Heegaard Splittings of noncompact 3-manifolds related to the Cantor Set. Sarah Kirker

Transcription

1 Heegaard Splittings of noncompact 3-manifolds related to the Cantor Set Sarah Kirker May 14, 2011

2 Contents 1 Introduction 2 2 A first look at Heegaard Splittings Extending our definitions Noncompact Heegaard Splittings Constructing Heegaard Splittings by Amalgamation A standardly embedded Cantor Set Description Description of S 3 K Constructing a Heegaard splitting of S 3 K Antoine s Necklace Description Description of S 3 A Constructing a Heegaard splitting of the complement of Antoine s Necklace The Horned Spheres Description of Antoine s Horned Sphere Description of Alexander s Horned Sphere Alternate construction of Alexander s Horned Sphere Challenges in constructing Heegaard splittings of the Horned Spheres Conclusion and Further Research Questions 26 7 Notation Index 26 1

3 1 Introduction In this project we will investigate a very interesting branch of 3-manifold topology, the field of Heegaard splittings. Heegaard splittings were first introduced by the Danish mathematician Poul Heegaard in They remained an important area in topology through the years, becoming particularly prominent in the 1960s when topologists Wolfgang Haken and Friedhelm Waldhausen studied them extensively. The generalization of this theory into the noncompact case was pioneered by Frohman and Meeks in their paper from the early 1990s. Recently, Heegaard splittings have been applied chiefly to study properties of complicated 3-manifolds. They have been used to construct different invariants of 3-manifolds, as well as in more specific theoretical areas, such as the study of minimal surfaces. In our paper we give a basic introduction to the field of Heegaard splittings for compact and noncompact cases. We also investigate a few examples of the applications of one of the accepted methods of creating a noncompact Heegaard splitting; amalgamation. One of the original goals of this project was to create Heegaard splittings of the horned spheres by amalgamation, however we were unable to achieve this. Instead we will outline some of the problems that we face in applying this method to these particular 3- manifolds and pose some research questions that we believe merit further study. 2 A first look at Heegaard Splittings Before we begin working with Heegaard splittings we need some fundamental definitions. These definitions apply to the basic, compact case and are an important first step in understanding the more general definitions that we will use throughout this paper. Informally, Heegaard splittings are the decompositions of (possibly complicated) 3-manifolds into handlebodies attached by a homeomorphism along their boundaries. By creating these splittings we can simplify problems about the geometry and properties of complicated 3-manifolds to questions about much simpler, and much better studied, handlebodies. Definition 1. A handlebody of genus g is an orientable 3-manifold obtained by attaching g disjoint 1 handles D 2 [0,1] to a 3-ball B 3 by using gluing homeomorphisms to attach D 2 {0} and D 2 {1} to g disjoint discs D 1,..., D g on B 3 as shown in Figure 1. 2

4 Figure 1: A genus two handlebody before (above) and after (below) the gluing homeomorphism Definition 2. A genus g Heegaard splitting of a compact 3-manifold M is a triple (H 1, H 2, h) where each H i is a handlebody of genus g and h : H 2 H 1 is a homeomorphism such that M = H 1 h H 2. It is helpful to look at a few examples in order to get a stronger grasp on the foundations of the theory of Heegaard splittings. Example 1. A Heegaard splitting of S 3 The simplest splitting of S 3 is the genus zero splitting created by gluing the boundaries of two 3-balls together. However, there are also many higher genus Heegaard splittings of S 3. The clearest way to present these higher genus splittings is in terms of the characteristic curves of a diagram. Definition 3. The characteristic curves of a diagram with a Heegaard splitting H 1 h H 2 are the images C i H 1 of the boundary of the disks D i (1 i g) under the homeomorphism h: H 2 H 1. Where D i is equal to the disks D 2 { 1 2 }, that is, the centers of the 1-handles of H 2. For our example we will take the Heegaard splitting where H 1 and H 2 are each tori, or genus one handlebodies. Then a splittings of S 3 in terms of characteristic curves is shown in Figure 2 We can also construct a higher genus splitting of S 3 by taking H 1 and H 2 to be genus 2 handlebodies as shown in Figure 3. 3

5 Figure 2: A genus one Heegaard splitting of S 3 Figure 3: A genus two Heegaard splitting of S 3 4

6 This example demonstrates a very important fact about Heegaard splittings: they are not unique. As in Example 1, one 3-manifold can have splittings of different genera. Furthermore, different 3-manifolds can have splittings of the same genera. Example 2. A Heegaard splitting of S 2 S 1 We can also create a different 3-manifold by changing the gluing map h relating H 1 and H 2. If we take H 1 and H 2 to be two genus one handlebodies, as we did in the first part of Example 1, and instead of gluing the meridian of H 2 to the longitude of H 1, we glue meridians to meridians as shown in Figure 4, we get S 2 S 1 instead of S 3. Figure 4: A genus one Heegaard splitting of S 2 S Extending our definitions For the first step in the broadening of this definition we will generalize the handlebody construction to a compression body. Definition 4. A compression body H is a connected 3-manifold constructed by taking a closed surface H and attaching 1-handles to H {1} H [0,1] See Figure 5 for a schematic picture. For notational purposes, we define + H as H - H. In order to ensure that this definition is truly a generalization, it is convention to consider a compression body with H = Ø to be a handlebody. The definition of Heegaard splittings has a corresponding generalization. Definition 5. For compression bodies H 1 and H 2 with + H 1 homeomorphic to + H 2 glue H 1 to H 2 along + H i = S to get a Heegaard splitting of the compact 3-manifold M = H 1 S H 2. 5

7 Figure 5: A schematic drawing of a compression body 2.2 Noncompact Heegaard Splittings So far in our discussion we have only dealt with compact 3-manifolds, however, Heegaard splittings are also applicable in a more general setting. By slightly modifying our definitions to apply more broadly we can extend the theory of Heegaard splittings into noncompact cases. Notice that every handlebody H as defined in Definition 1 contains a finite graph Σ such that H is a regular neighborhood of Σ. We call this Σ the spine of the handlebody. See Figure 6 for an example. Figure 6: A handlebody H with spine Σ To extend the definition of handlebodies to the noncompact case, we define handlebodies as follows: 6

8 Definition 6. A (possibly noncompact) handlebody H is the regular neighborhood of a graph properly embedded in R 3. The existence of these spines is what makes this definition an extension. This new definition allows us to take the neighborhoods of infinite graphs, which give us noncompact handlebodies. With this definition of handlebodies we can construct Heegaard splittings of noncompact 3-manifolds in the obvious way. Definition 7. A Heegaard splitting of a noncompact 3-manifold M is a triple (H 1, H 2, S) such that M=H 1 S H 2 and H 1 and H 2 are handlebodies as defined in Definition 6 and S = H 1 = H 2. A foundational theorem in this extension of the field of Heegaard splittings is that after allowing this generalization of the definition, every orientable 3-manifold has a Heegaard splitting. In this paper all of the most interesting manifolds dealt with will be non-compact so this result is an important step in legitimizing our future work. The basic idea behind the proof of this theorem rests on a very difficult result proved by Moise and Bing that every 3-manifold has a triangulation. The proof sketched below is the one found in [R], modified to include our new definition of Heegaard splittings. Theorem 1. Every orientable compact or open 3-manifold has a Heegaard splitting. Sketch of proof: Let M 1 be the 1-skeleton of the triangulation of an orientable 3-manifold M. Notice that M 1 is made up of tetrahedron. We call the vertices and edges of these tetrahedrons the 1-skeleton of M 1. Then let M 2 be the dual 1-skeleton of the barycentric subdivision, i.e. set M 2 equal to the union of the sets of vertices and edges in the subdivision which do not lie on a simplex that intersects M 1. Now take the simplicial neighborhoods of M 1 and M 2 and call them N 1 and N 2 respectively. To get an idea of what N 1 and N 2 look like for a single tetrahedron of the triangulation see Figure 7. Notice that both N i are thickened neighborhoods of a 1-skeleton. Clearly each N i i=1,2 is a 3-manifold such that N 1 N 2 is equal to M. Therefore we just need to show that N i are handlebodies. Since we defined N 1 and N 2 as neighborhoods of the graphs M 1 and M 2 we can refer to Definition 6 to make this completely clear. Therefore we have our Heegaard splitting, M = N 1 N 2 which is dependent only on the triangulation of M. The immediately relevant consequence of this theorem is that it is possible to construct Heegaard splittings of orientable open (noncompact without 7

9 Figure 7: Example of pieces of N 1 (left) and N 2 (right) boundary) 3-manifolds. Frohman and Meeks were the first to introduce this idea in their paper [FrMe] and Taylor explored it further in [T]. Example 3. Constructing a Heegaard splitting of R 3 To construct a Heegaard splitting of R 3, we notice that the upper half space and the lower half space are each homeomorphic to a closed regular neighborhood of rays along the z-axis. Thus for each of these spaces we can take sets homeomorphic to 3-balls around these rays to give us a genus zero Heegaard splitting see Figure 8. Figure 8: A Heegaard splitting of R 3 8

10 2.3 Constructing Heegaard Splittings by Amalgamation One process often used to create a Heegaard splitting of an open manifold is the process of amalgamation, which is to say by amalgamating Heegaard splittings of compact submanifolds. Because we use this method extensively in this paper, a thorough investigation into how it works is vital. First let us look at amalgamation in the compact case. Intuitively, compact amalgamation is an easy process to understand. Think of taking two 3-manifolds, each with their own Heegaard splitting, and using a specified gluing homeomorphism to connect them along their boundary components. If done correctly, this process yields a single 3-manifold with a Heegaard splitting composed of the pieces of the splittings of the original 3-manifolds. The process itself is a bit more complicated. To ensure that we get a Heegaard splitting of our newly created 3-manifold we must first make sure that our original splittings do not separate the boundary components. That is to say that all boundary components of each 3-manifold must be contained completely in the interior of one of the compression bodies. While this does not come into play in the compact case, it is crucial in the noncompact case, so we will just include it in our process. Then, within our new 3-manifold there will be distinct product regions of the set of boundary components cross I. To get our new Heegaard splitting we will run 1-handles through this product region. Start the handles from disjoint disks within the set of boundary components of one of the compression bodies, then attach them to the opposite boundary surface of the product region. These handles, together with the compression bodies of the original Heegaard splittings yield a Heegaard splitting of the amalgamated 3-manifold. For a more complete, technical description of the compact amalgamation process see [T]. To use this process of amalgamation to create a splitting of a noncompact manifold M it is first necessary to create an exhaustion {K i } of M. Definition 8. An exhaustion of a noncompact 3-manifold M is a sequence {K i } of compact connected 3-submanifolds such that K i int(k i+1 ) and M = i K i. To construct a splitting of M, we create splittings U i V i of each K i in such a way that the boundary components of each K i are contained completely in each V i. By amalgamating along these boundary components we can create a splitting U V of M in which we use the U i and V i to form U and V 9

11 respectively. Let L i = cl(k i+1 K i ). Then F i, from the discussion above, is equal to L i K i. To get K i+1 we amalgamate K i and each component of L i along a component of the surface F i. We let L i = X i Y i be a Heegaard splitting such that Y i contains all boundary components of L 1. See Figure 9 for a schematic illustration of this set-up. This figure shows the Heegaard surfaces that get fused into one during the amalgamation process i.e. the dashed lines separated by the thick straight lines, as well as the horizontal tubes that are incorporated into the 1-handles that we eventually use to connect our compression bodies. A comparison of Figure 9 and Figure 10 shows schematically what happens to these components during the process of amalgamation. To create a Heegaard splitting of K n, we amalgamate the compact splittings of K n 1 and L n 1. Figure 9: A schematic picture of M before we begin amalgamating For a detailed proof that this does in fact result in a Heegaard splitting as we have defined it see [T]. With this background in place, we can now move on to creating Heegaard splittings of more complicated manifolds. 10

12 Figure 10: A schematic picture of after the amalgamation 3 A standardly embedded Cantor Set 3.1 Description In R, the well known Cantor set K is a well known subset of the interval [0,1] created by removing the middle thirds of intervals. We first remove the open interval ( 1 3, 2 3 ), then the middle third of the remaining intervals, i.e. ( 1 9, 2 9 ) and ( 7 9, 8 9 ). We continue this process inductively removing 2n 1 intervals of length 1/3 n intervals at the nth step. For the first three steps see Figure 11. We define K as the points remaining after this process. That is to say that K = n=1 I n where I n is the union of the intervals remaining at step n. Figure 11 shows the first three steps of this construction. The standard embedding of K into S 3 is to think of S 3 as R 3 { } and take the x-axis as R. The complement, S 3 K, is a simply connected noncompact 3-manifold, and thus we can create a Heegaard splitting of it. 11

13 3.2 Description of S 3 K Figure 11: The Cantor set K in R To create a Heegaard splitting of the complement of K in S 3 by amalgamation we need an exhaustion of this manifold. To construct this exhaustion first let B i be the union of regular neighborhoods of each component of I n at each step n. See Figure 12 Figure 12: Regular neighborhoods B 1, B 2, B 3 of K in S3 12

14 Notice that at each step n, there are 2 n components of B i and each is homeomorphic to a 3-ball. Furthermore, each B i completely contains B i+1. Let B i be the closure of the space between B i and B i 1 i.e. cl(b i \B i 1 ) as seen in Figure 13. Then {B i (S3 K)} is an exhaustion of S 3 K. Figure 13: The component B i This is homeomorphic to a punctured 3-ball because it is equal to the complement of a set of 3-balls within a larger 3-ball. For the B i at each step n we will add 2 n 1 of these. 3.3 Constructing a Heegaard splitting of S 3 K To continue the amalgamation process we need a Heegaard splitting of each B i that does not separate the boundary components. To do this we construct the splitting as shown in Figure 14. Notice that the boundary components of each B i are contained in V i which is what we were looking for. Amalgamating these splittings is simple as we just map the boundary components of the nth stage to the boundary components of the (n 1)st stage as shown in Figure 15 Since each U i and V i is a genus zero handlebody, this construction gives us a genus zero Heegaard splitting of S 3 K. 13

15 Figure 14: A Heegaard splitting U i V i of B i Figure 15: The first four steps of the amalgamation of the Heegaard splitting of S 3 K. 14

16 4 Antoine s Necklace 4.1 Description The complement of the standard K in S 3 above is simply connected. In 1921, Louis Antoine proved that there existed a compact set in S 3 homeomorphic to the Cantor set that had a complement with nontrivial fundamental group. This set is known as Antoine s Necklace. To construct Antoines Necklace first we embed a chain of n, n 2 components into a solid torus V. Definition 9. A chain of n components is a link of n unknotted circles arranged as shown in Figure 16 inside a solid torus standardly embedded in S 3. Figure 16: A chain of n links To continue this construction, we thicken each of the chain components to form a chain C 1 of solid tori. In each of these n tori components we construct a smaller chain of solid tori embedded exactly as C 1 was in V. Call the union of these n smaller chains C 2. See Figure 17 for the first 3 steps. We continue this, constructing another chain within each component of C 2. Then we repeat countably many times to obtain a sequence C 1 C 2 C 3... with the diameters of the components C i 0 as i. We call the intersection of these chains 15

17 Figure 17: Step by step construction of Antoine s Necklace A = i=1 C i Notice that there are actually infinitely many versions of A, one for each n N, where n is equal to the number of components in each chain C i. To keep this distinction clear, we will differentiate between these versions by referring when necessary to a necklace with n components as A n and using A to refer to the entire set of A n. Antoine s Necklace is interesting on several levels, the first of which is that for large enough values of n, A n is homeomorphic to the Cantor set, K. Lemma 1. For large enough n, A n = K Proof: By the construction above we know that A = i=1 C i. From this fact, we can deduce that A n = i=1 X i where X 1 = {1, 2, 3,..., n} with the discrete topology. Similarly, and K = n=1 I n is homeomorphic to i=1 Y i where Y i is a two point set with the discrete topology. Since all functions from a set with the discrete topology are continuous, any bijection is a homeomorphism [DaVe]. To prove that Antoine s Necklace is not identical to the standard Cantor set we will just note that it is possible to show that S 3 K is not simply connected [DaVe], [R]. 16

18 4.2 Description of S 3 A To construct a Heegaard splitting of the complement of A in S 3 by amalgamation we need to define an exhaustion {S i }. Before we begin, note that it suffices to work with the case of A 2 in which n=2, i.e. each chain in A has only two links. Lemma 2. If we have a Heegaard splitting A=U V for A for n=2, then we can easily extend the construction to a splitting A = U V for A for any n. Sketch of proof: The key to this proof is that the Heegaard splittings of the components of the complement will all be identical. Therefore, the only difference in a splitting of A n and A n+1 is the number of boundary components. Since from our construction we know that the embeddings are identical at each step, the result of adding an extra link in the chain is to change the number of identical components of the complement at each step k from n k to (n + 1) k. However, this will cause an equal increase in the number of boundary components in step k 1, and the gluing map described below will still be applicable. With this in mind we will for the remainder of this section deal only with the case where n=2. The first step of this process it to identify the compact, connected components that make up the complement of A in S 3. The first building block is S 3 - V where V is a solid torus embedded in S 3. This is another solid torus and we will ignore it for now. For each of the kth remaining steps we need to add 2 k identical building blocks which we will denote as S i. Each of these S i is homeomorphic to the exterior of the link L in Figure 18. The set { n i=1 S i} n is an exhaustion of A n. To see why the exterior of L is the building block we want, we recall that the exterior of unknotted S 1 in S 3 is a solid torus. Thus the exterior of L is a solid torus minus a chain of two links. By the construction of Antoine s necklace given above it is easy to see that this is exactly what we want our building blocks to be. The next step is to create a Heegaard splitting of each of these components that we can later use in the amalgamation process. In order for the process to work properly we require a Heegaard splitting of the exterior of L that does not separate the boundary components of the resulting handlebodies/compression bodies. 17

19 Figure 18: Link L with exterior homeomorphic to each S i One way to achieve this is to find arcs such that when we add them to L we can transform L into a planar graph. By Definition 6 we know that the thickened neighborhood of the link itself will be a handlebody, which immediately gives us our H 1. Then, since the transformations we use to change L into a planar graph will induce an ambient isotopy of this handlebody, we can quote the well known result that the exterior of a planar graph in S 3 is a handlebody. By this result, we can take H 2 to be the exterior of our original graph. Thus, taking the thickened link L and its complement gives us a Heegaard splitting of our building block. Notice that we choose to add the arcs a 1 and a 2 into L as in Figure 19 below, because these fulfill our constraints and give us our Heegaard splitting. Figure 19: Link L with the arcs a 1 and a 2 added Figure 20 shows how the addition of these arcs allows us to transform L into a planar graph. 18

20 Figure 20: Transformation from the link L to a planar graph Now, to construct our desired handlebody and compression body, we thicken a neighborhood of L (a 1, a 2 ). This thickened neighborhood V i gives us the compression body, and the complement of V i the handlebody. Therefore, Figure 21 shows a Heegaard splitting of each building block S i, S i = U i V i. Notice that each of the S i has three boundary components b i1, b i2, b i3 corresponding to the three components of the original link L contained completely in the compression body V i. For A n, n 3 the we use a very similar process to create our splittings. In place of the Link L, we will have a link L that has n + 1 components (a chain of N components plus the extra components added as in the case of L). Then we add arcs a n between each of the components of the link until all but the last piece of the chain are joined. Now that we have our exhaustion and Heegaard splittings, we are ready to begin amalgamating. 19

21 Figure 21: Thickened link L with boundary components 4.3 Constructing a Heegaard splitting of the complement of Antoine s Necklace Remember, our goal is to amalgamate these splittings U i V i of each S i into a splitting U V of the entire 3-manifold S 2 A. To do this we will amalgamate along the boundary components in such a way that the resulting division remains a Heegaard splitting. Figure 22 shows the schematic picture of this setup. Notice that in Figure 22 two boundary components, b i2 and b i3 are represented by the same line, to ease the illustration of the gluing map. The Heegaard splitting in the schematic picture is exactly equal to the original splitting we constructed. To form it we thickened the boundary components then inserted tunnels corresponding to the arcs a 1 and a 2 between b i1 and b i2 and b i2 and b i3 respectively. Now to amalgamate these splittings, we glue along these boundary components associating b i1 with b i+12 and b i+13 as shown in Figure 23. Repeat this for each i. For the final step in the amalgamation, glue the single component V (which we were ignoring throughout the complicated portion of the process) to the first boundary component of S 1. This completes the amalgamation and leaves us with the Heegaard splitting U V along the surface created 20

22 Figure 22: Schematic picture of the unamalgamated Heegaard splitting Figure 23: A schematic picture of after the amalgamation 21

23 by amalgamating the boundary components. The resulting U and V are both noncompact and infinite genus handlebodies, so this process gives us an infinite genus Heegaard splitting of the complement of Antoine s necklace in S 3. 5 The Horned Spheres The complement of Antoine s Necklace is not the only manifold closely related to the Cantor set; two more examples are Antoine s Horned Sphere and Alexander s Horned Sphere. These horned spheres are inequivalent topological 2-spheres. They each bound a 3-ball in S 3. However, unlike the standard 2-sphere in 3-space they each have a complement that is not simply connected. This characteristic was the motivation for these constructions, as it makes them counterexamples to the Schoenflies Theorem in dimension three. J.W Alexander explained the construction of these spheres in his papers [A1] and [A2]. 5.1 Description of Antoine s Horned Sphere Antoine s Horned Sphere got its name from the close connection because of the large role Antoine s Necklace plays in its construction. To build Antoine s Horned Sphere, we begin with a copy of A n (recall from the previous section that this is a copy of Antoine s Necklace with a chain of n links). Next we attach a truncated cone B 0 to V by a homeomorphism h 0 taking the top of B 0 to a disk on the boundary of V. Figure 24: The first step in the construction of Antoine s Horned Sphere 22

24 Next add n miniature copies of B 0 attached to the top of B 0 and call their union B 1. Extend h 0 to h 1 which embeds each component of B 1 in cl(v - C 1 ) as B 0 was embedded in S 3, that is the top of each of the n components of B 1 is attached to a disk on the boundary of each of the n chain links of C 1 as shown in Figure 25. Figure 25: Constructing Antoine s Horned Sphere Since we constructed A in such a way that C i+1 lies in C i exactly how C 1 lies in V we can keep repeating this process inductively to embed each B i. At each step i, B i will have exactly n i components, corresponding to the n i chain link components of each C i. 5.2 Description of Alexander s Horned Sphere Alexander also conceived of another method of constructing a horned sphere that did not rely on using Antoine s Necklace. We begin with a solid torus 23

25 W. This is then cut along a meridian, leaving two disks δ 1 and δ 2. To each disk attach a solid tube to a smaller disk in the interior of δ i. We embed the tubes in such a way that they interlock as if they were links in a chain. Then, we cut along a meridian of each tube to get four more disks δ 3, δ 4, δ 5 and δ 6. We repeat this inductively, creating 2 n new disks at each step n. The resulting construction will be Figure 26. Figure 26: Alexander s Horned Sphere 5.3 Alternate construction of Alexander s Horned Sphere An equivalent way of constructing this horned sphere which may be easier to visualize is to instead construct the complement component-wise. To do this begin with a solid torus W, the complement of W in S 3. Then at each step n of the construction add 2 n copies of the space X shown in Figure

26 Figure 27: A component X i of the complement of Alexander s Horned Sphere These components clearly reflect the tubular construction from above which gives us an obvious gluing map. Simply glue these components in such a way that it takes the disk M to the meridians of the tubes cut in the first version of the construction. This construction yields the same results as the first. 5.4 Challenges in constructing Heegaard splittings of the Horned Spheres It seems upon first glance that the amalgamation process used to construct the splittings of the standardly embedded Cantor set and the complement of Antoine s necklace should work to create splittings of these manifolds. However, in each case while simply applying this method to these manifolds we run into trouble. For Antoine s Horned sphere the issue comes when we try to create an exhaustion. Because of the tubes that we added to the construction of Antoine s Necklace there is a concrete connection between each step of the construction. Therefore, the decomposition of the space into components is much more complicated than in the previous examples. In the case of Alexander s Horned sphere the issue is with the boundary components. In our constructions of the splittings of the complements of the standard Cantor set and Antoine s Necklace the number of components less than or equal to the number of components in the next step. Therefore, we can associate the boundary components without having any extra components. However, in the case of the Alexander Horned Sphere components this is not true. Consequently, the method of amalgamation we have been using does not work. 25

27 6 Conclusion and Further Research Questions The principal results of this project were the Heegaard splittings of the complements for the standardly embedded Cantor set and Antoines Necklace in S 3. The completion of this study also revealed that the process of amalgamation as it was used to construct the splittings of S 3 K and S 3 A does not work to create a splitting of the closely related Antoine s Horned Sphere and Alexander s Horned Sphere. This project did however open the door for further research into this area, in particular bringing up several interesting questions for further study of these particular Heegaard splittings. 1. In the field of Heegaard splittings there exists an idea called stabilization, or adding unnecessary handles to a Heegaard splitting. The Heegaard splitting of the complement of Antoine s Necklace given here is of infinite genus. Is there a way to create a simpler splitting? i.e. Is the splitting given here stabilized? 2. Although amalgamation does not work in the same way as above, we know that the horned spheres, as orientable 3-manifolds, have Heegaard splittings. Can we modify the amalgamation method or find a new method to construct them? 3. What do these Heegaard splittings tell us about the 3-manifolds they come from? Are there any surprising or illuminating conclusions we can draw from further study of these constructions? 7 Notation Index A list of some of the undefined notation used in this paper cl(x) - The closure of a set X. i.e. boundary. the interior of the set plus its S 3 - Compact 3-space. Created by gluing together the boundaries of two 3-balls, or by adding the point { } to R 3. I - The closed unit interval [0,1]. X h Y - The result of associating two spaces X and Y by the homeomorphism h : X Y. 26

28 References [A1] Alexander, J.W. Remarks on a Point Set Constructed by Antoine. Proceedings of the National Academy of Sciences of the United States of America Vol. 10, No. 1 (Jan. 15, 1924), pp [A2] Alexander, J.W. An example of a simply connected surface bounding a region which is not simply connected. Proceedings of the National Academy of Sciences of the United States of America Vol. 10, No. 1 (Jan. 15, 1924), pp [DaVe] Daverman, Robert J., and Gerard Venema. Embeddings in Manifolds. Providence, RI: American Mathematical Society, [FrMe] Frohman, Charles; Meeks, William H., III The topological uniqueness of complete one-ended minimal surfaces and Heegaard surfaces in R3. J. Amer. Math. Soc. 10 (1997), no. 3, [R] Rolfsen, Dale. Knots and Links. Berkeley, CA: Publish or Perish, [Sc] Scharlemann, Martin Heegaard splittings of 3-manifolds. Low dimensional topology, 2539, New Stud. Adv. Math., 3, Int. Press, Somerville, MA, 2003 [T] Taylor, Scott A. On non-compact Heegaard splittings. Algebr. Geom. Topol. 7 (2007),