Hopf invariant
An invariant of a homotopy class of mappings of topological spaces. It was first defined by H. Hopf ([1], [2]) for mappings of spheres .
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) [3].
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 [4]):
![]() |
The definition of the Hopf invariant can be generalized (see [5], [6]) to the case of mappings for
. In this case there is a decomposition
![]() | (*) |
where
![]() |
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 [6]).
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 [7]). If
is odd, then
(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 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
. 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 [7]). 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 [10]). Let
![]() |
be the suspension homomorphism. Then (see [10]). The Hopf invariant
can also be defined in terms of the Stiefel numbers (cf. Stiefel number) (see [11]): 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 [12]). 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.
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 ![]() |
[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 |
Hopf invariant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hopf_invariant&oldid=47270