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Hiro Lee Tanaka
Hiro Lee Tanaka
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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, have 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 refer to stability in (the titles ofand many others) their booksuse the word stability without hesitation. I feel like, by this time, it was common to use the word "stability" in both ways (though I wouldn't know, but I am not having been present)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.

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

  • 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 these phenomena?

As far as I can tell:

In the 90's and onward, Hovey, Schwede-Shipley, and Lurie, have used the second meaning to define stable (model, oo-) categories.

In the 60's, Adams and Boardman refer to stability in (the titles of) their books. I feel like, by this time, it was common to use the word "stability" in both ways (though I wouldn't know, not having been present).

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.

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.

Source Link
Hiro Lee Tanaka
Hiro Lee Tanaka
  • 3.4k
  • 27
  • 28

When was the word "stable" first used to describe stable homotopy theory?

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

  • 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 these phenomena?

As far as I can tell:

In the 90's and onward, Hovey, Schwede-Shipley, and Lurie, have used the second meaning to define stable (model, oo-) categories.

In the 60's, Adams and Boardman refer to stability in (the titles of) their books. I feel like, by this time, it was common to use the word "stability" in both ways (though I wouldn't know, not having been present).

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.