Editing Homotopy groups of spheres (section)

===Hopf fibrations===
The classical [[Hopf fibration]] is a [[fiber bundle]]:

:<math>S^1\hookrightarrow S^3\rightarrow S^2.</math>

The general theory of fiber bundles {{math|''F'' → ''E'' → ''B''}} shows that there is a [[homotopy group#Long exact sequence of a fibration|long exact sequence]] of homotopy groups

:<math> \cdots \to \pi_i(F) \to \pi_i(E) \to \pi_i(B) \to \pi_{i-1}(F) \to \cdots.</math>

For this specific bundle, each group homomorphism {{math|π<sub>''i''</sub>(''S''<sup>1</sup>) → π<sub>''i''</sub>(''S''<sup>3</sup>)}}, induced by the inclusion {{math|''S''<sup>1</sup> → ''S''<sup>3</sup>}}, maps all of {{math|π<sub>''i''</sub>(''S''<sup>1</sup>)}} to zero, since the lower-dimensional sphere {{math|''S''<sup>1</sup>}} can be deformed to a point inside the higher-dimensional one {{math|''S''<sup>3</sup>}}. This corresponds to the vanishing of {{math|π<sub>1</sub>(''S''<sup>3</sup>)}}. Thus the long exact sequence breaks into [[short exact sequence]]s,

:<math>0\rightarrow \pi_i(S^3)\rightarrow \pi_i(S^2)\rightarrow \pi_{i-1}(S^1)\rightarrow 0 .</math>

Since {{math|''S''<sup>''n''+1</sup>}} is a [[suspension (topology)|suspension]] of {{math|''S''<sup>''n''</sup>}}, these sequences are [[splitting lemma|split]] by the [[Freudenthal suspension theorem|suspension homomorphism]] {{math|π<sub>''i''−1</sub>(''S''<sup>1</sup>) → π<sub>''i''</sub>(''S''<sup>2</sup>)}}, giving isomorphisms

:<math>\pi_i(S^2)= \pi_i(S^3)\oplus \pi_{i-1}(S^1) .</math>

Since {{math|π<sub>''i''−1</sub>(''S''<sup>1</sup>)}} vanishes for {{mvar|i}} at least 3, the first row shows that {{math|π<sub>''i''</sub>(''S''<sup>2</sup>)}} and {{math|π<sub>''i''</sub>(''S''<sup>3</sup>)}} are isomorphic whenever {{mvar|i}} is at least 3, as observed above.

The Hopf fibration may be constructed as follows: pairs of complex numbers {{math|(''z''<sub>0</sub>,''z''<sub>1</sub>)}} with {{math|{{abs|''z''<sub>0</sub>}}<sup>2</sup> + {{abs|''z''<sub>1</sub>}}<sup>2</sup> {{=}} 1}} form a 3-sphere, and their ratios {{math|{{sfrac|''z''<sub>0</sub>|''z''<sub>1</sub>}}}} cover the [[Riemann sphere|complex plane plus infinity]], a 2-sphere. The Hopf map {{math|''S''<sup>3</sup> → ''S''<sup>2</sup>}} sends any such pair to its ratio.{{cn|date=February 2022}}

Similarly (in addition to the Hopf fibration <math>S^0\hookrightarrow S^1\rightarrow S^1</math>, where the bundle projection is a double covering), there are [[Hopf fibration#Generalizations|generalized Hopf fibrations]]

:<math>S^3\hookrightarrow S^7\rightarrow S^4</math>

:<math>S^7\hookrightarrow S^{15}\rightarrow S^8</math>

constructed using pairs of [[quaternion]]s or [[octonion]]s instead of complex numbers.{{sfn|Hatcher|2002}} Here, too, {{math|π<sub>3</sub>(''S''<sup>7</sup>)}} and {{math|π<sub>7</sub>(''S''<sup>15</sup>)}} are zero. Thus the long exact sequences again break into families of split short exact sequences, implying two families of relations.

:<math>\pi_i(S^4)= \pi_i(S^7)\oplus \pi_{i-1}(S^3) ,</math>

:<math>\pi_i(S^8)= \pi_i(S^{15})\oplus \pi_{i-1}(S^7) .</math>

The three fibrations have base space {{math|''S''<sup>''n''</sup>}} with {{math|''n'' {{=}} 2<sup>''m''</sup>}}, for {{math|''m'' {{=}} 1, 2, 3}}. A fibration does exist for {{math|''S''<sup>1</sup>}} ({{math|''m'' {{=}} 0}}) as mentioned above, but not for {{math|''S''<sup>16</sup>}} ({{math|''m'' {{=}} 4}}) and beyond. Although generalizations of the relations to {{math|''S''<sup>16</sup>}} are often true, they sometimes fail; for example,

:<math>\pi_{30}(S^{16})\neq \pi_{30}(S^{31})\oplus \pi_{29}(S^{15}) .</math>

Thus there can be no fibration

:<math>S^{15}\hookrightarrow S^{31}\rightarrow S^{16} ,</math>

the first non-trivial case of the [[Hopf invariant]] one problem, because such a fibration would imply that the failed relation is true.{{cn|date=February 2022}}