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THE TORUS TRICK DESTINE LEE C 1. Introduction 1 2. Preliminaries 3

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THE TORUS TRICK DESTINE LEE C 1. Introduction 1 2. Preliminaries 3 THE TORUS TRICK DESTINE LEE Abstract. Following Allen Hatcher’s The Kirby Torus Trick for Surfaces,I prove that every homeomorphism between smooth surfaces is isotopic to a diffeomorphism. This exposition assumes only that the reader has com- pleted first courses in topology and multivariable real analysis. Contents 1. Introduction 1 2. Preliminaries 3 2.1. Smooth Manifolds 3 2.2. Smooth Maps 5 2.3. Isotopies 8 3. Proving the Main Theorem 8 3.1. Handle Decompositions of Surfaces 9 3.2. Proof of the Main Theorem 12 4. Smoothing 2-handles 13 5. Smoothing 1-handles 15 6. Smoothing 0-handles 16 References 20 1. Introduction In a first course in smooth manifold theory, one quickly realizes that smooth manifolds are significantly more tame than their topological coun- terparts. For one, smooth manifolds are—loosely speaking—distinguished by a lack of sharp corners or cusps. However, the two results below reveal that the two worlds are perhaps not so far apart. E-mail address: [email protected]. Date: January 1, 2021. 1 THE TORUS TRICK 2 Theorem 1.1 (Theorem A). Every topological surface has a smooth structure. Theorem 1.2 (Theorem B). Every homeomorphism between smooth surfaces is isotopic to a diffeomorphism. Remark. Theorem 1.2 in particular holds in dimension 3 as well, but not 4 dimension 4; for instance, R admits uncountably many non-diffeomorphic smooth structures ([Mil20]). According to Theorem 1.1, every topological surface can be turned into a smooth surface, making questions about topological surfaces amenable to smooth techniques. Moreover, according to Theorem 1.2, this relationship is symbiotic; namely, homeomorphisms of smooth surfaces can be turned into diffeomorphisms, meaning that we can apply techniques normally re- served for topological surfaces (e.g. the Alexander trick) to smooth surfaces and still recover information about them as smooth surfaces. The objective of this paper is to present a reasonably self-contained proof of Theorem 1.2, following Hatcher’s approach in [Hat13]. Specifically, I as- sume only that the reader has completed first courses in topology and mul- tivariable real analysis. The paper is structured as follows. In Section 2, I introduce the language of smooth manifolds needed to decode the statement of Theorem 1.2. In Section 3, I sketch a proof of Theorem 1.2, but assuming the following result. Theorem 3.1 (Handle Smoothing Theorem). Let S be a smooth surface. Then: 2 • A topological embedding R ! S of a smooth 0-handle can be isotoped to a smooth embedding in a neighborhood of the origin, staying fixed outside a larger neighborhood of the origin. • A topological embedding I × R ! S of a smooth 1-handle which is a smooth embedding near @I × R can be isotoped to be a smooth em- bedding in a neighborhood of I × f0g, staying fixed outside a larger neighborhood of I × f0g and near @I × R. • A topological embedding D2 ! S of a smooth 2-handle which is a smooth embedding in a neighborhood of @D2 can be isotoped to be a smooth embedding on all of D2, staying fixed in a smaller neighbor- hood of @D2. The remaining sections are devoted to proving the three statements that make up the Handle Smoothing Theorem. By far the simplest case of the THE TORUS TRICK 3 three is that of 2-handles, presented in Section 4, but all three arguments follow the same blueprint. (1) Given a topological embedding h of a handle into a surface, adjust the handle by way of a self-diffeomorphism g to improve the smoothness properties of h. (2) Apply the Alexander trick in order to obtain an isotopy gt from the identity map to the diffeomorphism g. (3) Compose h with gt to obtain an isotopy h ◦ gt from h to h ◦ g, where h ◦ g has the desired smoothness properties. The primary hurdle in the 1-handle and 0-handle cases lies in manipulating the maps so that the Alexander trick is applicable. Section 5 presents the 1-handle case, and Section 6 presents the 0-handle case. 2. Preliminaries 2.1. Smooth Manifolds. In this section, I develop the basic language of smooth manifolds that we will need in order to understand the statement of Theorem 1.2. I begin by reminding the reader of the definition of topological manifold. Definition 2.1 (topological manifold). Let M be a topological space. If for some positive integer n, M is • Hausdorff, • second-countable, • and for every point p 2 M, there exists an open set U homeomorphic n to R and containing p, then we say M is a topological n-manifold. From here, we want to transition to the notion of a smooth manifold. The idea is we also want to be able to do calculus on manifolds, but for a general topological n-manifold M and a function f : M ! R, the smoothness of f is not well-defined, for f is not necessarily a map of Euclidean spaces. Of course, even when M is not an open subset of Euclidean space, it is (by n definition) locally homeomorphic to R , and so for a given point p 2 M, n we have a homeomorphism h : U ! R for some open neighborhood U of p. Using this, we might try and say f is differentiable if f ◦h−1 is differentiable, −1 and this now parses because f ◦h : h(U) ! R is a map of Euclidean spaces n defined on an open subset of R . However, the smoothness of f now depends THE TORUS TRICK 4 Figure 1. The two transition functions hαβ and hβα given by a pair of charts (Uα; hα); (Uβ; hβ). n on the choice of h : U ! R . In order to obtain a notion of smoothness that does not depend on such a choice, we introduce an additional layer of structure, namely that of an atlas. Definition 2.2 (atlas). Let M be a topological n-manifold. If p 2 M is a n point and U is an open neighborhood of p in M homeomorphic to R by n h : U ! R , we say (U; h) is a chart for M around p. In addition, if for a pair of charts (Uα; hα) and (Uβ; hβ) around p 2 M the two functions −1 hαβ : hα(Uα \ Uβ) ! hβ(Uα \ Uβ); hαβ(q) = (hβ ◦ hα )(q) and −1 hβα : hβ(Uα \ Uβ) ! hα(Uα \ Uβ); hβα(r) = (hα ◦ hβ )(r) are smooth, we say the two charts are compatible. (See Figure 1.) Finally, we say a family of charts f(Uα; hα)gα2A indexed by A is an atlas on M if the charts are pairwise compatible and the Uα form an open cover of M. THE TORUS TRICK 5 Notice it follows from the definitions that the transition functions hαβ and hβα for a pair of charts (Uα; hα); (Uβ; hβ) in an atlas are not only smooth, but in fact diffeomorphisms. As a result, we see that the smoothness of a real- valued function f : M ! R on a topological n-manifold M is now well-defined as long as we fix an atlas. In particular, given an atlas f(Uα; hα)gα2A, we can now say f is smooth at a point p 2 M if for some chart (Uα; hα) around p, the −1 map f ◦ hα : hα(Uα) ! R is smooth. While we just made a choice of chart, the choice of chart in fact does not matter, because if (Uβ; hβ) is another chart around p, then −1 −1 f ◦ hα is smooth at hα(p) () (f ◦ hα ) ◦ hβα is smooth at hβ(p) −1 −1 () (f ◦ hα ) ◦ (hα ◦ hβ ) is smooth at hβ(p) −1 () f ◦ hβ is smooth at hβ(p): Thus, we see that the additional structure of an atlas overcomes the diffi- culty encountered when trying to define smoothness in the context of gen- eral topological manifolds. This leads us to the notion of smooth manifold. Definition 2.3 (smooth manifold). A smooth n-manifold is a topological n- manifold together with an atlas. For example, consider the 2-sphere. We have seen that the 2-sphere is a topological manifold; the local homeomorphisms are given by projecting each of the six open hemispheres onto the obvious planes and radially reparametrizing. One can verify that the same pairings of open sets and homeomorphisms give rise to an atlas, and so the 2-sphere is a smooth 2- manifold. Alternatively, one can define an atlas consisting of just two charts using stereographic projection. The way this works is as follows. First, we take the north pole N = (0; 0; 1) as our projection point, and for any other point p 2 S2 consider the line passing through N and p, and map p to the point of intersection between this line and the xy-plane. This defines for us 2 2 a homeomorphism from S n fNg ! R . Likewise, we can do the same with the south pole S = (0; 0; −1) as the projection point, and together, these two charts form an atlas. 2.2. Smooth Maps. The previous section was motivated by the search for a well-defined notion of a smooth real-valued function on a manifold. In order to achieve this, we developed some machinery, and as an added bonus, we can also define what it means for a map of smooth manifolds to be smooth. THE TORUS TRICK 6 Figure 2. Stereographic projection of the 2-sphere onto the plane, with the north pole as the projection point.
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