One of the things I really want to do with the Surgery for Amateurs project is to make a reasonably plausible presentation of surgery theory for topological manifolds, as well as for smooth and/or PL. But this requires one to come to grips with an inconvenient truth: to set up the basic machinery (handle decompositions and such like) which even allows surgery theory to get going in the topological category, one needs as input the whole development in the smooth or PL categories. The “classical” way that this comes about is via the Kirby torus trick, though a more modern development via controlled surgery is also possible.
A very nice preprint by Hatcher (see below) helps demystify the torus trick by applying it to a low-dimensional problem. The theorem in question is that every (closed, for simplicity) topological surface can be given a compatible smooth structure. (As usual the theorem comes along with various relative versions and uniqueness (up to isotopy) to accompany the existence statement.) The use of the torus trick means that the argument only depends on the differential topology of the plane and not on difficult \(C^0\) facts like the topological Schoenflies theorem (which is needed in the classical approach to the smoothability of surfaces).
Hatcher deduces the existence of smooth structures from the following
Handle Smoothing Theorem Let \(S\) be a smooth surface and \(p\in\{0,1,2\}\). Any topological embedding of \(D^p\times {\mathbb R}^{2-p}\) into \(S\) which is smooth in a neighborhood of \(\partial D^p \times {\mathbb R}^{2-p}\) can be isotoped to a smooth embedding in a neighborhood of \(D^p \times \{0\}\), staying fixed near \(\partial D^p \times {\mathbb R}^{2-p}\) and outside a larger neighborhood of \(D^p\times \{0\}\).
(Note that the final clause is vacuous when \(p=2\).) Here is a sketch of the argument leading from this theorem to the existence of smooth structures: Cover the surface\(S\) by finitely many domains of coordinate charts and inductively extend the smooth structure one chart at a time. To do the inductive step, one needs to know that if \(W\subseteq S\) is an open subset which has already been smoothed, and \(U\subseteq S\) is the domain of a coordinate chart \(h\colon U\to {\mathbb R}^2\), then the embedding
\[k_0= h^{-1}\colon {\mathbb R}^2\to S \]
can be changed by an isotopy \(k_t\) that remains constant outside \(h(W)\) in such a way that the new embedding \(k_1\) is smooth on \(h(W)\). To construct such an isotopy, triangulate the open set \(h(W)\subseteq{\mathbb R}^2\) by simplices whose size tends to zero at the boundary. Then apply the handle smoothing theorem first to small balls around all the 0-simplices, then to neighborhoods of the 1-simplices, and finally to what’s left. The “small simplex” condition ensures that the resultant isotopies extend (by the identity) on the rest of \({\mathbb R}^2\).
So how is the handle smoothing theorem to be proved? Hatcher gives separate arguments for the three cases, but the only one that involves the torus trick is the 0-handle case (\(p=0\)), so that’s what I’ll concentrate on here. The basic structure of the argument is the following.
Look at the topological embedding \(h\colon {\mathbb R}^2 \to S\), and use it to pull back the smooth structure of \(S\) to some smooth structure on \({\mathbb R}^2\); let \(X\) denote \({\mathbb R}^2\) equipped with this (potentially exotic) smooth structure. The identity map \({\mathbb R}^2\to X\) is then a (potentially wild) homeomorphism.
Fact 1 There are no exotic smooth structures on \({\mathbb R}^2\).
Thus the identity map \({\mathbb R}^2\to X\) becomes identified with a homeomorphism from \({\mathbb R}^2\) to itself.
Fact 2 Any self-homeomorphism \(g\) of \({\mathbb R}^2\) can be written as a composite \( g_1\circ g_2\), where \(g_1\) is a diffeomorphism and \(g_2\) is a bounded homeomorphism, i.e., \( |g(x)-x|\) is uniformly bounded.
When \({\mathbb R}^2\) is identified with the interior of a 2-disk, any bounded homeomorphism extends continuously (as the identity) to the boundary and exterior of the disk. Using this trick, it is easy to deduce the 0-Handle Smoothing Theorem from Facts 1 and 2.
The torus trick is used to prove these facts (actually, to prove slightly weaker versions of them, but still sufficient for what is needed). I will explain in the followup post.
References
Hatcher, Allen. “The Kirby Torus Trick for Surfaces.” arXiv:1312.3518 [math] (December 12, 2013). http://arxiv.org/abs/1312.3518.
Kirby, Robion C, and Laurence C Siebenmann. Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton, N.J.: Princeton University Press, 1977.
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