Hopf fibration

A locally trivial fibration $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ for $ n = 2, 4, 8 $. This is one of the earliest examples of locally trivial fibrations, introduced by H. Hopf in [1]. These mappings induce trivial mappings in homology and cohomology; however, they are not homotopic to the null mapping, which follows from the fact that their Hopf invariant is non-trivial. The creation of the mappings requires the so-called Hopf construction.

Let $ X \star Y $ be the join of two spaces $ X $ and $ Y $, which has natural coordinates $ \langle x, t, y\rangle $, where $ x \in X $, $ t \in [ 0, 1] $, $ y \in Y $. Here, for example, $ X \star S ^ {0} = SX $, where $ SX $ is the suspension of $ X $. The Hopf construction $ \mathfrak H $ associates with a mapping $ f: X \times Y \rightarrow Z $ the mapping $ \mathfrak H ( f ): X \star Y \rightarrow SZ $ given by $ \mathfrak H ( f ) \langle x, t, y\rangle = \langle f ( x, y), t \rangle $.

Suppose that mappings $ \mu _ {n} : S ^ {n - 1 } \times S ^ {n - 1 } \rightarrow S ^ {n - 1 } $ are defined for $ n = 2, 4, 8 $ by means of multiplications: in the complex numbers for $ n = 2 $, in the quaternions for $ n = 4 $, and in the Cayley numbers for $ n = 8 $. Then $ S ^ {n - 1 } \star S ^ {n - 1 } = S ^ {2n - 1 } $, and the Hopf mapping is defined as

$$ \mathfrak H _ {n} = \ \mathfrak H ( \mu _ {n} ): \ S ^ {2n - 1 } \rightarrow S ^ {n} . $$

The Hopf mapping $ \mathfrak H _ {n} $, $ n = 2, 4, 8 $, is a locally trivial fibration with fibre $ S ^ {n - 1 } $. If $ f: S ^ {n - 1 } \times S ^ {n - 1 } \rightarrow S ^ {n - 1 } $ is a mapping of bidegree $ ( d _ {1} , d _ {2} ) $, then the Hopf invariant of the mapping $ \mathfrak H ( f ) $ is $ d _ {1} d _ {2} $. In particular, the Hopf invariant of the Hopf fibration is 1.

Sometimes the Hopf fibration is defined as the mapping $ f: S ^ {2n + 1 } \rightarrow \mathbf C P ^ {n} $ given by the formula $ ( z _ {0} \dots z _ {n} ) \rightarrow [ z _ {0} : \dots : z _ {n} ] $, $ z _ {i} \in \mathbf C $. This mapping is a locally trivial fibration with fibre $ S ^ {1} $. For $ n = 1 $ one obtains the classical Hopf fibration $ f: S ^ {3} \rightarrow S ^ {2} $.

References

[1]	H. Hopf, "Ueber die Abbildungen von Sphären niedriger Dimension" Fund. Math. , 25 (1935) pp. 427–440
[2]	D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)