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Home , Torus

THE TORUS TRICK

DESTINE LEE

Abstract. Following ’s The Kirby Torus Trick for Surfaces,I prove that every between smooth surfaces is isotopic to a diffeomorphism. This exposition assumes only that the reader has com- pleted first courses in and multivariable real analysis.

Contents

1. Introduction 1 2. Preliminaries 3 2.1. Smooth 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 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 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, 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 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 × {0}, staying fixed outside a larger neighborhood of I × {0} 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 properties of h.

(2) Apply the Alexander trick in order to obtain an isotopy gt from the identity 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 .

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 ∈ 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 , it is (by n definition) locally homeomorphic to R , and so for a given point p ∈ 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 ∈ 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 , if for a pair of charts (Uα, hα) and (Uβ, hβ) around p ∈ 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 {(Uα, hα)}α∈A 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 {(Uα, hα)}α∈A, we can now say f is smooth at a point p ∈ 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-. 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 . 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 ∈ S2 consider the line passing through N and p, and map p to the point of intersection between this line and the xy-. This defines for us 2 2 a homeomorphism from S \{N} → 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.

Definition 2.4 (smooth map). Let M be a smooth m-manifold, and let N be a smooth n-manifold. A map f : M → N is said to be smooth at a point p ∈ M if for some chart (U, g) for M around p and another chart (V, h) for N around f(p) the induced map f˜ : g(U) → h(N) defined by f˜(q) = (h◦f ◦g−1)(q) for each q ∈ U is smooth at g(p). In addition, we say f is smooth (without qualification) if f is smooth at p for all p ∈ M, and we say f is a diffeomorphism if f is a smooth bijection and f −1 is smooth.

Remark. Equivalently, we could have defined the induced map f˜in the above definition to be the map such that the diagram

f U V

g h f˜ g(U) h(V ) commutes.

Next, I introduce two important classes of smooth maps that will play a role in the discussion to follow.

Definition 2.5 (smooth ). Let M be an m-manifold, N an n- manifold, and f : M → N a smooth map. Additionally, let p ∈ M be any point. As in the definition of a smooth map, we can consider the map m n f˜ : R → R induced by a choice of two charts, the first a chart for M around p, and the other a chart for N around f(p). THE TORUS TRICK 7

Figure 3. The pumpkin Yr for r = 1.25 (rendered in GeoGebra).

m n We say f is an immersion at p if the total derivative Df : R → R is injective. In addition, we say f is an immersion (without qualification) if f is an immersion at every point in M.

Remark. One can verify that whether f : M → N is an immersion at a point p does not depend on the choice of charts.

Definition 2.6 (smooth embedding). Let M be an m-manifold, N an n- manifold, and f : M → N a smooth map. We say f is a smooth embedding if f is simultaneously a topological embedding and a smooth immersion.

Up until this point, the discussion has centered around functions on smooth manifolds. However, smoothness has fundamental consequences for the of these spaces as well. The following example illustrates the difference between topological manifolds and smooth manifolds. 3 Let Yr be the set of points in R at distance r > 0 away from the 3 2 2 C = {(x, y, z) ∈ R : x + y = 1, z = 0}.

When r < 1, Yr is a torus with major radius 1 and minor radius r. However, if r is taken to 1, the tube thickens, and eventually the inner equator collapses to a single point at the origin. As r is taken beyond 1, the point separates

into two, and Yr develops two cusps, one above and one below the origin. The result is the contour of an exceptionally smooth pumpkin, rendered in Figure 3 for r = 1.25.

The point of this is that one can ask when Yr is a topological manifold, and when Yr is a smooth manifold, or more precisely, when Yr admits a smooth 3 structure such that the inclusion of Yr into R is a smooth embedding. It turns out that Yr is a topological manifold for all r 6= 1. Intuitively, Yr fails to be a topological manifold when r = 1 because no matter how far in THE TORUS TRICK 8 you zoom, any small neighborhood of the origin looks like two copies of the plane glued together at a single point. On the other hand, Yr is a smooth manifold precisely when r ∈ (0, 1), i.e., when Yr is a torus. The obstruction to smoothness when r > 1 is—roughly speaking—a consequence of the fact that no open neighborhood of either of the two cusps can be realized as the graph of a smooth function.

2.3. Isotopies. Finally, much of what follows will be concerned with de- forming maps that are nice in some topological sense into maps that are nice in some smooth sense. For instance, the main theorem is concerned with perturbing homeomorphisms into diffeomorphisms. I conclude this section with the following definition, which makes precise the relevant no- tion of perturbation.

Definition 2.7 (isotopy). Let M be an m-manifold, and N an n-manifold. A map F : [0, 1] × M → N is an isotopy if F is a such that for each t ∈ [0, 1], Ft is a topological embedding. Along the same lines, if f : M → N and g : M → N are a pair of topological , and f = F0 and g = F1 for some isotopy F : [0, 1] × M → N, then f and g are said to be isotopic.

As every isotopy is a homotopy, we can transfer over our intuition for homotopy from . Vaguely, an isotopy is a way to wiggle the image of one topological embedding into the image of another while avoiding self-intersections.

3. Proving the Main Theorem

In the remainder of this document, we restrict ourselves to the setting of smooth 2-manifolds, i.e., smooth surfaces. Recall that smooth manifolds are also topological manifolds, and so we can consider not just diffeomor- phisms of smooth surfaces, but also homeomorphisms of smooth surfaces. In doing so, we discard much of the structure that defines smooth surfaces, and one might fear that upon doing so, one cannot recover much about the smooth surfaces that were originally under study. However, Theorem 1.2 reveals that homeomorphisms of smooth surfaces are never too far away from being diffeomorphisms.

Theorem 1.2 (Main Theorem). Every homeomorphism between smooth sur- faces is isotopic to a diffeomorphism. THE TORUS TRICK 9

Figure 4. First row: a 0-handle (left) and a 2-handle (right). Second row: a 1-handle. You can think of the red edges as being adhesive. These are the edges along which each handle is attached.

The goal of this section to prove Theorem 1.2. However, although we now have the language to understand the statement of Theorem 1.2, the proof I present requires yet more machinery, namely the Handle Smoothing The- orem. I begin with an informal introduction to the theory of handles for smooth surfaces. After this, I present the statement of the Handle Smooth- ing Theorem, and finally, I use this result to sketch a proof of Theorem 1.2.

3.1. Handle Decompositions of Surfaces. This section is an informal pre- sentation of the theory of handle decompositions. The idea is that I can construct any smooth surface using three basic building blocks: 0-handles, 1-handles, and 2-handles. Each of these handles is pictured in Figure 4. The process for such a construction is perhaps best illustrated by exam- ple, so consider the torus. In the following, I decompose the torus as one 0-handle, two 2-handles, and one 2-handle. We begin with a single 0-handle, now realized as a square instead of a , as in Figure 5. To a pair of opposite edges on this 0-handle, I attach a single 1-handle, which I imagine as a long strip with adhesive edges. The result is an object resembling a wristwatch, with the 0-handle telling the time. THE TORUS TRICK 10

Figure 5. Stage one of the construction: square.

Figure 6. Stage two of the construction: wristwatch.

To the remaining pair of edges on the 0-handle, I attach yet another 1- handle, but in such way that the time is still visible, as opposed to producing a pair of handcuffs. The result is Figure 7. At last, we can begin to see the torus take .

Figure 7. Stage three of the construction: not-handcuff.

Finally, we attach the 2-handle, which like the 0-handle we started with, I realize as a rectangle. This completes the torus.

Remark. To see that the missing area completed by the 2-handle is indeed a rectangle, it may help to picture the object in Figure 8 as a life-size THE TORUS TRICK 11

Figure 8. Stage four of the construction: ring buoy. buoy levitating before you. Next, rotate the buoy so that what was the watch display is now facing away from you, and reach your hands into the donut hole from below with your palms facing you. If from this position you grab the inner side of the buoy with your hands and slide your hands left and right, you should feel a rectangle.

Let us now reflect upon what just happened. We began the whole process with a 0-handle, and this is a natural choice because 0-handles cannot be attached to anything else. After that, we attached 1-handles to the 0-handle, and in the process created a skeleton for the torus. It was at this point that we began to see the torus take shape. Finally, we completed the torus with the addition of a 2-handle. In general, any smooth surface can be constructed in a similar manner, by beginning with some number of 0-handles, attaching some number of 1-handles, and finishing the job with some number of 2-handles. For the purpose of stating the Handle Smoothing Theorem, I make rig- orous the notion of handle. A k-handle can be described as a smooth 2- 1 k 2−k manifold D × R , and is attached along the of the first factor Dk. For each k = 0, 1, 2, this translates to: 2 • a 0-handle is a copy of R ; • a 1-handle is a copy of I × R, where I = [−1, 1], and is attached along the edges {−1} × R and {1} × R;

1 k 2−k Technically, D × R is not a manifold for k > 0, but a manifold with boundary. This distinction will not be important for our discussion, so instead, I refer the curious reader to §22 in [Tu10]. THE TORUS TRICK 12

2 2 • a 2-handle is a copy of D , the unit disk in R , and is attached along its boundary. In the example of the torus, we saw that the end result of a handle decom- position is a collection of (smooth) embeddings of handles. However, we can also consider topological embeddings of handles. The following result lays out conditions under which the smoothness properties of such embeddings can be improved by isotopy.

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 × {0}, staying fixed outside a larger neighborhood of I × {0} 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.

3.2. Proof of the Main Theorem. With the Handle Smoothing Theorem in hand, the main theorem follows with relative ease. I sketch the proof below. Let S and T be smooth surfaces and let f : S → T be a homeomorphism. The central idea is as follows. Every smooth surface has a , so I fix a triangulation of S. The surface S is now covered with vertices, edges, and faces, each of which can be thickened to a 0-handle, 1-handle, and 2-handle, respectively. Restricting f to any one of these handles then yields a topological embedding of that handle into T . I can then inductively apply the Handle Smoothing Theorem to incrementally enlarge (by isotopy) the region on which f is a smooth embedding. Start from the vertices. Because S is a smooth manifold, around each 2 vertex p ∈ S, there is an open neighborhood U homeomorphic to R . Take U to be a 0-handle, and take f|U : U → T to be the corresponding embedding. Finally, apply the Handle Smoothing Theorem for 0-handles to smooth out f in a smaller open neighborhood of p. THE TORUS TRICK 13

Figure 9. Left: a triangulation of S and a 0-handle centered at each vertex. The 0-handles are outlined by the dashed lines. Right: the result of applying the Handle Smoothing Theorem to each vertex. The green highlights delineate the regions on which f is now a smooth embedding.

Having performed the above procedure for each vertex, consider now any one of the edges. Because f is now smooth near each vertex, f is smooth near the two endpoints. Thicken the edge so that the edge now lies on the center {0} × R of a 1-handle, but taking care to ensure f is still smooth near the ends. We can now apply the Handle Smoothing Theorem to smooth out f in an open neighborhood of the edge.

Figure 10. Left: Continuing from Figure 9, we choose an edge and thicken it to a 1-handle. Right: the result of applying the Handle Smoothing Theorem to the chosen handle. Again, the green highlights delineate the regions on which f is a smooth embedding.

Having performed the above procedure for each edge, we now smooth f on each face. However, f is already smooth around the vertices and edges, and hence near the boundary of each face, which we now view as a 2-handle. Apply the Handle Smoothing Theorem once more to smooth out f on each face. When this process is complete, f will be a smooth embedding on all of S, and hence a diffeomorphism. 

4. Smoothing 2-handles

Recall the setting. We are given a surface S and a topological embedding h : D2 → S of the unit disk, and h is a smooth embedding in a neighborhood THE TORUS TRICK 14

Figure 11. Continuing from Figure 10, we consider each face as a 2-handle and apply the Handle Smoothing Theorem to isotope f into a smooth embedding on the entire face. This completes the proof.

of the boundary ∂D2. The goal is to isotope h into a smooth embedding, but because h is already a smooth embedding near the boundary, we want to perform this isotopy without changing what h does in a neighborhood of the boundary. Because h is an embedding, I can via h pull back the smooth structure of S to a smooth structure on D2 by pulling back charts on S. Of course, 2 2 D also has a smooth structure induced from R by the inclusion map, and this may or may not agree with the smooth structure induced by h. To distinguish between these two spaces as smooth manifolds, I denote by D2 2 2 the disk with the smooth structure induced from R and by Dh the disk with the smooth structure induced by h. In general, the smooth structure 2 on a subset K ⊆ R induced by inclusion is said to be the standard smooth structure on K. In this language, D2 has the standard smooth structure. Recall that h : D2 → S is a smooth embedding near ∂D2, say in an open 2 neighborhood U. This implies that Dh is actually standard in U. To see this, ˆ 2 2 let h : Dh → S be the same map as h but defined instead on Dh. Both h and hˆ are smooth embeddings on U, so when I restrict h and hˆ to U and take the ˆ−1 ˆ−1 composition h ◦ h : Uh → U, I obtain a diffeomorphism. However, h ◦ h is just the identity, so what I have just shown is that the identity map Uh → U 2 2 is a diffeomorphism. That is, the smooth structures of D and Dh coincide near the boundary. Now, it is a fact that for any two smooth structures on the disk standard near the boundary, there exists a diffeomorphism from one disk to the other that is the identity in a neighborhood of the boundary. We can apply this 2 2 here to obtain a diffeomorphism g : D → Dh. Notice that the composition of h and g is a smooth embedding, for hˆ is a smooth embedding and g is a THE TORUS TRICK 15 diffeomorphism. g◦h D2 S g hˆ 2 Dh Thus, g ◦ h is precisely the kind of map that we would like to isotope h to. To achieve this isotopy, I use what is known as the Alexander trick, quoted below. A proof can be found in [Smi17].

Lemma 4.1 (Alexander Trick). Suppose g : I2 → I2 is a homeomorphism of I2 that fixes the boundary. Then g is isotopic to the identity.

Of course, the unit disk is homeomorphic to I2 and diffeomorphisms are 2 2 homeomorphisms, so I can apply the trick to obtain an isotopy gt : Dh → D from the identity map to g. Then g0 ◦ h = h, and g1 ◦ h = g ◦ h is the desired map. Moreover, the composition of an isotopy and a homeomorphism is again an isotopy, and so the assertion is proved.

5. Smoothing 1-handles

We are given a surface S and a topological embedding h : I × R → S into S of a 1-handle which is a smooth embedding near ∂I × R, the two edges along which the handle is attached. The goal is to isotope h to be a smooth embedding in a neighborhood of I × {0} (i.e., the line running along the center of the strip from one adhesive edge to the other) but without perturbing the map near ∂I × R (where h is already a smooth embedding) or outside some larger neighborhood of I × {0}. First, pull back by h the smooth structure on S to a smooth structure on I × R. As in the 2-handle case, I denote the space I × R equipped with this nonstandard smooth structure by (I × R)h to distinguish it from the same 2 set equipped with the standard smooth structure of R . It is a fact that every smooth structure on I × R standard near ∂I × R is diffeomorphism to the standard smooth structure on I × R via a diffeomorphism that fixes a neighborhood of ∂I×R. As such, I have a diffeomorphism g : I×R → (I×R)h. At this stage of the argument, what we did in the 2-handle case is invoke the Alexander trick to obtain an isotopy gt. However, the domain here is not a disk but an infinite strip, and so somehow we want to manipulate the diffeomorphism g to be a diffeomorphism of something to which we can apply the Alexander trick. This time, we use I2. THE TORUS TRICK 16

Figure 12. The embedding e : I × R → I × R folds in the top and bottom edges of the 1-handle to form the left and right sides of the square.

Let e : I × R → I × R be the topological embedding illustrated in Figure 5. In particular, note e fixes I × {0} and fits the entire image into I2 \ ({0} × ∂I), the square with two points removed. Through this embedding, the homeomorphism g induces a homeomorphism of the image of e, and this 2 2 homeomorphism can in turn be extended to a homeomorphism G : R → R by the identity outside of the interior of the square. Because G fixes the square, I can now apply the Alexander trick to the restriction of G to I2 2 to obtain an isotopy Gt that, perturbing only I and fixing everything else, goes from the identity map to G. Notice that for all t, Gt fixes the strip I ×R. Restrict Gt to I ×R to obtain an isotopy on the handle. Next, notice that for all t, this restriction of Gt fixes an open neighborhood of ∂I × R, and so h ◦ Gt is smooth near the two edges. Finally, when t = 1, G1 is exactly g in a neighborhood of I × {0}, making h ◦ G1 a smooth embedding. Together, these observations verify that h ◦ Gt is the desired isotopy.

6. Smoothing 0-handles

2 We are given a surface S and a topological embedding h : R → S into S of a 0-handle. The goal is to isotope h to be a smooth embedding in a neighborhood of the origin. As in the preceding two cases, I can pull back by h the smooth structure 2 2 on S to a smooth structure on R , yielding a smooth manifold Rh on which h is a smooth embedding. Our experience in the higher-dimensional cases 2 2 suggests the next step is to produce a diffeomorphism g : Rh → R . Indeed, such a diffeomorphism necessarily exists because up to diffeomorphism, 2 there is only one smooth structure on R . However, this path quickly leads THE TORUS TRICK 17

Figure 13. Handle decomposition of the torus.

2 to a dead end, for one cannot apply the Alexander trick to R . We could try 2 emulating our solution to the 1-handle case, by radially reparametrizing R to the disk, applying the Alexander trick, and extending by the identity to 2 the remainder of R . However, this fails because radial reparametrization produces only a self-diffeomorphism of the interior of the disk, and the Alexander trick requires a self-diffeomorphism of the entire disk. To resolve this, we could try extending this diffeomorphism to the boundary by the identity, but doing so may not yield a continuous map if |g(x) − x| is not bounded. On the other hand, if |g(x) − x| is bounded, then we can proceed as we did in the 1-handle case. This inspires the following approach: I manipulate g so that |g(x) − x| is bounded, and I do so using what is known as the Kirby torus trick. 0 2 The first step of the trick is to immerse the punctured torus T into R 2 0 and pull back the smooth structure on R to a smooth structure on T . This immersion is constructed as follows. Start with the handle decomposition of the torus. We saw in Problem 1 of Homework 7 that the punctured torus is homotopy-equivalent to the wedge of two . By thinking of the 2- handle as a fattened puncture and the two 1-handles as fattened copies of S1 \ {−1}, we can consider T 0 as being just the 0-handle and two 1-handles and immerse T 0 by embedding each handle separately, and thus, we have our immersion. Inevitably the images of the two 1-handles overlap, so we get something like Figure 14. I now make two claims without proof. The first is that the smooth struc- ture on T 0 induced by the map T 0 → S extends to the entire torus. To

distinguish this torus from the standard torus, I denote the former by TS and the latter simply by T . The second claim is that, up to diffeomorphism, there is only one smooth torus. It follows that there is a diffeomorphism

g : T → TS . What we would like is for this map g to lift to a diffeomorphism THE TORUS TRICK 18

Figure 14. T 0 immersed in the plane. g˜ : R → RS fixing integer points, but in order for this to happen, we need to make some adjustments to g.

Without loss of generality, I assume that the point x0 = (1, 1) is fixed, and I may do so by composing g with rotations of each factor of S1 as needed. 2πit 2πit More explicitly, if g(x0) = (e 1 , e 2 ), then composing g with

−2πit1 −2πit2 (z1, z2) 7→ (z1e , z2e )

will produce a diffeomorphism fixing x0. This alone means that the lift of g sends integer points to integer points, but we do not yet have that integer points are fixed.

Consider now what g˜ does to integer points. Because x0 is fixed, we can consider g as a map (T, x0) → (TS , x0) of pointed spaces. We then get the ∼ 2 induced map g∗ : π1(T, x0) → π1(TS , x0). We have seen that π1(T, x0) = Z , 2 and so I identify the two generators of π1(TS , x0) with (1, 0) and (0, 1) of Z . 2 Then for any element (u, v) ∈ Z ,

g˜(u, v) = g∗(u, v) = u · g∗(1, 0) + v · g∗(0, 1),

2 and this reveals that as a map on Z , g˜ is given by a A ∈ M2×2(Z). Moreover, because g is a homeomorphism, g˜ is invertible, and so we actu- ally have that A is invertible. Now take the inverse. Since A−1 is a self- 2 −1 −1 diffeomorphism of R , A descends to a self-diffeomorphism A of the torus. Therefore, composing g with A−1 creates a self-diffeomorphism of

the torus that still fixes x0, but that also lifts to a map fixing all integer points. It follows from the discussion just above that we can assume without loss of generality that g˜ : R → RS fixes the integer . The reason this is desirable for our purposes is because this implies we can bound |g˜(x)−x| by THE TORUS TRICK 19

2 some constant; that is, g˜ never moves points in R too far away from their 2 original position. To see this, begin by observing that for any (u, v) ∈ R , g˜(u, v) =g ˜(u − buc, v − bvc) +g ˜(buc, bvc). (6.1)

Because g˜ fixes the integer lattice, g˜(buc, bvc) = (buc, bvc), so |g˜(u, v) − (u, v)| ≤ |g˜(u, v) − (buc, bvc)| + |(buc, bvc) − (u, v)| √ ≤ |g˜(u − buc, v − bvc)| + 2.

It now suffices to bound |g˜(u − buc, v − bvc)|, but g˜(u − buc, v − bvc) is in the image of the unit square [0, 1]2 under g˜. Since the unit square is compact and g˜ is continuous, this image is bounded, and the claim follows. Notice also what can happen if g˜ does not fix integer points. For example, consider what happens when g˜(u, v) = (2u, 2v). We would then have |g˜(u, v) − (u, v)| = |(u, v)|,

which is evidently unbounded, even though g˜ is the lift of a diffeomorphism of the torus. Remember that the reason for all this work into ensuring |g˜(x) − x| is bounded is ultimately so we can invoke the Alexander trick. As planned, I obtain via radial reparametrization a diffeomorphism G from the interior of the unit disk to itself. Next, I can extend this diffeomorphism to the entire disk by defining G to be the identity on the boundary. Now apply

the Alexander trick to obtain an isotopy Gt from the identity to G. Finally, 2 extend Gt to the entire handle R by defining Gt to be the identity outside of the disk as well, and for all t. If we retrace our steps, we find that we are at the tail end of the argument. Specifically, observe that near the origin, G takes a neighborhood of the 2 2 origin in R and maps it onto a neighborhood of the origin in Rh. Thus, when I compose G with h, I see that h ◦ G is a smooth embedding near the origin.

This means the composition h◦Gt is an isotopy from h to another topological embedding that is a smooth embedding near the origin, as desired. THE TORUS TRICK 20

Acknowledgements

This document is a final project for the Fall 2020 offering of MATH 4051 Topology taught by Prof. Mike Miller at Columbia University. The author extends his gratitude to Prof. Miller not only for his valuable input through- out the process of preparing this document, but also for introducing him to Hatcher’s paper and guiding him through what would otherwise have been a hopeless undertaking.

References

[Hat13] Allen Hatcher, The kirby torus trick for surfaces, 2013. [Mil20] Mike Miller, Private Communication, 2020. [Smi17] H. L. Smith, On continuous representations of a square upon itself., Annals of Math- ematics 19 (1917), no. 2, 137–141. [Tu10] L.W. Tu, An introduction to manifolds, Universitext, Springer New York, 2010.