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If you didn't know anything about stabilization phenomena in algebraic topology and were trying to discover/prove theorems about the homotopy theory of spaces, what clues would point you toward results such as Freudenthal suspension or the existence of stable homotopy groups of spheres?

References suggest that Freudenthal originally stated his result in this 1938 Paper, although I'm unable to find an English translation. This was published only a few short years after the discovery of the Hopf fibration, so I find it pretty surprising that not only would there have been clear notions of $\pi_{\geq 2}$ at the time, but also enough evidence to suggest looking for things like the suspension map or stable homotopy groups.

Analogous stabilization phenomena do seem to occur elsewhere in mathematics: for instance, vector bundles that become isomorphic after taking Whitney sums with trivial bundles. From there, it may not be that much of a leap to suppose that something similar might work for fibrations.

However, it also seems that Freudenthal's paper predated results like this, and so historically, perhaps the flow of ideas was the other way around. What other results might have motivated his suspension theorem? Or in retrospect, what are some signs that such a thing might have worked and been useful?

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    $\begingroup$ Perhaps homology? The folk stories that I've heard about the early days of the homotopy groups have to do with expectations being set by what was known about homology. $\endgroup$ – Jeff Strom Jun 3 at 3:01
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    $\begingroup$ +1 to the way the first paragraph is written.. $\endgroup$ – Praphulla Koushik Jun 3 at 3:19
  • $\begingroup$ I would guess the intuition comes from general position (Sard's theorem is also from around that time) The preimage of a point is a manifold in the domain (for the hopf map this preimage is a circle). If the codimension is not too large, this can be knotted, but if the codimension is large, everything can be unknotted. Stabilization does not change the preimage of a regular value, but does increase the codimension. So one might expect that one can unknot everything. $\endgroup$ – Thomas Rot Jun 3 at 11:03
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    $\begingroup$ @JeffStrom : But wouldn't reasoning by analogy with homology lead you to expect that higher homotopy groups of spheres would all be trivial? Once you realized that was false, why would you expect that reasoning by analogy with homology would be a reliable guide to the behavior of higher homotopy groups? $\endgroup$ – Timothy Chow Jun 4 at 13:39
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    $\begingroup$ @TimothyChow whenever an intuition like this fails, the natural next step is to ask how badly it fails. So: “Ok, homotopy groups are not homology groups; what properties do they have in common? How can we compare them?” $\endgroup$ – Jeff Strom Jun 4 at 20:16
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For those whose German is shaky or non-existent, it is fun to copy and paste a couple of paragraphs of Freudenthal's paper into Google translate. The answer to your question emerges. His paper is concerned with the interplay of the then new Hopf invariant and "suspension" - "Einhängung" in German, and possibly named first in this paper. Another thing that seems to be named first here is the notion of "$k$-stem" ("$k$-Stamm").

In modern terms, he is exploring the exactness of the EHP sequence: his Satz I says that the kernel of $H$ (= Hopf) is the image of $E$ (= Einhängung), his Satz II is telling us that homotopy groups stabilize in the usual way, and his Satz III is showing that the first stable stem is $\mathbb Z/2$, with a nonzero element represented by the suspension of any map with odd Hopf invariant.

His methods seem to consist of a careful analysis using simplicial approximation as a key tool. And this would be the answer to your question: anyone exploring such questions finds themselves thinking about general position, how we build up spaces, etc. To a modern student, I would observe that the stable range can be seen by considering the difference between the wedge of two $n$-spheres and their product: one needs to attach a $2n$-disk using a map from a $2n-1$-sphere.

His paper is even more impressive when one remembers that it was written under the darkening cloud of Nazism.

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    $\begingroup$ Are all those hyphens really meant to be doubled? $\endgroup$ – LSpice Jun 4 at 21:04
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    $\begingroup$ It is definitely the case that Freudenthal invented suspension and named it Einhängung. He referred to it as the `big fish' that he had caught. When asked what mathematical result he was most proud of, this was his answer. $\endgroup$ – Wilberd van der Kallen Jun 16 at 6:58
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$\newcommand{\R}{\mathbb R}\newcommand{\inj}{\hookrightarrow}$This will be an anachronistic answer, because it was discovered a bit later, but: Whitney proved that every $n$-manifold embeds in $\R^N$ for $N$ large enough, and Wu showed in 1958 that for $N\ge 2n+2$, all such embeddings are isotopic. This leads to a few interesting stability phenomena: most notably, that every manifold has canonically the data of the isomorphism type of the normal bundle to the embedding $M\inj\R^N$, up to direct sums with trivial bundles. (And this leads to stable vector bundles, another stabilization in algebraic topology…)

How might this have led to Freudenthal's theorem? One upshot is that is bordism groups of immersions stabilize, and in high codimension are just abstract bordism groups. Thom's work on bordisms showed that bordism groups of immersions are homotopy groups of certain spaces, called Thom spaces, and the Thom space for $n$-manifolds in $\R^{N+1}$ is the suspension of the Thom space for $n$-manifolds in $\R^N$. So there are two different reasons these homotopy groups stabilize (the numbers don't quite match: Freudenthal's theorem is sharper). But in some alternate history, where Whitney and Wu's work was earlier, one could imagine people asking, “so the homotopy groups of Thom spaces stabilize, what about everything else?”

(If you modified this by asking for $M\inj\R^N$ to be equipped with a trivialization of its normal bundle, then the Thom space is a sphere, so this provides another description/proof of the stable homotopy groups of the spheres.)

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