I learned about the push-pull construction from a video lecture by Freedman in which it is explained starting around 39:08. It is somewhat long and technical to describe in detail, but the main idea is that it is a technique to control an homeomorphism in one linear direction.

The example application is to show that if $X$ and $Y$ are compact metric spaces such that $X\times\Bbb R$ is homeomorphic to $Y\times\Bbb R$, then $X\times S^1$ and $Y\times S^1$ are also homeomorphic. The way this is done is by taking an homeomorphism $h\colon X\times\Bbb R\to Y\times\Bbb R$ and using the push-pull construction to modify it to another homeomorphism between the two spaces which is periodic in the $\Bbb R$ coordinate, so that it descends to an homeomorphism $X\times S^1\to Y\times S^1$.

Morse used the construction in 1960 to establish the Schoenflies conjecture "Every bicollared embedding $S^d\to\Bbb R^{d+1}$ extends to an embedding $D^{d+1}\to\Bbb R^{d+1}$". In particular he used the push-pull construction to show that every such bicollared embedding admits a so called flat spot after composing with a self-homeomorphism of $\Bbb R^{d+1}$, and the conjecture had already been established by Mazur for bicollared embeddings with a flat spot.

However Freedman mentions that the push-pull construction was not introduced by Morse, and in fact that it was already well known by the time he used to to settle Schoenflies's conjecture, hence my question: who invented the push-pull construction?

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    $\begingroup$ There is a more detailed printed version of Freedman's lectures, where he mentions that it was "common knowledge", that "by the time Morse had completed Mazur’s argument, Brown had already given an independent proof", and ends up "giving some more applications, probably also due to Brown, of push-pull" (oddly enough, all his previous applications are associated with Morse). Stern mentions in the MathSciNet review of Kirby-Siebenmann (1977) that Brown's proof "marked the advent of push-pull topology". $\endgroup$
    – Conifold
    Apr 19 '20 at 9:31

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