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An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([[#References|[1]]], [[#References|[2]]]) for mappings of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480001.png" />.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480002.png" /> be a continuous mapping. By transition, if necessary, to a homotopic mapping, one may assume that this mapping is simplicial with respect to certain triangulations of the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480003.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480004.png" />. Then the Hopf invariant is defined as the [[Linking coefficient|linking coefficient]] of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480005.png" />-dimensional disjoint submanifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480006.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480007.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480008.png" /> for any distinct <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h0480009.png" />.
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The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800010.png" /> determines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800011.png" />, and the image of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800012.png" /> under the homomorphism
+
An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([[#References|[1]]], [[#References|[2]]]) for mappings of spheres  $  f: S ^ {2n - 1 } \rightarrow S  ^ {n} $.
  
<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/h048/h048000/h04800013.png" /></td> </tr></table>
+
Let  $  f: S ^ {2n - 1 } \rightarrow S  ^ {n} $
 +
be a continuous mapping. By transition, if necessary, to a homotopic mapping, one may assume that this mapping is simplicial with respect to certain triangulations of the spheres  $  S  ^ {n} $
 +
and  $  S ^ {2n - 1 } $.  
 +
Then the Hopf invariant is defined as the [[Linking coefficient|linking coefficient]] of the  $  ( n - 1) $-
 +
dimensional disjoint submanifolds  $  f ^ { * } ( a) $
 +
and  $  f ^ { * } ( b) $
 +
in  $  S ^ {2n - 1 } $
 +
for any distinct  $  a, b \in S  ^ {n} $.
  
coincides with the Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800014.png" /> (here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800015.png" /> is the Hurewicz homomorphism) [[#References|[3]]].
+
The mapping  $  f: S ^ {2n - 1 } \rightarrow S  ^ {n} $
 +
determines an element  $  [ f] \in \pi _ {2n - 1 }  ( S  ^ {n} ) $,
 +
and the image of the element  $  [ f] $
 +
under the homomorphism
  
Suppose now that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800016.png" /> is a mapping of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800017.png" /> and that a form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800018.png" /> is a generator of the integral cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800019.png" />. For such a form one may take, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800021.png" /> is the volume element on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800022.png" /> in some metric (for example, in the metric given by the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800023.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800024.png" /> is the volume of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800025.png" />. Then the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800026.png" /> is closed and it is exact because the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800027.png" /> is trivial. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800028.png" /> for some form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800029.png" />. A formula for the computation of the Hopf invariant is (see [[#References|[4]]]):
+
$$
 +
\pi _ {2n - 1 }  ( S  ^ {n} )  = \
 +
\pi _ {2n - 2 }  ( \Omega S  ^ {n} ) \rightarrow ^ { h }  \
 +
H _ {2n - 2 }  ( \Omega S  ^ {n} ) =  \mathbf Z
 +
$$
  
<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/h048/h048000/h04800030.png" /></td> </tr></table>
+
coincides with the Hopf invariant  $  H ( f  ) $(
 +
here  $  h $
 +
is the Hurewicz homomorphism) [[#References|[3]]].
  
The definition of the Hopf invariant can be generalized (see [[#References|[5]]], [[#References|[6]]]) to the case of mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800032.png" />. In this case there is a decomposition
+
Suppose now that  $  f:  S ^ {2n - 1 } \rightarrow S  ^ {n} $
 +
is a mapping of class  $  C  ^ {2} $
 +
and that a form  $  \Omega \in \Lambda  ^ {n} S  ^ {n} $
 +
is a generator of the integral cohomology group  $  H  ^ {n} ( S  ^ {n} , \mathbf Z ) $.
 +
For such a form one may take, for example,  $  \Omega = ( dV)/(  \mathop{\rm vol}  S  ^ {n} ) $,
 +
where  $  dV $
 +
is the volume element on  $  S  ^ {n} $
 +
in some metric (for example, in the metric given by the imbedding  $  S  ^ {n} \subset  \mathbf R ^ {n + 1 } $),
 +
and  $  \mathop{\rm vol}  S  ^ {n} $
 +
is the volume of the sphere  $  S  ^ {n} $.
 +
Then the form  $  f ^ { * } ( \Omega ) \in \Lambda  ^ {n} S ^ {2n - 1 } $
 +
is closed and it is exact because the group  $  H  ^ {n} ( S ^ {2n - 1 } , \mathbf Z ) $
 +
is trivial. Thus,  $  f ^ { * } ( \Omega ) = d \theta $
 +
for some form  $  \theta \in \Lambda ^ {n - 1 } S ^ {2n - 1 } $.
 +
A formula for the computation of the Hopf invariant is (see [[#References|[4]]]):
  
<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/h048/h048000/h04800033.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
H ( f  )  = \
 +
\int\limits _ {S ^ {2n - 1 } }
 +
\theta \wedge d \theta .
 +
$$
 +
 
 +
The definition of the Hopf invariant can be generalized (see [[#References|[5]]], [[#References|[6]]]) to the case of mappings  $  f: S  ^ {m} \rightarrow S  ^ {n} $
 +
for  $  m \leq  4n - 4 $.  
 +
In this case there is a decomposition
 +
 
 +
$$ \tag{* }
 +
\pi _ {m} ( S  ^ {n} \lor S  ^ {n} )  = \
 +
\pi _ {m} ( S  ^ {n} ) \oplus
 +
\pi _ {m} ( S  ^ {n} ) \oplus
 +
\pi _ {m} ( S ^ {2n - 1 } ) \oplus
 +
\mathop{\rm ker}  k _ {*} ,
 +
$$
  
 
where
 
where
  
<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/h048/h048000/h04800034.png" /></td> </tr></table>
+
$$
 +
k _ {*} : \pi _ {m + 1 }
 +
( S  ^ {n} \times S  ^ {n} ,\
 +
S  ^ {n} \lor S  ^ {n} )  \rightarrow \
 +
\pi _ {m + 1 }  ( S  ^ {2n} )
 +
$$
  
is the homomorphism induced by the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800035.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800036.png" /> be the mapping given by contracting the equator of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800037.png" /> to a point. Then the Hopf invariant is defined as the homomorphism
+
is the homomorphism induced by the projection $  k: ( S  ^ {n} \times S  ^ {n} , S  ^ {n} \lor S  ^ {n} ) \rightarrow ( S  ^ {n} ,  \mathop{\rm pt} ) $.  
 +
Let $  g: S  ^ {n} \rightarrow S  ^ {n} \lor S  ^ {n} $
 +
be the mapping given by contracting the equator of the sphere $  S  ^ {n} $
 +
to a point. Then the Hopf invariant is defined as the homomorphism
  
<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/h048/h048000/h04800038.png" /></td> </tr></table>
+
$$
 +
H: \pi _ {m} ( S  ^ {n} )  \rightarrow  \pi _ {m} ( S ^ {2n - 1 } )
 +
$$
  
under which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800039.png" /> is transformed to the projection of the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800040.png" /> onto the direct summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800041.png" /> in the decomposition (*). Since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800042.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800043.png" /> one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800044.png" /> of the homomorphisms
+
under which $  [ f] \in \pi _ {m} ( S  ^ {n} ) $
 +
is transformed to the projection of the element $  [ g \circ f] \in \pi _ {m} ( S  ^ {n} \lor S  ^ {n} ) $
 +
onto the direct summand $  \pi _ {m} ( S ^ {2n - 1 } ) $
 +
in the decomposition (*). Since $  \pi _ {2n - 1 }  ( S ^ {2n - 1 } ) = \mathbf Z $,  
 +
for $  m = 2n - 1 $
 +
one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite $  H  ^ {*} $
 +
of the homomorphisms
  
<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/h048/h048000/h04800045.png" /></td> </tr></table>
+
$$
 +
\pi _ {m} ( S  ^ {n} )  \rightarrow ^ { {g _ *} } \
 +
\pi _ {m} ( S  ^ {n} \lor S  ^ {n} )  \rightarrow ^ { p }  \
 +
$$
  
<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/h048/h048000/h04800046.png" /></td> </tr></table>
+
$$
 +
\rightarrow ^ { p }  \pi _ { m + 1 } ( S  ^ {n} \times S  ^ {n}
 +
, S  ^ {n} \lor S  ^ {n} )  \rightarrow ^ { {k _ * } }  \pi _ {m + 1 }  ( S  ^ {2n} ),
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800047.png" /> is the projection of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800048.png" /> onto the direct summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800049.png" />, and the homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800051.png" /> are described above. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800052.png" /> the Hopf–Whitehead invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800053.png" /> and the Hopf–Hilton invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800054.png" /> are connected by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800055.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800056.png" /> is the [[Suspension|suspension]] homomorphism (see [[#References|[6]]]).
+
where $  p $
 +
is the projection of the group $  \pi _ {m} ( S  ^ {n} \lor S  ^ {n} ) $
 +
onto the direct summand $  \pi _ {m + 1 }  ( S  ^ {n} \times S  ^ {n} , S  ^ {n} \lor S  ^ {n} ) $,  
 +
and the homomorphisms $  g _ {*} $
 +
and $  k _ {*} $
 +
are described above. For $  m \leq  4n - 4 $
 +
the Hopf–Whitehead invariant $  H $
 +
and the Hopf–Hilton invariant $  H  ^ {*} $
 +
are connected by the relation $  H  ^ {*} = S \circ H $,  
 +
where $  S: \pi _ {m} ( S ^ {2n - 1 } ) \rightarrow \pi _ {m + 1 }  ( S  ^ {2n} ) $
 +
is the [[Suspension|suspension]] homomorphism (see [[#References|[6]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800057.png" /> be a mapping and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800058.png" /> be its cylinder (cf. [[Mapping cylinder|Mapping cylinder]]). Then the cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800059.png" /> has as homogeneous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800060.png" />-basis a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800061.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800063.png" />. Here the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800064.png" /> holds (see [[#References|[7]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800065.png" /> is odd, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800066.png" /> (because multiplication in cohomology is skew-commutative).
+
Let $  f: S ^ {2n - 1 } \rightarrow S  ^ {n} $
 +
be a mapping and let $  C _ {f} $
 +
be its cylinder (cf. [[Mapping cylinder|Mapping cylinder]]). Then the cohomology space $  H  ^ {*} ( C _ {f} , S ^ {2n - 1 } ) $
 +
has as homogeneous $  \mathbf Z $-
 +
basis a pair $  \{ a, b \} $
 +
with $  \mathop{\rm dim}  a = n $
 +
and $  \mathop{\rm dim}  b = 2n $.  
 +
Here the relation $  a  ^ {2} = H ( f  ) b $
 +
holds (see [[#References|[7]]]). If $  n $
 +
is odd, then $  H ( f  ) = 0 $(
 +
because multiplication in cohomology is skew-commutative).
  
There is (see [[#References|[8]]]) a generalization of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800067.png" /> be the semi-exact homotopy functor in the sense of Dold (see [[#References|[9]]]), given on the category of finite CW-complexes and taking values in a certain Abelian category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800068.png" />. Then the mapping of complexes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800069.png" /> determines an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800070.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800071.png" /> is the set of morphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800072.png" />. The Hopf–Adams invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800073.png" /> is defined when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800076.png" /> is the corresponding suspension mapping. In this case the sequence of cofibrations
+
There is (see [[#References|[8]]]) a generalization of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. [[Generalized cohomology theories|Generalized cohomology theories]]). Let $  k $
 +
be the semi-exact homotopy functor in the sense of Dold (see [[#References|[9]]]), given on the category of finite CW-complexes and taking values in a certain Abelian category $  A $.  
 +
Then the mapping of complexes $  f: X \rightarrow Y $
 +
determines an element $  f ^ { * } = d ( f  ) \in  \mathop{\rm Hom} ( k ( Y), k ( X)) $,  
 +
where $  \mathop{\rm Hom} $
 +
is the set of morphisms in $  A $.  
 +
The Hopf–Adams invariant $  e ( f  ) $
 +
is defined when $  f ^ { * } = 0 $
 +
and $  d ( Sf  ) = 0 $,  
 +
where $  Sf: SX \rightarrow SY $
 +
is the corresponding suspension mapping. In this case the sequence of cofibrations
  
<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/h048/h048000/h04800077.png" /></td> </tr></table>
+
$$
 +
X  \rightarrow ^ { f }  \
 +
Y  \rightarrow ^ { f }  \
 +
Y \cup _ {f} CX  \rightarrow ^ { j }  \
 +
SX  \mathop \rightarrow \limits ^ {-}  Sf \
 +
SY
 +
$$
  
corresponds to an exact sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800078.png" />:
+
corresponds to an exact sequence in $  A $:
  
<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/h048/h048000/h04800079.png" /></td> </tr></table>
+
$$
 +
0 \leftarrow  k ( X)  \leftarrow ^ { {i _ *} } \
 +
k ( Y \cup _ {f} CX)  \leftarrow ^ { {j _ *} } \
 +
k ( SX)  \leftarrow  0,
 +
$$
  
which determines the Hopf–Adams–Steenrod invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800080.png" />.
+
which determines the Hopf–Adams–Steenrod invariant $  e ( f  ) = \mathop{\rm Ext}  ^ {1} ( k ( Y), k ( X)) $.
  
In the case of the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800081.png" /> taking values in the category of modules over the [[Steenrod algebra|Steenrod algebra]] modulo 2, one obtains the Hopf–Steenrod invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800082.png" /> of a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800083.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800084.png" /> (see [[#References|[7]]]). The cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800085.png" /> has as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800086.png" />-basis a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800089.png" />, and then
+
In the case of the functor $  k = H  ^ {*} ( - ;  \mathbf Z _ {2} ) $
 +
taking values in the category of modules over the [[Steenrod algebra|Steenrod algebra]] modulo 2, one obtains the Hopf–Steenrod invariant $  H _ {2} ( f  ) \in \mathbf Z $
 +
of a mapping $  f: S  ^ {m} \rightarrow S  ^ {n} $
 +
for $  m > n $(
 +
see [[#References|[7]]]). The cohomology space $  H  ^ {*} ( C _ {f} , S  ^ {m} ;  \mathbf Z _ {2} ) $
 +
has as $  \mathbf Z _ {2} $-
 +
basis a pair $  \{ a, b \} $
 +
with $  \mathop{\rm dim}  a = n $
 +
and $  \mathop{\rm dim}  b = m + 1 $,  
 +
and then
  
<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/h048/h048000/h04800090.png" /></td> </tr></table>
+
$$
 +
Sq ^ {m - n + 1 } a  = \
 +
H _ {2} ( f  ) b.
 +
$$
  
The Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800091.png" /> modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800092.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800093.png" /> is a prime number) is defined as the composite of the mappings
+
The Hopf invariant $  H _ {p} $
 +
modulo $  p $(
 +
where $  p $
 +
is a prime number) is defined as the composite of the mappings
  
<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/h048/h048000/h04800094.png" /></td> </tr></table>
+
$$
 +
\pi _ {2pn} ( S ^ {2n + 1 } ) _ {(} p)  \mathop \rightarrow \limits ^  \approx  \
 +
\pi _ {2pn - 2 }  ( \Omega  ^ {2} S ^ {2n + 1 } ) _ {(} p) \rightarrow
 +
$$
  
<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/h048/h048000/h04800095.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
\pi _ {2pn - 2 }  ( \Omega  ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) \rightarrow
 +
$$
  
<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/h048/h048000/h04800096.png" /></td> </tr></table>
+
$$
 +
\rightarrow \
 +
H _ {2pn - 2 }  ( \Omega  ^ {2} S ^ {2n +
 +
1 } , S ^ {2n - 1 } ) _ {(} p)  = \mathbf Z /p,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800097.png" /> is the localization by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h04800098.png" /> of the pair of spaces (see [[#References|[10]]]). Let
+
where $  ( X, Y) _ {p} $
 +
is the localization by $  p $
 +
of the pair of spaces (see [[#References|[10]]]). Let
  
<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/h048/h048000/h04800099.png" /></td> </tr></table>
+
$$
 +
S: \pi _ {4n - 1 }  ( S  ^ {2n} )  \rightarrow \
 +
\pi _ {4n} ( S ^ {2n + 1 } )
 +
$$
  
be the suspension homomorphism. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000100.png" /> (see [[#References|[10]]]). The Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000101.png" /> can also be defined in terms of the Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) (see [[#References|[11]]]): If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000102.png" /> is a closed equipped manifold and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000103.png" />, then the characteristic Stiefel–Whitney number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000104.png" /> of the normal bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000105.png" /> is the same as the Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000106.png" /> of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000107.png" /> that is a representative of the class of equipped cobordisms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000108.png" />.
+
be the suspension homomorphism. Then $  H _ {2} ( Sf  ) = H _ {2} ( f  ) $(
 +
see [[#References|[10]]]). The Hopf invariant $  H ( f  ) $
 +
can also be defined in terms of the Stiefel numbers (cf. [[Stiefel number|Stiefel number]]) (see [[#References|[11]]]): If $  M ^ {n - 1 } $
 +
is a closed equipped manifold and if $  M ^ {n - 1 } = \partial  V $,  
 +
then the characteristic Stiefel–Whitney number $  w _ {n} ( \nu ) [ V, M] $
 +
of the normal bundle $  \nu $
 +
is the same as the Hopf invariant $  H _ {2} ( f  ) $
 +
of the mapping $  f: S ^ {n + r - 1 } \rightarrow S  ^ {r} $
 +
that is a representative of the class of equipped cobordisms of $  M ^ {n - 1 } $.
  
The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000110.png" /> (see [[#References|[12]]]). From the form of the differentials of this spectral sequence it follows that
+
The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants $  q _ {i} :   \mathop{\rm ker}  q _ {i - 1 }  \rightarrow E _  \infty  ^ {i,*} $
 +
and $  q _ {0} : \pi _ {*}  ^ {S} \rightarrow E _  \infty  ^ {0,*} $(
 +
see [[#References|[12]]]). From the form of the differentials of this spectral sequence it follows that
  
<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/h048/h048000/h048000111.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm Ext} _ {AU} ^ {i, * }
 +
( \Omega _ {U} , \Omega _ {U} )  \supset \
 +
E _  \infty  ^ {i, * } ,\ \
 +
i = 0, 1, 2, 3
 +
$$
  
(where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000112.png" /> is the ring of complex point cobordisms); therefore, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000113.png" />, the invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000114.png" /> lie in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000115.png" /> and are called the Hopf–Novikov invariants. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000116.png" /> one obtains the Adams invariant.
+
(where $  \Omega _ {U} $
 +
is the ring of complex point cobordisms); therefore, for $  i = 0, 1, 2, 3 $,
 +
the invariants $  q _ {i} $
 +
lie in $  \mathop{\rm Ext} _ {AU}  ^ {i,*} ( \Omega _ {U} , \Omega _ {U} ) $
 +
and are called the Hopf–Novikov invariants. For $  i = 1 $
 +
one obtains the Adams invariant.
  
The values that a Hopf invariant can take are not arbitrary. For example, for a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000117.png" /> the Hopf invariant is always 0. The Hopf invariant modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000118.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000119.png" />, is trivial, except when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000120.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000121.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000122.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000123.png" />. On the other hand, for any even number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000124.png" /> there exists a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000125.png" /> with Hopf invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000126.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000127.png" /> is arbitrary). For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000128.png" /> there exists mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000129.png" /> with Hopf invariant 1.
+
The values that a Hopf invariant can take are not arbitrary. For example, for a mapping $  f: S ^ {4n + 1 } \rightarrow S ^ {2n + 1 } $
 +
the Hopf invariant is always 0. The Hopf invariant modulo $  p $,  
 +
$  H _ {(} p) : \pi _ {2mp} ( S ^ {2m + 1 } ) \rightarrow \mathbf Z _ {p} $,  
 +
is trivial, except when $  p = 2 $,
 +
$  m = 1, 2, 4 $
 +
and $  p > 2 $,  
 +
$  m = 1 $.  
 +
On the other hand, for any even number $  k $
 +
there exists a mapping $  f: S ^ {4n - 1 } \rightarrow S  ^ {2n} $
 +
with Hopf invariant $  k $(
 +
$  n $
 +
is arbitrary). For $  n = 1, 2, 4 $
 +
there exists mappings $  f: S ^ {4n - 1 } \rightarrow S  ^ {2n} $
 +
with Hopf invariant 1.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,  "Ueber die Abbildungen der dreidimensionalen Sphäre auf die Kügelfläche"  ''Math. Ann.'' , '''104'''  (1931)  pp. 639–665</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  J.-.P. Serre,  "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.H.C. Whitehead,  "An expression of the Hopf invariant as an integral"  ''Proc. Nat. Acad. Sci. USA'' , '''33'''  (1937)  pp. 117–123</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.H.C. Whitehead,  "A generation of the Hopf invariant"  ''Ann. of Math. (2)'' , '''51'''  (1950)  pp. 192–237</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Hilton,  "Suspension theorem and generalized Hopf invariant"  ''Proc. London. Math. Soc. (3)'' , '''1''' :  3  (1951)  pp. 462–493</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N. Steenrod,  "Cohomologies invariants of mappings"  ''Ann. of Math. (2)'' , '''50'''  (1949)  pp. 954–988</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Adams,  "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000130.png" />"  ''Topology'' , '''5'''  (1966)  pp. 21–71</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Dold,  "Halbexakte Homotopiefunktoren" , Springer  (1966)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the view point of cobordism theories"  ''Math. USSR-Izv.'' , '''4''' :  1  (1967)  pp. 827–913  ''Izv. AKad. Nauk SSSR Ser. Mat.'' , '''31''' :  4  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J.F. Adams,  "On the non-existence of elements of Hopf invariant one"  ''Ann. of Math.'' , '''72'''  (1960)  pp. 20–104</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Hopf,  "Ueber die Abbildungen der dreidimensionalen Sphäre auf die Kügelfläche"  ''Math. Ann.'' , '''104'''  (1931)  pp. 639–665</TD></TR><TR><TD valign="top">[2]</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">[3]</TD> <TD valign="top">  J.-.P. Serre,  "Groupes d'homotopie et classes de groupes abéliens"  ''Ann. of Math.'' , '''58''' :  2  (1953)  pp. 258–294</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.H.C. Whitehead,  "An expression of the Hopf invariant as an integral"  ''Proc. Nat. Acad. Sci. USA'' , '''33'''  (1937)  pp. 117–123</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.H.C. Whitehead,  "A generation of the Hopf invariant"  ''Ann. of Math. (2)'' , '''51'''  (1950)  pp. 192–237</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Hilton,  "Suspension theorem and generalized Hopf invariant"  ''Proc. London. Math. Soc. (3)'' , '''1''' :  3  (1951)  pp. 462–493</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  N. Steenrod,  "Cohomologies invariants of mappings"  ''Ann. of Math. (2)'' , '''50'''  (1949)  pp. 954–988</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  J. Adams,  "On the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h048/h048000/h048000130.png" />"  ''Topology'' , '''5'''  (1966)  pp. 21–71</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A. Dold,  "Halbexakte Homotopiefunktoren" , Springer  (1966)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Husemoller,  "Fibre bundles" , McGraw-Hill  (1966)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  R.E. Stong,  "Notes on cobordism theory" , Princeton Univ. Press  (1968)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the view point of cobordism theories"  ''Math. USSR-Izv.'' , '''4''' :  1  (1967)  pp. 827–913  ''Izv. AKad. Nauk SSSR Ser. Mat.'' , '''31''' :  4  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  J.F. Adams,  "On the non-existence of elements of Hopf invariant one"  ''Ann. of Math.'' , '''72'''  (1960)  pp. 20–104</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([1], [2]) for mappings of spheres $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $.

Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ be a continuous mapping. By transition, if necessary, to a homotopic mapping, one may assume that this mapping is simplicial with respect to certain triangulations of the spheres $ S ^ {n} $ and $ S ^ {2n - 1 } $. Then the Hopf invariant is defined as the linking coefficient of the $ ( n - 1) $- dimensional disjoint submanifolds $ f ^ { * } ( a) $ and $ f ^ { * } ( b) $ in $ S ^ {2n - 1 } $ for any distinct $ a, b \in S ^ {n} $.

The mapping $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ determines an element $ [ f] \in \pi _ {2n - 1 } ( S ^ {n} ) $, and the image of the element $ [ f] $ under the homomorphism

$$ \pi _ {2n - 1 } ( S ^ {n} ) = \ \pi _ {2n - 2 } ( \Omega S ^ {n} ) \rightarrow ^ { h } \ H _ {2n - 2 } ( \Omega S ^ {n} ) = \mathbf Z $$

coincides with the Hopf invariant $ H ( f ) $( here $ h $ is the Hurewicz homomorphism) [3].

Suppose now that $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ is a mapping of class $ C ^ {2} $ and that a form $ \Omega \in \Lambda ^ {n} S ^ {n} $ is a generator of the integral cohomology group $ H ^ {n} ( S ^ {n} , \mathbf Z ) $. For such a form one may take, for example, $ \Omega = ( dV)/( \mathop{\rm vol} S ^ {n} ) $, where $ dV $ is the volume element on $ S ^ {n} $ in some metric (for example, in the metric given by the imbedding $ S ^ {n} \subset \mathbf R ^ {n + 1 } $), and $ \mathop{\rm vol} S ^ {n} $ is the volume of the sphere $ S ^ {n} $. Then the form $ f ^ { * } ( \Omega ) \in \Lambda ^ {n} S ^ {2n - 1 } $ is closed and it is exact because the group $ H ^ {n} ( S ^ {2n - 1 } , \mathbf Z ) $ is trivial. Thus, $ f ^ { * } ( \Omega ) = d \theta $ for some form $ \theta \in \Lambda ^ {n - 1 } S ^ {2n - 1 } $. A formula for the computation of the Hopf invariant is (see [4]):

$$ H ( f ) = \ \int\limits _ {S ^ {2n - 1 } } \theta \wedge d \theta . $$

The definition of the Hopf invariant can be generalized (see [5], [6]) to the case of mappings $ f: S ^ {m} \rightarrow S ^ {n} $ for $ m \leq 4n - 4 $. In this case there is a decomposition

$$ \tag{* } \pi _ {m} ( S ^ {n} \lor S ^ {n} ) = \ \pi _ {m} ( S ^ {n} ) \oplus \pi _ {m} ( S ^ {n} ) \oplus \pi _ {m} ( S ^ {2n - 1 } ) \oplus \mathop{\rm ker} k _ {*} , $$

where

$$ k _ {*} : \pi _ {m + 1 } ( S ^ {n} \times S ^ {n} ,\ S ^ {n} \lor S ^ {n} ) \rightarrow \ \pi _ {m + 1 } ( S ^ {2n} ) $$

is the homomorphism induced by the projection $ k: ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) \rightarrow ( S ^ {n} , \mathop{\rm pt} ) $. Let $ g: S ^ {n} \rightarrow S ^ {n} \lor S ^ {n} $ be the mapping given by contracting the equator of the sphere $ S ^ {n} $ to a point. Then the Hopf invariant is defined as the homomorphism

$$ H: \pi _ {m} ( S ^ {n} ) \rightarrow \pi _ {m} ( S ^ {2n - 1 } ) $$

under which $ [ f] \in \pi _ {m} ( S ^ {n} ) $ is transformed to the projection of the element $ [ g \circ f] \in \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ onto the direct summand $ \pi _ {m} ( S ^ {2n - 1 } ) $ in the decomposition (*). Since $ \pi _ {2n - 1 } ( S ^ {2n - 1 } ) = \mathbf Z $, for $ m = 2n - 1 $ one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite $ H ^ {*} $ of the homomorphisms

$$ \pi _ {m} ( S ^ {n} ) \rightarrow ^ { {g _ *} } \ \pi _ {m} ( S ^ {n} \lor S ^ {n} ) \rightarrow ^ { p } \ $$

$$ \rightarrow ^ { p } \pi _ { m + 1 } ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) \rightarrow ^ { {k _ * } } \pi _ {m + 1 } ( S ^ {2n} ), $$

where $ p $ is the projection of the group $ \pi _ {m} ( S ^ {n} \lor S ^ {n} ) $ onto the direct summand $ \pi _ {m + 1 } ( S ^ {n} \times S ^ {n} , S ^ {n} \lor S ^ {n} ) $, and the homomorphisms $ g _ {*} $ and $ k _ {*} $ are described above. For $ m \leq 4n - 4 $ the Hopf–Whitehead invariant $ H $ and the Hopf–Hilton invariant $ H ^ {*} $ are connected by the relation $ H ^ {*} = S \circ H $, where $ S: \pi _ {m} ( S ^ {2n - 1 } ) \rightarrow \pi _ {m + 1 } ( S ^ {2n} ) $ is the suspension homomorphism (see [6]).

Let $ f: S ^ {2n - 1 } \rightarrow S ^ {n} $ be a mapping and let $ C _ {f} $ be its cylinder (cf. Mapping cylinder). Then the cohomology space $ H ^ {*} ( C _ {f} , S ^ {2n - 1 } ) $ has as homogeneous $ \mathbf Z $- basis a pair $ \{ a, b \} $ with $ \mathop{\rm dim} a = n $ and $ \mathop{\rm dim} b = 2n $. Here the relation $ a ^ {2} = H ( f ) b $ holds (see [7]). If $ n $ is odd, then $ H ( f ) = 0 $( because multiplication in cohomology is skew-commutative).

There is (see [8]) a generalization of the Hopf–Steenrod invariant in terms of a generalized cohomology theory (cf. Generalized cohomology theories). Let $ k $ be the semi-exact homotopy functor in the sense of Dold (see [9]), given on the category of finite CW-complexes and taking values in a certain Abelian category $ A $. Then the mapping of complexes $ f: X \rightarrow Y $ determines an element $ f ^ { * } = d ( f ) \in \mathop{\rm Hom} ( k ( Y), k ( X)) $, where $ \mathop{\rm Hom} $ is the set of morphisms in $ A $. The Hopf–Adams invariant $ e ( f ) $ is defined when $ f ^ { * } = 0 $ and $ d ( Sf ) = 0 $, where $ Sf: SX \rightarrow SY $ is the corresponding suspension mapping. In this case the sequence of cofibrations

$$ X \rightarrow ^ { f } \ Y \rightarrow ^ { f } \ Y \cup _ {f} CX \rightarrow ^ { j } \ SX \mathop \rightarrow \limits ^ {-} Sf \ SY $$

corresponds to an exact sequence in $ A $:

$$ 0 \leftarrow k ( X) \leftarrow ^ { {i _ *} } \ k ( Y \cup _ {f} CX) \leftarrow ^ { {j _ *} } \ k ( SX) \leftarrow 0, $$

which determines the Hopf–Adams–Steenrod invariant $ e ( f ) = \mathop{\rm Ext} ^ {1} ( k ( Y), k ( X)) $.

In the case of the functor $ k = H ^ {*} ( - ; \mathbf Z _ {2} ) $ taking values in the category of modules over the Steenrod algebra modulo 2, one obtains the Hopf–Steenrod invariant $ H _ {2} ( f ) \in \mathbf Z $ of a mapping $ f: S ^ {m} \rightarrow S ^ {n} $ for $ m > n $( see [7]). The cohomology space $ H ^ {*} ( C _ {f} , S ^ {m} ; \mathbf Z _ {2} ) $ has as $ \mathbf Z _ {2} $- basis a pair $ \{ a, b \} $ with $ \mathop{\rm dim} a = n $ and $ \mathop{\rm dim} b = m + 1 $, and then

$$ Sq ^ {m - n + 1 } a = \ H _ {2} ( f ) b. $$

The Hopf invariant $ H _ {p} $ modulo $ p $( where $ p $ is a prime number) is defined as the composite of the mappings

$$ \pi _ {2pn} ( S ^ {2n + 1 } ) _ {(} p) \mathop \rightarrow \limits ^ \approx \ \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } ) _ {(} p) \rightarrow $$

$$ \rightarrow \ \pi _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) \rightarrow $$

$$ \rightarrow \ H _ {2pn - 2 } ( \Omega ^ {2} S ^ {2n + 1 } , S ^ {2n - 1 } ) _ {(} p) = \mathbf Z /p, $$

where $ ( X, Y) _ {p} $ is the localization by $ p $ of the pair of spaces (see [10]). Let

$$ S: \pi _ {4n - 1 } ( S ^ {2n} ) \rightarrow \ \pi _ {4n} ( S ^ {2n + 1 } ) $$

be the suspension homomorphism. Then $ H _ {2} ( Sf ) = H _ {2} ( f ) $( see [10]). The Hopf invariant $ H ( f ) $ can also be defined in terms of the Stiefel numbers (cf. Stiefel number) (see [11]): If $ M ^ {n - 1 } $ is a closed equipped manifold and if $ M ^ {n - 1 } = \partial V $, then the characteristic Stiefel–Whitney number $ w _ {n} ( \nu ) [ V, M] $ of the normal bundle $ \nu $ is the same as the Hopf invariant $ H _ {2} ( f ) $ of the mapping $ f: S ^ {n + r - 1 } \rightarrow S ^ {r} $ that is a representative of the class of equipped cobordisms of $ M ^ {n - 1 } $.

The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants $ q _ {i} : \mathop{\rm ker} q _ {i - 1 } \rightarrow E _ \infty ^ {i,*} $ and $ q _ {0} : \pi _ {*} ^ {S} \rightarrow E _ \infty ^ {0,*} $( see [12]). From the form of the differentials of this spectral sequence it follows that

$$ \mathop{\rm Ext} _ {AU} ^ {i, * } ( \Omega _ {U} , \Omega _ {U} ) \supset \ E _ \infty ^ {i, * } ,\ \ i = 0, 1, 2, 3 $$

(where $ \Omega _ {U} $ is the ring of complex point cobordisms); therefore, for $ i = 0, 1, 2, 3 $, the invariants $ q _ {i} $ lie in $ \mathop{\rm Ext} _ {AU} ^ {i,*} ( \Omega _ {U} , \Omega _ {U} ) $ and are called the Hopf–Novikov invariants. For $ i = 1 $ one obtains the Adams invariant.

The values that a Hopf invariant can take are not arbitrary. For example, for a mapping $ f: S ^ {4n + 1 } \rightarrow S ^ {2n + 1 } $ the Hopf invariant is always 0. The Hopf invariant modulo $ p $, $ H _ {(} p) : \pi _ {2mp} ( S ^ {2m + 1 } ) \rightarrow \mathbf Z _ {p} $, is trivial, except when $ p = 2 $, $ m = 1, 2, 4 $ and $ p > 2 $, $ m = 1 $. On the other hand, for any even number $ k $ there exists a mapping $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ with Hopf invariant $ k $( $ n $ is arbitrary). For $ n = 1, 2, 4 $ there exists mappings $ f: S ^ {4n - 1 } \rightarrow S ^ {2n} $ with Hopf invariant 1.

References

[1] H. Hopf, "Ueber die Abbildungen der dreidimensionalen Sphäre auf die Kügelfläche" Math. Ann. , 104 (1931) pp. 639–665
[2] H. Hopf, "Ueber die Abbildungen von Sphären niedriger Dimension" Fund. Math. , 25 (1935) pp. 427–440
[3] J.-.P. Serre, "Groupes d'homotopie et classes de groupes abéliens" Ann. of Math. , 58 : 2 (1953) pp. 258–294
[4] J.H.C. Whitehead, "An expression of the Hopf invariant as an integral" Proc. Nat. Acad. Sci. USA , 33 (1937) pp. 117–123
[5] J.H.C. Whitehead, "A generation of the Hopf invariant" Ann. of Math. (2) , 51 (1950) pp. 192–237
[6] P. Hilton, "Suspension theorem and generalized Hopf invariant" Proc. London. Math. Soc. (3) , 1 : 3 (1951) pp. 462–493
[7] N. Steenrod, "Cohomologies invariants of mappings" Ann. of Math. (2) , 50 (1949) pp. 954–988
[8] J. Adams, "On the groups " Topology , 5 (1966) pp. 21–71
[9] A. Dold, "Halbexakte Homotopiefunktoren" , Springer (1966)
[10] D. Husemoller, "Fibre bundles" , McGraw-Hill (1966)
[11] R.E. Stong, "Notes on cobordism theory" , Princeton Univ. Press (1968)
[12] S.P. Novikov, "The methods of algebraic topology from the view point of cobordism theories" Math. USSR-Izv. , 4 : 1 (1967) pp. 827–913 Izv. AKad. Nauk SSSR Ser. Mat. , 31 : 4 (1967) pp. 855–951
[13] J.F. Adams, "On the non-existence of elements of Hopf invariant one" Ann. of Math. , 72 (1960) pp. 20–104
How to Cite This Entry:
Hopf invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_invariant&oldid=11380
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article