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A locally trivial [[Fibration|fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479801.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479802.png" />. This is one of the earliest examples of locally trivial fibrations, introduced by H. Hopf in [[#References|[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|Hopf invariant]] is non-trivial. The creation of the mappings requires the so-called Hopf construction.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479803.png" /> be the [[Join|join]] of two spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479805.png" />, which has natural coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479806.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479807.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h0479809.png" />. Here, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798011.png" /> is the [[Suspension|suspension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798012.png" />. The Hopf construction <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798013.png" /> associates with a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798014.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798015.png" /> given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798016.png" />.
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Suppose that mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798017.png" /> are defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798018.png" /> by means of multiplications: in the complex numbers for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798019.png" />, in the quaternions for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798020.png" />, and in the Cayley numbers for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798021.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798022.png" />, and the Hopf mapping is defined as
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A locally trivial [[Fibration|fibration]]  $  f: S ^ {2n - 1 } \rightarrow S  ^ {n} $
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for $  n = 2, 4, 8 $.  
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This is one of the earliest examples of locally trivial fibrations, introduced by H. Hopf in [[#References|[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|Hopf invariant]] is non-trivial. The creation of the mappings requires the so-called Hopf construction.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798023.png" /></td> </tr></table>
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Let  $  X \star Y $
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be the [[Join|join]] of two spaces  $  X $
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and  $  Y $,
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which has natural coordinates  $  \langle  x, t, y\rangle $,
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where  $  x \in X $,
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$  t \in [ 0, 1] $,
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$  y \in Y $.
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Here, for example,  $  X \star S  ^ {0} = SX $,
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where  $  SX $
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is the [[Suspension|suspension]] of  $  X $.
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The Hopf construction  $  \mathfrak H $
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associates with a mapping  $  f: X \times Y \rightarrow Z $
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the mapping  $  \mathfrak H ( f  ): X \star Y \rightarrow SZ $
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given by  $  \mathfrak H ( f  ) \langle  x, t, y\rangle = \langle  f ( x, y), t \rangle $.
  
The Hopf mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798025.png" />, is a locally trivial fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798027.png" /> is a mapping of bidegree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798028.png" />, then the Hopf invariant of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798029.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798030.png" />. In particular, the Hopf invariant of the Hopf fibration is 1.
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Suppose that mappings  $  \mu _ {n} :  S ^ {n - 1 } \times S ^ {n - 1 } \rightarrow S ^ {n - 1 } $
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are defined for  $  n = 2, 4, 8 $
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by means of multiplications: in the complex numbers for  $  n = 2 $,
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in the quaternions for  $  n = 4 $,  
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and in the Cayley numbers for  $  n = 8 $.  
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Then  $  S ^ {n - 1 } \star S ^ {n - 1 } = S ^ {2n - 1 } $,  
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and the Hopf mapping is defined as
  
Sometimes the Hopf fibration is defined as the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798031.png" /> given by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798033.png" />. This mapping is a locally trivial fibration with fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798034.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798035.png" /> one obtains the classical Hopf fibration <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h047/h047980/h04798036.png" />.
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$$
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\mathfrak H _ {n}  = \
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\mathfrak H ( \mu _ {n} ): \
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S ^ {2n - 1 }  \rightarrow  S  ^ {n} .
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$$
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The Hopf mapping  $  \mathfrak H _ {n} $,
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$  n = 2, 4, 8 $,
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is a locally trivial fibration with fibre  $  S ^ {n - 1 } $.
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If  $  f:  S ^ {n - 1 } \times S ^ {n - 1 } \rightarrow S ^ {n - 1 } $
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is a mapping of bidegree  $  ( d _ {1} , d _ {2} ) $,
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then the Hopf invariant of the mapping  $  \mathfrak H ( f  ) $
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is  $  d _ {1} d _ {2} $.
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In particular, the Hopf invariant of the Hopf fibration is 1.
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Sometimes the Hopf fibration is defined as the mapping $  f: S ^ {2n + 1 } \rightarrow \mathbf C P  ^ {n} $
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given by the formula $  ( z _ {0} \dots z _ {n} ) \rightarrow [ z _ {0} : \dots : z _ {n} ] $,  
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$  z _ {i} \in \mathbf C $.  
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This mapping is a locally trivial fibration with fibre $  S  ^ {1} $.  
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For $  n = 1 $
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one obtains the classical Hopf fibration $  f: S  ^ {3} \rightarrow S  ^ {2} $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,  "Ueber die Abbildungen von Sphären niedriger Dimension"  ''Fund. Math.'' , '''25'''  (1935)  pp. 427–440</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,  "Ueber die Abbildungen von Sphären niedriger Dimension"  ''Fund. Math.'' , '''25'''  (1935)  pp. 427–440</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


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