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The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things:

  • Homotopy groups stabilize after taking suspensions (Freudenthal suspension theorem), or
  • Cofiber sequences are fiber sequences (e.g., you've inverted the loops-suspension functors).

Does anybody know the first instances in which the words "stable" or "stability" were used to describe each of these phenomena?

As far as I can tell:

In the 90's and onward, Hovey, Schwede-Shipley, and Lurie, used the second meaning to define stable (model, oo-) categories. I don't know of any earlier references in which this second meaning was pinched out as the meaning of stability, though there may have been many.

In the 60's, Adams and Boardman (and many others) use the word stability without hesitation. I feel like, by this time, it was common to use the word "stability", but I am not sure which of the above two phenomenon the word "stability" was intended to capture. Perhaps both.

In 1938, Freudenthal observed the first phenomenon. I would venture to say this might have been the beginning of "stable" phenomena in homotopy theory, but I don't know enough German to see if he even used the word "stable" in his paper.

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    $\begingroup$ Maybe the answer is somewhere in: math.uiuc.edu/K-theory/0321/history.pdf (But probably better to just wait for Peter to come along and answer anyway...) $\endgroup$
    – Drew Heard
    Commented May 8, 2014 at 7:53
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    $\begingroup$ In his paper, Freudenthal does not use the word "stable" (="stabil" in German) or any similar word. $\endgroup$
    – nsrt
    Commented May 8, 2014 at 8:21
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    $\begingroup$ One early reference is Bott, An application of the Morse theory to the topology of Lie groups (1956). It is not in Spanier, Whitehead, A first approximation to homotopy theory (1953), where the introduce the Spanier-Whitehead category. 1962 it was well-known enough that Whitehead could call his ICM-talk "Some aspects of stable homotopy theory." $\endgroup$
    – Lennart Meier
    Commented May 8, 2014 at 13:15
  • $\begingroup$ Earlier references: Thom uses it in 'Quelques propriétés globales des variétés différentiables' (1954) for the homotopy groups of his "Thom spectrum" and Serre uses it in 'Cohomologie modulo 2 des complexes d'Eilenberg-MacLane' (1953) for the stable (co)homology groups of Eilenberg-MacLane spaces (later spectra). You also find the word 'stable' in Serre's 'Homologie Singuliere Des Espaces Fibres', but there it is stable under other operations than suspension-related ones. $\endgroup$
    – Lennart Meier
    Commented May 8, 2014 at 13:21
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    $\begingroup$ @HiroLeeTanaka There is no such word. The closest is that he defines that $\pi_dS^e$ "belongs to the k-stem", where $k=d-e$. $\endgroup$
    – nsrt
    Commented May 12, 2014 at 8:18

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I like Lennart's comment so, Drew, I won't try to say much more. Stability in the first sense was certainly well understood by the very early 1950's, and it would be hard to be certain of a ``first'' reference. The second sense had to come later since cofiber sequences and fiber sequences were only formalized in 1958 (Puppe) and 1960 (Nomura); the comparison map $\Sigma Ff \longrightarrow Cf$ relating them came a bit later, I believe. Both senses were thoroughly understood before 1964 (when Boardman and I got our theses).

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