So how are appropriate versions of Facts 1 and 2 (from the previous post) to be proved, and what has this all got to do with the torus? The Euclidean space \({\mathbb R}^2\) that appears is a little piece of overlap on which something has to be modified (smoothed) while keeping it constant near infinity. There is no harm in making the Euclidean space smaller (i.e. passing to an open subset) in order to accomplish this.
But the punctured torus \({\mathbb T}^2 \setminus \{pt\}\) can be immersed in \({\mathbb R}^2\). In fact, the punctured torus is just a disc with two handles attached, and this can be immersed in the plane if we allow the two handles to cross over one another (the crossing explains why we don’t get an embedding).
In the set-up of the Handle Smoothing Theorem, \({\mathbb R}^2 \) is equipped with a smooth structure which is pulled back via a topological embedding into a smooth surface \(S\). This smooth structure, call it \(\mathcal S\), can then be pulled back to the punctured torus. Let \(T’\) denote the punctured torus equipped with the pulled-back coarse structure.
At this point Hatcher quotes two facts from the theory of smooth surfaces. (In higher dimensional applications of the torus trick, this is where we would appeal to non-simply-connected smooth or PL surgery.)
Fact 3 The smooth structure \(T’\) on the punctured torus extends to a smooth structure on the unpunctured torus; call \(T\) the unpunctured torus equipped with this smooth structure.
Fact 4 The (potentially) “exotic” torus \(T\) is in fact diffeomorphic to the standard torus \({\mathbb T}^2\).
Accepting these for now, let us write \(R\) for the universal cover of \(T\), a potentially exotic \({\mathbb R}^2\). Notice that a small open set around 0 in \(R\) is identified (smoothly) with a small open set in the copy of \({\mathbb R}^2\) with which we started the discussion. Let \(\phi\) be a diffeomorphism from \(T\) to \({\mathbb T}^2\). We can write
\[ \phi = \phi_1 \circ \phi_2 \]
where \(\phi_2\) is a diffeomorphism \(T\to {\mathbb T}^2 \) acting trivially on the fundamental group and \(\phi_1\) is given by an element of \(GL(2,{\mathbb Z}) \). If we lift \(\phi_2\) to a periodic diffeomorphism of universal covers then it preserves the integer lattice and thus is bounded. This is Fact 2 – and though it is applied to a different copy of \({\mathbb R}^2\) than the one we originally started from, we can restrict attention to a small open set where the two copies agree and that is sufficient.
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