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I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in manifold theory, to denote a diffeomorphism of a product manifold $M \times I$ that's the identity on all but the one boundary face $M \times \{1\}$.

Doing a MathSciNet search I see pseudo-isotopy appearing in some detail in Milnor's h-cobordism notes (1965), where he mentions the synonym "I-cobordism" was used by Munkres and concordance by Hirsch. Without the hyphen pseudoisotopy appears in a 1959 paper of R.H.Bing, but he appears to use the term as if it were already in common usage. A search for concordance gives an even earlier paper, 1948 by Youngs.

None of these papers give the appearance of being first-usage, as the term appears to be used too casually. Perhaps I am misinterpreting one of these authors. There are some sources MathSciNet does not know, like the Soviet journals of the time. Perhaps the term originated in one of those unindexed journals, or some other non-English language journal. That said, the community was small so perhaps the decision to use the terminology was made collectively, at a conference.

I would presume the connection between pseudo-isotopy and the h-cobordism theorem was made by one of Smale, Whitney or Milnor as it was solidified by the time Milnor's lecture notes were written, and this was a perspective favoured by Whitney.

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  • $\begingroup$ I believe Bing used the term pseudo-isotopy in a different sense from what you have in mind. In Bing's sense a pseudo-isotopy is a 1-parameter family of maps $f_t$, $0\leq t\leq 1$, which are homeomorphisms for $t<1$ but with $f_1$ allowed to collapse nontrivial subspaces to points. $\endgroup$
    – Allen Hatcher
    Commented May 20, 2023 at 1:34
  • $\begingroup$ @AllenHatcher: Cerf does not appear to be using the language of pseudoisotopy in 1962/63, so perhaps it wasn't standardized until Milnor's notes? $\endgroup$
    – Ryan Budney
    Commented May 20, 2023 at 2:07
  • $\begingroup$ I’m voting to close this question because it seems better for HSM. $\endgroup$
    – LSpice
    Commented May 20, 2023 at 3:29

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An immediate search on Google Books shows that "pseudo-isotopy" in the modern sense occurs in the notes of Zeeman's seminar on Combinatorial topology in Berkeley vol 2 (1963) pp 41-43; not being able to access the full text, I cannot say who was the author of the talk!

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Going along with Gael Meigniez's reference, but presumably more accessible, is an article by Hudson and Zeeman, On combinatorial isotopy, Publications mathématiques de l’I.H.É.S., tome 19 (1964), p. 69-94. It defines pseudo-isotopy (hyphenated) more broadly for embeddings. Curiously, it seems that the name concordance has been retained for embeddings.

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MathSciNet refers to a paper of Bing from 1959 with the term "pseudo-isotopy" in the math review:

Bing, R. H. The cartesian product of a certain nonmanifold and a line is E^4. Ann. of Math. (2)70(1959), 399–412.

The reviewer seems to indicate that the term was already in use by then.

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