Let 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 and . Then the Hopf invariant is defined as the linking coefficient of the -dimensional disjoint submanifolds and in for any distinct .
The mapping determines an element , and the image of the element under the homomorphism
coincides with the Hopf invariant (here is the Hurewicz homomorphism) .
Suppose now that is a mapping of class and that a form is a generator of the integral cohomology group . For such a form one may take, for example, , where is the volume element on in some metric (for example, in the metric given by the imbedding ), and is the volume of the sphere . Then the form is closed and it is exact because the group is trivial. Thus, for some form . A formula for the computation of the Hopf invariant is (see ):
is the homomorphism induced by the projection . Let be the mapping given by contracting the equator of the sphere to a point. Then the Hopf invariant is defined as the homomorphism
under which is transformed to the projection of the element onto the direct summand in the decomposition (*). Since , for one obtains the usual Hopf invariant. The generalized Hopf invariant is defined as the composite of the homomorphisms
where is the projection of the group onto the direct summand , and the homomorphisms and are described above. For the Hopf–Whitehead invariant and the Hopf–Hilton invariant are connected by the relation , where is the suspension homomorphism (see ).
Let be a mapping and let be its cylinder (cf. Mapping cylinder). Then the cohomology space has as homogeneous -basis a pair with and . Here the relation holds (see ). If is odd, then (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 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 . Then the mapping of complexes determines an element , where is the set of morphisms in . The Hopf–Adams invariant is defined when and , where is the corresponding suspension mapping. In this case the sequence of cofibrations
corresponds to an exact sequence in :
which determines the Hopf–Adams–Steenrod invariant .
In the case of the functor taking values in the category of modules over the Steenrod algebra modulo 2, one obtains the Hopf–Steenrod invariant of a mapping for (see ). The cohomology space has as -basis a pair with and , and then
The Hopf invariant modulo (where is a prime number) is defined as the composite of the mappings
where is the localization by of the pair of spaces (see ). Let
be the suspension homomorphism. Then (see ). The Hopf invariant can also be defined in terms of the Stiefel numbers (cf. Stiefel number) (see ): If is a closed equipped manifold and if , then the characteristic Stiefel–Whitney number of the normal bundle is the same as the Hopf invariant of the mapping that is a representative of the class of equipped cobordisms of .
The Adams–Novikov spectral sequence makes it possible to construct higher Hopf invariants. Namely, one defines inductively the invariants and (see ). From the form of the differentials of this spectral sequence it follows that
(where is the ring of complex point cobordisms); therefore, for , the invariants lie in and are called the Hopf–Novikov invariants. For one obtains the Adams invariant.
The values that a Hopf invariant can take are not arbitrary. For example, for a mapping the Hopf invariant is always 0. The Hopf invariant modulo , , is trivial, except when , and , . On the other hand, for any even number there exists a mapping with Hopf invariant ( is arbitrary). For there exists mappings with Hopf invariant 1.
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Hopf invariant. A.V. Shokurov (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hopf_invariant&oldid=11380