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My basic question is: What is Kirby's torus trick and why did it solve so many problems?

I can get a glimmer of it from looking at Kirby's original paper, "Stable Homeomorphisms and the Annulus Conjecture," and its mathscinet review. However, it is a little bit unclear to me what exactly the torus trick was. It seems that there are two important ideas: the first is that by lifting along ever higher covers one may make obstructions to surgery vanish, and the second is that one may pull back differential structures to tori by immersing them in $\mathbb{R}^n$ and forming diagrams like:

$$ \require{AMScd} \begin{CD} {T^n-D^n}@>{id}>> \widetilde{T^n-D^n}\\ @VVV @VVV \\ \mathbb{R}^n @>{g}>> \mathbb{R}^n \end{CD} $$

I looked for a reference that would summarize the situation but I was unable to find one. If there some paper that gives a summary of the trick as well as the context to understand it, I would be grateful to know it. I was hoping to find a succinct summary on Wikipedia but the torus trick link on Kirby's page is sadly red. If someone could give an example of the sort of problem that the torus trick is good at solving that would also be great.

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I highly recommend Allen Hatcher's preprint in which he employs the torus trick for the pedestrian task of proving existence and uniqueness-up-to-isotopy of smooth structures on every 2-manifold. The issue of surgery obstructions is avoided in this case by simple facts about the 2-dimensional torus, laying bare the torus trick itself.

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As a complement to more scholarly references, you might enjoy this video recording of a delightful interview with Kirby. In the section labeled "The Torus Trick," he describes the process which brought him to discover the trick, which I found both mathematically and historically interesting, though it's a bit brief. Some of the surrounding sections of the video contain further interesting information about the state of topology at the time.

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