with the integers is [[abuse of notation|often written]] as an equality: thus {{math|π<sub>1</sub>(''S''<sup>1</sup>) {{=}} Z}}.<ref>See, e.g., {{harvnb|Homotopy type theory|2013}}, Section 8.1, "<math display=inline>\pi_1(S^1)</math>".</ref>

==={{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]]).

==={{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}}

==={{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}===
All the interesting cases of homotopy groups of spheres involve mappings from a higher-dimensional sphere onto one of lower dimension. Unfortunately, the only example which can easily be visualized is not interesting: there are no nontrivial mappings from the ordinary sphere to the circle. Hence, {{math|π<sub>2</sub>(''S''<sup>1</sup>) {{=}} 0}}. This is because {{math|''S''<sup>1</sup>}} has the real line as its [[universal cover]] which is contractible (it has the homotopy type of a point). In addition, because {{math|''S''<sup>2</sup>}} is simply connected, by the [[Homotopy lifting property|lifting criterion]],{{sfn|Hatcher|2002|p=61}} any map from {{math|''S''<sup>2</sup>}} to {{math|''S''<sup>1</sup>}} can be lifted to a map into the real line and the nullhomotopy descends to the downstairs space (via composition).

==={{math|π<sub>3</sub>(''S''<sup>2</sup>) {{=}} Z}}===

[[Image:Hopf Fibration.png|right|thumb|The [[Hopf fibration]] is a nontrivial mapping of the 3-sphere to the 2-sphere, and generates the third homotopy group of the 2-sphere. Each colored circle maps to the corresponding point on the 2-sphere shown bottom right.]]

The first nontrivial example with {{math|''i'' > ''n''}} concerns mappings from the [[3-sphere]] to the ordinary 2-sphere, and was discovered by [[Heinz Hopf]], who constructed a nontrivial map from {{math|''S''<sup>3</sup>}} to {{math|''S''<sup>2</sup>}}, now known as the [[Hopf fibration]].{{sfn|Hopf|1931}} This map [[Generating set of a group|generates]] the homotopy group {{math|π<sub>3</sub>(''S''<sup>2</sup>) {{=}} Z}}.{{sfn|Walschap|2004|p=90}}