# 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)
How to Cite This Entry:
Hopf fibration. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_fibration&oldid=47269
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article