The idea of this note comes from the chapter , H. Hopf, W.K. Clifford, F. Klein written by H. Samelson in I.M. James’ book, History of Topology, and more precisely from the quoted letter from Hopf to Freudhental:
In case you are still interested in the question of the [homotopy] classes of maps of the 3-sphere onto the 2-sphere I want to tell you that I now can answer this question: there exist infinitely many classes. Namely there is a class invariant of the given map the counter image of consists of finitely many simple closed oriented polygons , , …, and likewise the counter image of consists of polygons , , …, . If denotes the linking number of and , then is independent of and of the approximation and does not change under continuous change of the map. For every there exists maps. Whether to every there is only one map, I do not know. If the whole is not covered by the image, then is . A consequence is that one cannot sweep the line elements on continuously into a point.
Our aim is to developp this nice geometric argument following problems 13, 14 and 15 of Milnor’s book, Topology from differential viewpoint. Thus, our main result is:
Theorem: The group is infinite.
In fact, it may be proved that . As a consequence that may be non-trivial when is that the homotopy group is not determined by the CW complex structure, unlike homology and cohomology groups.
The first step is to define the linking number of two submanifolds of with total dimension . For that purpose, we use degree theory; for more information, see Milnor’s book.
Definition: Let be a smooth map between two -dimensional smooth compact oriented manifolds. If is a regular point, let be if is orientation-preserving and otherwise. The degree of with respect to a regular value is defined by
Property 1: The degree does not depend on the regular value .
In particular, property 1 allows us to define the degree of a smooth map without reference to any regular value.
Property 2: If two smooth maps and are smoothly homotopic, then .
Property 3: Let , be two smooth manifolds and be a smooth map. If there exists a smooth manifold whose boundary is and such that extends to a smooth map , then .
Properties 1 and 2 are fundamental in degree theory; property 3 is a rather technical lemma used to prove the two previous properties, but it will be useful later.
Definition: Let be two submanifolds of total dimension . The linking number is defined as the degree of the linking map
Property 4: .
Proof. Let be the linking maps
and let for convenience
Then . It is an easy exercice to show that when is a diffeomorphism, and that . So
Property 5: If bounds an oriented manifold disjoint from , then .
Proof. If is an oriented submanifold such that , then the map
extends the linking map , and , hence according to property 3.
If and are two knots in , it is possible to compute from a regular diagram.
More precisely, if and is the plane normal to , then the projection of on is a regular diagram if is a regular value of ; moreover, is exactly the crossing points of when is over . Therefore, where (resp. ) is the number of crossing points where is orientation-preserving (resp. orientation-reversing); on , such crossing points correspond respectively to
For example, the linking number computed on the following diagramm is two:
Before introducing Hopf invariant, let us mention some results about cobordism, needed in the sequel:
Some cobordism theory:
Definition: Two oriented compact manifolds are cobordant if there exists an oriented compact manifold , called a cobordism, such that where denotes with the reversed orientation.
Lemma 1: Let be a smooth map and be a regular value. If is a regular value sufficently close to , then there exists a cobordism between and .
Proof. If is the set of singular points of , is compact so there exists an open neighborhood of containing only regular values. Let .
Let be a great circle from to , and let be a rotation whose restriction to is a rotation sending to . In particular, if , defines an isotopy between and . Let
Because , is a regular value of and finally to since
Thus, defines a cobordism between and .
Lemma 2: If are smoothly homotopic and is a regular value for both, then there exists a cobordism between and .
Proof. Let be a homotopy between and . Let be a regular value of close to so that and , and and , are cobordant (using lemma 1). Then defines a cobordism between and . Lemma 2 follows since cobordism is an equivalent relation.
From now on, let be a smooth map and be two regular values. Applying a stereographic projection, we may view and as submanifolds of . Then the linking number is well-defined.
Claim 1: The linking number does not depend on the stereographic projection.
Proof. A stereographic projection is an orientation-preserving diffeomorphism, so if and are two stereographic projections, then induces an orientation-preserving diffeomorphism between and , hence
Claim 2: The linking number is locally constant as a function of .
Proof. Let the mapping maps
According to lemma 1, there exists a cobordism between and when is close to . Let
For convenience, let be the restriction of on , and (resp. ) be the restriction of on (resp. on ). According to property 3,
If are the obvious diffeomorphisms
then (), hence
But is orientation-preserving whereas is not. Therefore,
Claim 3: If are regular values of and , then
Proof. Let be the homotopy
is well-defined since implies
a contradiction. Therefore, and are smoothly homotopic. Using lemma 2, we prove in the same way as for claim 2 that the degrees of the associated linking maps are equal so that
Property 6: The linking number depends only on the homotopy class of .
Proof. According to claim 1, does not depend on the stereographic projection. According to claim 2 and property 4, the linking number does not depend on the regular values and by connectedness of . Finally, if is a smooth homotopy between and , then we deduce that
from claim 3 and property 4, taking a sequence satisfying for all .
Definition: We define the Hopf invariant of as the linking number for some regular values .
We just showed that Hopf invariant is a homotopic invariant.
Property 7: If and , then .
Proof. Let be two regular values of and , and let and . The inclusions
induce an orientation-preserving diffeomorphism between
so, if , we have:
The sphere can be viewed as the unit sphere in . Noticing that the intersection between and any complexe line is a circle, we can say that is covered by a family of circles indexed by .
More precisely, if then belongs to the complex line ; to this line (in ) is associated the complex number . Finally, if denotes the diffeomorphism induced by the stereographic projection, we get a point of the sphere .
Definition: Hopf map induces the Hopf fibration .
It is possible to visualize the decomposition of by circles in using stereographic projection. It gives something like that, a decomposition in concentric torii each covered by their Villarceau circles:
are two circles in . If is the stereographic projection with respect to , then , so the previous circles become
In fact, it is just a Hopf link; precisely, if and denotes the projections of the two previous circles on the plane , that is
then where is over at one point, and vice-versa at the other. Therefore, (be careful to the orientations!).
On the other hand, for every constant map . Indeed, let
where is the unit two-dimensional disk viewed as a subspace of ; in particular, since is contractible, is homotopic to a constant map and . But the circles and (viewed in thanks to the stereographic projection with respect to ) are clearly unlinked, hence according to property 5.
Moreover, using property 7, we deduce that the applications
define a family of pairwise non-homotopic maps, since . We just proved our main theorem!
Hopf fibration in higher dimensions:
The associated maps and induce respectively the Hopf fibrations and .
It can be shown (for example in Hatcher’s book, Algebraic topology, using a homological interpretation of Hopf invariant) that , hence:
Theorem: , and are infinite.
However, Hopf fibrations do not exist in other dimensions since a finite-dimensional real division algebra has dimension one, two, four or eight.