It follows from surgery theory that in dimension $\geq 5$ every closed PL manifold homotopy equivalent to a torus has a finite cover which is PL homeomorphic to a torus. This is an important ingredient for Kirby's torus trick, and thus for the theory of topological manifolds.

In the introduction to The Topological Classification of Stratified Spaces, Weinberger refers to an alternative proof of this fact:

I should mention that nowadays using some amazing constructions of Edwards and Chapman, one can prove this result about tori by pure geometry, without invoking surgery.

However, he gives no further explanation or references. Can someone point me to a reference or give an explanation of this argument?

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    $\begingroup$ I believe that he returns to this in 9.4.1. I guess you should be suspicious of this answer because he doesn't give a forward reference in the introduction. There he cites Chapman, Approximation results in topological manifolds, 1981. See the introduction for the discussion of the avoidance of surgery. $\endgroup$ Aug 6 '20 at 20:15

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