# Hopf invariant

An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf (, ) 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) .

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 ):

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

The definition of the Hopf invariant can be generalized (see , ) 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 ).

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 ). If $n$ is odd, then $H ( f ) = 0$( because multiplication in cohomology is skew-commutative).

There is (see ) 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 ), 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 ). 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 ). 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 ). The Hopf invariant $H ( f )$ can also be defined in terms of the Stiefel numbers (cf. Stiefel number) (see ): 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 ). 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.

How to Cite This Entry:
Hopf invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_invariant&oldid=47270
This article was adapted from an original article by A.V. Shokurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article