The idea of this note comes from the chapter \pi_3(S^2), 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 S^3 onto the 2-sphere S^2 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 x consists of finitely many simple closed oriented polygons P_1, P_2, …, P_a and likewise the counter image of y consists of polygons Q_1, Q_2, …, Q_b. If v_{ij} denotes the linking number of P_i and Q_j, then \sum_{i,j} v_{ij}= \gamma is independent of x, ~ y and of the approximation and does not change under continuous change of the map. For every \gamma there exists maps. Whether to every \gamma there is only one map, I do not know. If the whole S^2 is not covered by the image, then \gamma is =0. A consequence is that one cannot sweep the line elements on S^2 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 \pi_3(\mathbb{S}^2) is infinite.

In fact, it may be proved that \pi_3(\mathbb{S}^2) \simeq \mathbb{Z}. As a consequence that \pi_n(\mathbb{S}^m) may be non-trivial when n>m 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 \mathbb{R}^{k+1} with total dimension k. For that purpose, we use degree theory; for more information, see Milnor’s book.

Definition: Let f : M \to N be a smooth map between two n-dimensional smooth compact oriented manifolds. If p \in M is a regular point, let \mathrm{sign} (df(p)) be +1 if df(p) is orientation-preserving and -1 otherwise. The degree of f with respect to a regular value q \in N is defined by

\displaystyle \mathrm{deg}(f,q)= \sum\limits_{p \in f^{-1}(q)} \mathrm{sign}(df(p)).

Property 1: The degree \mathrm{deg}(f,q) does not depend on the regular value q.

In particular, property 1 allows us to define the degree \mathrm{deg}(f) of a smooth map f without reference to any regular value.

Property 2: If two smooth maps f and g are smoothly homotopic, then \mathrm{deg}(f)= \mathrm{deg}(g).

Property 3: Let M, N be two smooth manifolds and f : M \to N be a smooth map. If there exists a smooth manifold X whose boundary is M and such that f extends to a smooth map X \to N, then \mathrm{deg}(f)=0.

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.

Linking number:

Definition: Let M,N \subset \mathbb{R}^{k+1} be two submanifolds of total dimension m+n=k. The linking number l(M,N) is defined as the degree of the linking map

\lambda : \left\{ \begin{array}{ccc} M \times N & \to & \mathbb{S}^k \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right..

Property 4: l(M,N)= (-1)^{(m+1)(n+1)} l(N,M).

Proof. Let \lambda_1, \lambda_2 be the linking maps

\lambda_1 : \left\{ \begin{array}{ccc} M \times N & \to & \mathbb{S}^k \\ (a,b) & \mapsto & \frac{a-b}{ \| a - b \| } \end{array} \right. and \lambda_2 : \left\{ \begin{array}{ccc} N \times M & \to & \mathbb{S}^k \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right.,

and let for convenience

I : \left\{ \begin{array}{ccc} M \times N & \to & N \times M \\ (a,b) & \mapsto & (b,a) \end{array} \right..

Then \lambda_1 = - \lambda_2 \circ I. It is an easy exercice to show that \deg(f \circ g)= \deg(f) \cdot \deg(g) when g is a diffeomorphism, and that \deg(-f)= (-1)^{k+1} \deg(f). So

l(M,N)= \deg(\lambda_1)= (-1)^{k+1} \cdot \deg(I) \cdot \deg(\lambda_2)= (-1)^{m+n+1} (-1)^{mn} l(N,M). \square

Property 5: If M bounds an oriented manifold disjoint from N, then l(M,N)=0.

Proof. If X \subset \mathbb{R}^{k+1} is an oriented submanifold such that \partial X = M, then the map

\Lambda : \left\{ \begin{array}{ccc} X \times N & \to & \mathbb{S}^k \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right.

extends the linking map \lambda : M \times N \to \mathbb{S}^k, and \partial (X \times N)= \partial X \times N= M \times N, hence l(M,N)= \deg(\lambda)=0 according to property 3. \square

If M and N are two knots in \mathbb{R}^3, it is possible to compute l(M,N) from a regular diagram.

More precisely, if v \in \mathbb{S}^2 and P is the plane normal to v, then the projection of M \cup N on P is a regular diagram D if v is a regular value of \lambda; moreover, \lambda^{-1}(v) is exactly the crossing points of D when M is over N. Therefore, l(M,N)=r-s where r (resp. s) is the number of crossing points where \lambda is orientation-preserving (resp. orientation-reversing); on D, such crossing points correspond respectively to

Linking number

For example, the linking number computed on the following diagramm is two:

Knot

Before introducing Hopf invariant, let us mention some results about cobordism, needed in the sequel:

Some cobordism theory:

Definition: Two oriented compact manifolds M, N are cobordant if there exists an oriented compact manifold X, called a cobordism, such that \partial X= M \coprod (-N) where -N denotes N with the reversed orientation.

Lemma 1: Let f : M \to \mathbb{S}^p be a smooth map and y \in \mathbb{S}^p be a regular value. If z \in \mathbb{S}^p is a regular value sufficently close to y, then there exists a cobordism X \subset M \times [0,1] between f^{-1}(y) and f^{-1}(z).

Proof. If C is the set of singular points of f, f(C) is compact so there exists an open neighborhood V of y containing only regular values. Let z \in V.

Let \gamma : [0,1] \to \mathbb{S}^p be a great circle from y to z, and let r_t be a rotation whose restriction to \mathrm{span}(y,z) is a rotation sending y to \gamma(t). In particular, if r_0= \mathrm{Id}, (r_t) defines an isotopy between \mathrm{Id} and r_1. Let

F : \left\{ \begin{array}{ccc} M \times [0,1] & \to & \mathbb{S}^p \\ (x,t) & \mapsto & r_t \circ f(x) \end{array} \right..

Because r_t^{-1}(z) \in V, z is a regular value of F(t, \cdot) and finally to F since

\displaystyle dF(m,s)= r_s \circ df(m) + \frac{\partial r_t}{\partial t}_{|t=s} \circ f(m) dt.

Thus, F^{-1}(z) \subset M \times [0,1] defines a cobordism between f^{-1}(z) and f^{-1}(r_1^{-1}(z))=f^{-1}(y). \square

Lemma 2: If f,g : M \to \mathbb{S}^p are smoothly homotopic and y \in \mathbb{S}^p is a regular value for both, then there exists a cobordism X \subset M \times [0,1] between f^{-1}(y) and g^{-1}(y).

Proof. Let H : M \times [0,1] \to \mathbb{S}^p be a homotopy between f and g. Let z be a regular value of H close to y so that f^{-1}(y) and f^{-1}(z), and g^{-1}(y) and g^{-1}(z), are cobordant (using lemma 1). Then H^{-1}(z) defines a cobordism between f^{-1}(z) and g^{-1}(z). Lemma 2 follows since cobordism is an equivalent relation. \square

Hopf invariant:

From now on, let f : \mathbb{S}^{2p-1} \to \mathbb{S}^p be a smooth map and y \neq z be two regular values. Applying a stereographic projection, we may view f^{-1}(y) and f^{-1}(z) as submanifolds of \mathbb{R}^{2p-1}. Then the linking number l(f^{-1}(y),f^{-1}(z)) is well-defined.

Claim 1: The linking number l(f^{-1}(y),f^{-1}(z)) does not depend on the stereographic projection.

Proof. A stereographic projection is an orientation-preserving diffeomorphism, so if p and q are two stereographic projections, then q \circ p^{-1} \times q \circ p^{-1} induces an orientation-preserving diffeomorphism between pf^{-1}(y) \times pf^{-1}(z) and qf^{-1}(y) \times qf^{-1}(z), hence

l(pf^{-1}(y),pf^{-1}(z))= l(qf^{-1}(y),qf^{-1}(z)). \square

Claim 2: The linking number l(f^{-1}(y),f^{-1}(z)) is locally constant as a function of y.

Proof. Let the mapping maps

\lambda_1 : \left\{ \begin{array}{ccc} f^{-1}(y) \times f^{-1}(z) & \to & \mathbb{S}^{2p-2} \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right. and \lambda_2 : \left\{ \begin{array}{ccc} f^{-1}(x) \times f^{-1}(z) & \to & \mathbb{S}^{2p-2} \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right..

According to lemma 1, there exists a cobordism X \subset \mathbb{R}^{2p-1} \times [0,1] between f^{-1}(y) and f^{-1}(x) when x is close to y. Let

\Phi : \left\{ \begin{array}{ccc} X \times f^{-1}(z) & \to & \mathbb{S}^{2p-2} \times [0,1] \\ (a,t,b) & \to & \left( \frac{a-b}{ \| a-b \| },t \right) \end{array} \right..

For convenience, let \partial \Phi be the restriction of \Phi on \partial X \times f^{-1}(z), and \partial_1\Phi (resp. \partial_2 \Phi) be the restriction of \partial \Phi on f^{-1}(x) \times \{ 0 \} \times f^{-1}(z) (resp. on f^{-1}(y) \times \{ 1 \} \times f^{-1}(z)). According to property 3,

0 = \deg(\partial \Phi)= \deg( \partial_1 \Phi)+ \deg(\partial_2 \Phi).

If \varphi_1, \varphi_2 are the obvious diffeomorphisms

\varphi_1 : f^{-1}(x) \times \{ 0 \} \times f^{-1}(z) \to f^{-1}(x) \times f^{-1}(z)

and

\varphi_2 : f^{-1}(y) \times \{1\} \times f^{-1}(z) \to f^{-1}(y) \times f^{-1}(z),

then \partial_i \Phi= \lambda_i \circ \varphi_i (i=1,2), hence

0= \deg( \lambda_1 ) \cdot \deg(\varphi_1)+ \deg( \lambda_2) \cdot \deg(\varphi_2).

But \varphi_1 is orientation-preserving whereas \varphi_2 is not. Therefore,

l( f^{-1}(y), f^{-1}(z))= \deg( \lambda_1)= \deg(\lambda_2)= l(f^{-1}(x), f^{-1}(z)). \square

Claim 3: If y \neq z are regular values of g : \mathbb{S}^{2p-1} \to \mathbb{S}^p and \| f-g \|_{\infty} <1, then

l(f^{-1}(y),f^{-1}(z)) = l(g^{-1}(y), f^{-1}(z)).

Proof. Let H : \mathbb{S}^{2p-1} \times [0,1] \to \mathbb{S}^p be the homotopy

\displaystyle H(x,t)= \frac{(1-t)f(x)+tg(x)}{ \| (1-t)f(x)+g(x) \| }.

H is well-defined since (1-t)f(x)+tg(x)=0 implies

1= \| f(x) \| =t \| f(x)-g(x) \| \leq \| f-g \|_{\infty} <1,

a contradiction. Therefore, f and g 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

l(f^{-1}(y) , f^{-1}(z))= l(g^{-1}(y),f^{-1}(z)). \square

Property 6: The linking number l(f^{-1}(y),f^{-1}(z)) depends only on the homotopy class of f.

Proof. According to claim 1, l(f^{-1}(y),f^{-1}(z)) does not depend on the stereographic projection. According to claim 2 and property 4, the linking number does not depend on the regular values y and z by connectedness of \mathbb{S}^{2p-1}. Finally, if H is a smooth homotopy between f and g, then we deduce that

l(f^{-1}(y),f^{-1}(z))= l(g^{-1}(y),g^{-1}(z))

from claim 3 and property 4, taking a sequence 0=t_0 < t_1 < \cdots < t_k=1 satisfying \| H(t_i , \cdot ) - H ( t_{i+1} , \cdot ) \|_{\infty} < 1 for all 0 \leq i \leq k-1. \square

Definition: We define the Hopf invariant H(f) of f as the linking number l(f^{-1}(y),f^{-1}(z)) for some regular values y \neq z.

We just showed that Hopf invariant is a homotopic invariant.

Property 7: If f : \mathbb{S}^{2p-1} \to \mathbb{S}^p and g : \mathbb{S}^p \to \mathbb{S}^p, then H(g \circ f )= H(f) \cdot \deg(g)^2.

Proof. Let y \neq z be two regular values of g \circ f and g, and let g^{-1}(y)= \{ y_1, \dots, y_r \} and g^{-1}(z)= \{ z_1, \dots, z_s \}. The inclusions

f^{-1}(y_i) \times f^{-1}(z_i) \hookrightarrow (g \circ f)^{-1}(y) \times (g \circ f)^{-1} (z)

induce an orientation-preserving diffeomorphism between

\displaystyle \coprod\limits_{i=1}^r \coprod\limits_{j=1}^s \left( \mathrm{sign}(dg(y_i)) f^{-1}(y_i) \right) \times \left( \mathrm{sign}(dg(z_j)) f^{-1}(z_j) \right)

and

\displaystyle (g \circ f)^{-1}(y) \times (g \circ f )^{-1}(z),

so, if \lambda_{ij} : \left\{ \begin{array}{ccc} f^{-1}(y_i) \times f^{-1}(z_j) & \to & \mathbb{S}^{2p-2} \\ (a,b) & \mapsto & \frac{a-b}{ \| a-b \| } \end{array} \right., we have:

\begin{array}{lcl} \deg(\lambda) & = & \displaystyle \sum\limits_{i=1}^r \sum\limits_{j=1}^s \mathrm{sign}(dg(y_i)) \mathrm{sign}(dg(z_j)) \cdot \underset{= H(f)}{\underbrace{\deg(\lambda_{ij})}} \\ \\ & = & \displaystyle H(f) \left( \sum\limits_{i=1}^r \mathrm{sign}(dg(y_i)) \right) \left( \sum\limits_{j=1}^s \mathrm{sign}(dg(z_j)) \right) \\ \\ & = & H(f) \cdot \deg(g)^2 \hspace{1cm} \square \end{array}

Hopf fibration:

The sphere \mathbb{S}^3 can be viewed as the unit sphere \{ (z_1,z_2) \mid |z_1|^2+ |z_2|^2=1 \} in \mathbb{C}^2 \simeq \mathbb{R}^4. Noticing that the intersection between \mathbb{S}^3 and any complexe line is a circle, we can say that \mathbb{S}^3 is covered by a family of circles \mathbb{S}^1 indexed by \mathbb{C}P^1 \simeq \mathbb{S}^2.

More precisely, if (x,y,z,t) \in \mathbb{S}^3 \subset \mathbb{R}^4 then (x+iy,z+it) \in \mathbb{S}^3 \subset \mathbb{C}^2 belongs to the complex line (z+it)z_2= (x+iy)z_1; to this line (in \mathbb{C}P^1) is associated the complex number \frac{x+iy}{z+it} \in \mathbb{C} \cup \{ \infty\}. Finally, if h : \mathbb{S}^2 \to \mathbb{C} \cup \{\infty\} denotes the diffeomorphism induced by the stereographic projection, we get a point h^{-1} \left( \frac{x+iy}{z+it} \right) of the sphere \mathbb{S}^2.

Definition: Hopf map \pi : \left\{ \begin{array}{ccc} \mathbb{S}^3 & \to & \mathbb{S}^2 \\ (x_1,x_2,x_3,x_4) & \mapsto & h^{-1} \left( \frac{x_1+ix_2}{x_3+ix_4} \right) \end{array} \right. induces the Hopf fibration \mathbb{S}^1 \to \mathbb{S}^3 \overset{\pi}{\longrightarrow} \mathbb{S}^2.

It is possible to visualize the decomposition of \mathbb{S}^3 by circles in \mathbb{R}^3 using stereographic projection. It gives something like that, a decomposition in concentric torii each covered by their Villarceau circles:

Hopf

Very nice animations about Hopf fibration can be found in Etienne Ghys, Joe Leys and Aurélien Alvarèz’s movie, Dimensions, and in Niles Johnson’s lecture, Visualizations of Hopf fibration. Now,

\pi^{-1}(1,0,0)= \{ (x,y,x,y) \mid x^2+y^2= \frac{1}{2} \} and \pi^{-1}(-1,0,0) = \{ (x,y,-x,-y) \mid x^2+y^2= \frac{1}{2} \}

are two circles in \mathbb{S}^3. If \varphi : \mathbb{S}^3 \subset \mathbb{R}^4 \to \mathbb{R}^3 is the stereographic projection with respect to (0,0,0,1), then \varphi(x,y,z,t)= \left( \frac{x}{1-t}, \frac{y}{1-t}, \frac{z}{1-t} \right), so the previous circles become

\left\{ \left( \frac{x}{1-y}, \frac{y}{1-y}, \frac{x}{1-y} \right) \mid x^2+y^2= \frac{1}{2} \right\} and \left\{ \left( \frac{x}{1+y}, \frac{y}{1+y}, - \frac{x}{1+y} \right) \mid x^2+y^2= \frac{1}{2} \right\}.

In fact, it is just a Hopf link; precisely, if C_1 and C_2 denotes the projections of the two previous circles on the plane z=0, that is

C_1=\left\{ \left( \frac{x}{1-y}, \frac{y}{1-y} \right) \mid x^2+y^2= \frac{1}{2} \right\} and C_2= \left\{ \left( \frac{x}{1+y}, \frac{y}{1+y} \right) \mid x^2+y^2= \frac{1}{2} \right\},

then C_1 \cap C_2= \left\{ \left( \pm \frac{1}{\sqrt{2}}, 0 \right) \right\} where C_1 is over C_2 at one point, and vice-versa at the other. Therefore, H(\pi)=1 (be careful to the orientations!).

On the other hand, H(f)=0 for every constant map f : \mathbb{S}^3 \to \mathbb{S}^2. Indeed, let

p : \left\{ \begin{array}{ccc} \mathbb{S}_3 & \to & D^2 \\ (x,y,z,t) & \mapsto & (z,t) \end{array} \right.

where D^2 is the unit two-dimensional disk viewed as a subspace of \mathbb{S}^2; in particular, since D^2 is contractible, p is homotopic to a constant map f and H(p)=H(f). But the circles p^{-1}(0,0)= \{ (x,y,0) \mid x^2+y^2=1 \} and p^{-1}(1/2,0)= \{ (x,y,1/2) \mid x^2+y^2= 3/4 \} (viewed in \mathbb{R}^3 thanks to the stereographic projection with respect to (0,0,0,1)) are clearly unlinked, hence H(p)=0 according to property 5.

Moreover, using property 7, we deduce that the applications

p_n : \left\{ \begin{array}{ccc} \mathbb{S}^{3} \subset \mathbb{C}^2 & \to & \mathbb{S}^2= \mathbb{C} \cup \{ \infty \} \\ (z_1,z_2) & \mapsto & \left( z_1/z_2 \right)^n \end{array} \right.

define a family of pairwise non-homotopic maps, since H(p_n)= n^2. We just proved our main theorem!

Hopf fibration in higher dimensions:

Hopf fibration \mathbb{S}^1 \to \mathbb{S}^3 \to \mathbb{S}^2 is obtained using Hopf map \left\{ \begin{array}{ccc} \mathbb{S}^3 \subset \mathbb{C}^2 & \to & \mathbb{S}^2= \mathbb{C} \cup \{ \infty \} \\ (z_1,z_2) & \mapsto & z_1/z_2 \end{array} \right.. But the same thing can be done by replacing \mathbb{C} with any real division algebra, for example the quaternions \mathbb{H} or the octonions \mathbb{O}.

The associated maps p : \mathbb{S}^7 \subset \mathbb{H}^2 \to \mathbb{S}^4 = \mathbb{H} \cup \{ \infty \} and q : \mathbb{S}^{15} \subset \mathbb{O}^2 \to \mathbb{S}^8 = \mathbb{O} \cup \{ \infty \} induce respectively the Hopf fibrations \mathbb{S}^3 \longrightarrow \mathbb{S}^7 \overset{p}{\longrightarrow} \mathbb{S}^4 and \mathbb{S}^7 \longrightarrow \mathbb{S}^{15} \overset{q}{\longrightarrow} \mathbb{S}^8.

It can be shown (for example in Hatcher’s book, Algebraic topology, using a homological interpretation of Hopf invariant) that H(p)=H(q)=1, hence:

Theorem: \pi_3(\mathbb{S}^2), \pi_7(\mathbb{S}^4) and \pi_{15}(\mathbb{S}^8) 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.

 

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