Hopf fibration and $\pi_3(\mathbb{S}^2)$

Question

I am interested in Hopf's original argument showing that $\pi_3(\mathbb{S}^2)$ is non-trivial (using his fibration). It should be exposed in his paper Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche, but unfortunately I do not read German.

Do you know a translation or a reference following the same argument?

Nota Bene: I am aware that the argument has been generalized with Hopf invariant (as exposed in Hatcher's book), but I am really interested in the original approach.

Answer 1

Hopf's original argument is essentially (well, in spirit) unchanged and appears in several other sources. For example, the textbook by Milnor "Topology from the differentiable viewpoint" or Guillemin and Pollack's "Differential Topology". These also give you plenty of preliminary stepping stones before proving the result.

I haven't read much of Hopf's work but my impression is his writing is fairly easy to read, once you get past the German.

Answer 2

For completeness, I describe problems 13, 14 and 15 of Milnor's book, Topology from differentiable viewpoint, as suggested by Ryan Budney. The argument seems to be near to the text mentionned by Grigory M in comments.

Let $M,N \subset \mathbb{R}^{k+1}$ be two compact, oriented, boundaryless submanifolds with total dimension $m+n=k$. The linking number $l(M,N)$ is defined as the degree of the linking map $$\lambda : M \times N \to \mathbb{S}^k, ~ (x,y) \mapsto \frac{x-y}{\| x-y \|}.$$

• If $M$ bounds an oriented manifold $X$ disjoint from $N$, $l(M,N)=0$.

If $y \neq z$ are regular values for a map $f : \mathbb{S}^{2p-1} \to \mathbb{S}^p$, the linking number $l(f^{-1}(y),f^{-1}(z))$ is well defined.

• This linking number is locally constant as a function of $y$.

• If $y$ and $z$ are regular values of $g$ also, where $$\| f(x)-g(x) \| < \| y-z\|$$ for all $x$, then $$l(f^{-1}(y),f^{-1}(z))= l(g^{-1}(y),f^{-1}(z))=l(g^{-1}(y),g^{-1}(z)).$$

• The linking number $l(f^{-1}(y),f^{-1}(z))$ depends only on the homotopy class of $f$, and does not depend on the choice of $y$ and $z$.

This integer $H(f)= l(f^{-1}(y),f^{-1}(z))$ is called the Hopf invariant of $f$.

The Hopf fibration $\pi : \mathbb{S}^3 \to \mathbb{S}^2$ is defined by $$\pi(x_1,x_2,x_3,x_4)=h^{-1} \left( \frac{x_1+ix_2}{x_3+ix_4} \right)$$ where $h$ denotes stereographic projection to the complex plane.

• Then $H(\pi)=1$.

We deduce that $\pi \in \pi_3(\mathbb{S}^2)$ is essential.

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Nota Bene 1: In order to define the linking number $l(f^{-1}(y),f^{-1}(z))$, it is necessary to view $f^{-1}(y)$ and $f^{-1}(z)$ as subspaces of $\mathbb{R}^{2p-1}$ via a stereographic projection (of course, the linking number does not depend on the chosen projection).

Nota Bene 2: Some facts about cobordism are needed to prove the second and third points (namely, lemmas 2 and 3 of §7).