Algebraic Topology – Hopf Fibration and $\pi_3(\mathbb{S}^2)$

algebraic-topologyhomotopy-theoryhopf-fibrationreference-request

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.

Best Answer

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.

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).

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