What We Don't Know About 4-Dimensions

What we know and what we don’t know about 4-dimensions

My plan for this lecture is to tell you a little about what we do know and something of what we don’t know about 4-dimensional spaces.

1. Manifolds, dimensions and coordinates I must digress first to tell you what I mean by a 4-dimensional space. To this end, let me start with the notion of a manifold. This is a space that at small scales looks flat and featureless. Imagine the surface of a ball. If you look at it at very small scales, you won’t see that it is curved. To an ant, it seems to all intents and purposes like the surface of a table. Figure 1 below depicts a surface in space, an example of a manifold. A region that looks flat has been magnified.

Figure 1.

When I say featureless, I mean featureless in the sense that the straight line or a flat plane or any Euclidean space is featureless. There is nothing to distinguish one point from another or one direction from another. Figure 2 depicts the line and the plane, the two smallest Euclidean spaces.

Figure 2.

A technical definition has a manifold being a space whose points have neighborhoods that can be identified with a Euclidean space. The number of coordinates of the relevant Euclidean space is the dimension of the manifold. Said differently, the dimension a given manifold is the number of local

1 coordinates that you need to specify your position in a neighborhood of any one of its points. For example, if you were to live on a circle, then you would need to give but one coordinate to specify your position. You would need only to tell your friends the angle of your abode from some reference point. See Figure 3; the house is 138º counter clockwise from the reference point.

Figure 3.

The circle is an example of a 1-dimensional manifold. It is denoted by S1; the S standing for sphere (not sircle) and the 1 being the dimension. The surface of the earth is an example of a 2-dimensional manifold. You need to give locally 2 parameters--latitude and longitude--to completely specify the position of a point. For example, the position of this Math Forum building is 18º 19´ 42´´ north lattitude and 109º 24´ 37´´ east longitude. If I wanted to give someone directions as to where to find this lecture room, I would also have to give them a third coordinate, this being the height over the surface of the earth. According to Google earth, this lecture hall is 153 meters about sea level. So, we need not two, but three coordinates to describe our world because we can move in three directions, not two. Now suppose that someone wanted to actually see my lecture. If this is the case, then they would have to also know a fourth coordinate, this being the time of the lecture, December 19, 2013 at 6:30 GMT which is 14:30 local time. This illustrates the fact that our observable universe is 4-dimensional, one needs 3 spatial coordinates and a fourth temporal coordinate to completely specify the position of any given point. In general, a 4-dimensional manifold is characterized by the property that each of its points has a neighborhood that looks like 4-dimensional Euclidean space. This means that each point of this neighborhood is completely specified by four numbers and that any given set of four numbers specifies a point. These 4 numbers are the coordinates of the points. Note however that the coordinates that are used for a neighborhood of a chosen point can differ from those that are used near some other point. If there were just one

2 coordinate system that worked everywhere, then the manifold would be either the whole of 4-dimensional Euclidean space, or a part of 4-dimensional Euclidean space. This fact that one coordinate set may not work over the whole manifold signals the existence of structure at large scales. This is true in any dimension. The case of the circle is perhaps the simplest example whereby the Euclidean coordinate given by the counter-clockwise angle is only locally single valued. Go around the circle once and the angle increases by 360º. This is a manifestation of the fact that the circle looks like the 1- dimensional Euclidean space only at small scales. It has a non-Euclidean structure at large scales. By way of a second example, the coordinates of latitude and longitude for the surface of the earth don’t work as local coordinates near the north and south poles. You will need to use a different set of local coordinates near these two points. This is a manifestation of the fact that the surface of the earth has non-Euclidean structure at large scales.

2. The large scale shapes of the earth and the universe The earth is a solid sphere and its surface is just like the surface of a ball. Modulo relatively small bumps (i.e hills and mountains), the surface of the earth looks to our eyes like a flat surface, the 2-dimensional Euclidean space. Indeed, many people in antiquity thought the earth was flat. Even so, the ancient Greeks already knew that the earth was round. There is an annotated version of a lecture course given by a Greek savant by the name of Gemino. The annotated version is by James Evans and J. Lennart Berggren and it is called Gemino’s “Introduction to the Phenomena”A Translation and Study of a Hellenistic Survey of Astronomy (Princeton University Press 2006). Gemino was the equivalent of a college professor and his lectures were meant to teach his students about the ‘phenomena’, the phenomena being astronomy and geometry. In Hellenistic times, astronomy and geometry were taught together as aspects the same thing. Gemino in his lectures gives three pieces of evidence for the earth being round. The first is the following fact: On the summer solstice, if you put a vertical stake in the ground at Alexandria in Egypt at noon local time and do the same thing due north in central Anatolia (modern Turkey) and then somewhat farther north, you will find that the shadows pointing north have different lengths. The distances between the stakes and the shadow lengths can be used to compute the curvature of the earth. See Figure 4.

Figure 4.

The second piece of evidence can be seen when there is an eclipse of the moon. The shadow of the earth on the moon is curved. The Greeks knew that an eclipse is due to the earth getting between the sun and the moon. The third piece of evidence concerns elephants. If you go west from Alexandria towards the Atlantic ocean, you will first get to lands where there are elephants. The famous army of Hannibal (from Carthage in what is now Lybia) crossed the Alps into Italy coming from Spain with African elephants. Meanwhile, if you go east from Alexandria, across Anatolia towards India, you will again get to lands with elephants. For example, Alexander from Macedon fought against armies with Indian elephants during his campains in what is now Pakistan. If there are elephants to the west and elephants to the east and none in between, then the east and west must be connected around the back. Gemino makes a rather amazing prediction in his lectures from the fact that the earth is round. There was much commerce between the far flung parts of Europe and Asia during Hellenistic times, and as a consequence the Greeks knew that the climate got ever colder as you went to the north. For example, they knew that there are lands in the far north where there is perpetual ice and snow. They also knew that the climate got ever hotter to the south. In particular, the climate in the Sahara Desert was such that it was a barrier to travel and thus to knowledge. Gemino makes the bold statement that if you were to cross the Sahara, you would again get to lands with temperate climate, and then eventually to lands with perpetual snow and cold as in the far north. He then conjectures that the hypothetical southern temperate lands could be populated with people, people whom we (the Greeks) know nothing about. The surface of the earth is an example of a manifold that is curved in the large so as to have the shape of the boundary of a ball. This is to say that it is a 2-dimensional sphere. As I said, our universe is a 4-dimensional manifold, and this begs the question as

4 to what is its shape in the large? This is a question for astronomers to sort out. Even so, it is a mathematical question as to the list of all possible 4-dimensional spaces.

3. Lists of manifolds When talking about the listing manifolds, it is important to distinguish between the intrinsic structure and the shape that appears when a manifold is viewed as a subset of some larger, ambient space. By way of an example, the circle can be depicted in the plane in its standard round form and also with various bumps as in the left and right hand drawing of Figure 5. From an intrinsic point of view, both drawings are depictions of the same circle since in both cases, any given point is specified by an angular coordinate.

Figure 5

By the same token, the standard unknotted circle in 3-space and a knotted circle in the 3- dimensional Euclidean space are both standard circles. What differs is the identification as subspaces in the 3-dimensional Euclidean space.

Figure 6

By way of a second example, the surface of a perfectly round ball and the surface of the earth are both 2-dimensional spheres even as the latter has hills and valleys. The hills and valleys are not intrinsic to the sphere; they are the manifestation of sphere’s appearance as a subspace of the larger 3-dimensional Euclidean space. The key point here is that when I talk momentarily about a list for manifolds of a given dimension, I am not going to distinguish extrinsic features that arise from distinct appearances in some higher dimensional space. With this important point understood, consider now ‘listing’ the manifolds of a given dimension.

THE 0-DIMENSIONAL CASE: The 0-dimensional list is depicted in Figure 7.

• • • • • • • • • • • ······ • 1 point 2 points 3 points 4 points n points, and so on

Figure 7

Note in this regard that a point is 0-dimensional. If your universe were a point, then you wouldn’t have to give any number to specify your neighbors. Everyone would be living at the same place. Keep in mind with regards to Figure 7 that a manifold consisting of 2 points is also 0-dimensional since locally, no number is needed. Of course, this isn’t the case globally since you do have to say which of the two points you live on. This example illustrates the distinction between a manifold with connected manifold and a manifold with multiple components. I will henceforth restrict to manifolds that are connected.

THE 1-DIMENSIONAL CASE: The list of 1-dimensional manifolds has two elements, these being the line and the circle. They are depicted in Figure 8.

Figure 8

Keep in mind that I am considering only intrinsic differences; thus a perfectly round circle and an ellipse, for example, are considered identical. By the same token, an infinite line and an open interval in the real numbers are the same. The proof that the line and the circle are the only examples of connected, 1-dimensional manifolds is little more than a calculus exercise. The examples of the line and the circle illustrate the distinction between compact manifolds and noncompact manifolds. The circle is compact and the line is not. A manifold is compact if every sequence of points has a subsequence that limits to a point in the manifold. To keep things manageable, I will henceforth talk only about compact manifolds.

6 THE 2-DIMENSIONAL CASE: The list of the 2-dimensional (compact and connected) manifolds has two parts. The manifolds from the first part of the list are the surfaces in 3-dimensional Euclidean space that depicted below in Figure 9.

Figure 9

This left-most surface on this list is the 2-dimensional sphere, the next one is called the 2- dimensional torus. It is the surface of solid tube. The next one after the torus is called the 2-holed torus. Then there are tori with more and more holes. The torus can also be depicted as the set of pairs points from the circle. This is to say that each point in the torus is uniquely determined by two angular variables as depicted in Figure 10.

Figure 10

The standard circle is denoted by S1 and the torus is denoted by S1 × S1. This illustrates the fact that manifolds of smaller dimension can be used as building blocks for manifolds of higher dimension by taking the larger dimensional manifold to be the space of pairs of points in a given smaller dimensional manifold just as the space of pairs of points in the circle define the 2-dimensional torus manifold. The manifolds from the second of part of the list of 2-dimensional manifolds are harder to depict because they can not be drawn in a nice way as subsets of 3-dimensional Euclidean space. In any event, my description starts with the Möbius band which is depicted in 3-dimensional Euclidean space below in Figure 11.

Figure 11

Notice that the boundary of the Möbius band is a circle, albeit not the standard round depiction of the circle in 3-dimensional Euclidean space. See the next figure.

Figure 12

As it turns out, the boundary of the 2-dimensional disk is also a circle. This being the case, the boundary of the disk can be glued to the boundary of the Möbius band with the result being a 2-dimensional, compact, connected manifold. This manifold is called the real projective plan. See the next figure. It is the first manifold on the second part of the list of 2-dimensional manifolds.

Figure 13

To obtain the second manifold on the second part of the 2-dimensional manifold list, glue two Möbius bands together by identifying their respective boundary circles. The resulting manifold is called the Klein bottle. The third manifold on the list is obtained as follows: Cut out a disk from the 2-dimensional torus. This is depicted below in Figure 14.

Figure 14

You will notice that the boundary of the result is also a circle. This being the case, it can also be glued to the boundary of the Möbius band. The result of doing so is the second of the manifolds on Part 2 of the list of 2-dimensional manifolds. The remaining manifolds

8 on the list are obtained from those on Part 1’s list in Figure 9 by this same operation of cutting out a disk and replacing the latter by the Möbius band. This operation of cutting and then gluing to construct new manifolds from old ones is called surgery. The surgery operation plays a central role in the subsequent discussion. The proof that the list of 2-dimensional manifolds is as I just describe can be had using (in part) the theory of complex functions.

THE 3-DIMENSIONAL CASE: The list of compact, connected 3-dimensional manifolds was only recently determined. This was done by Grisha Pereleman ([P1-P3]. A conjecture for the list was made in 1982 by William Thurston [T]. The conjecture was called the geometrization conjecture. Perelman proved this conjecture using work of Richard Hamilton. See [MF] for an introduction and overview of all of this work. The now proved geometrization conjecture asserted that any given compact oriented 3- manifold could be constructed in an essentially unique way by gluing together parts from a list of eight basic kinds. The gluing was is to be done along certain specified two dimensional boundaries that are either 2-dimensional spheres or 2-dimensional tori. I won’t say more about the details. What follows instead depicts some of the simpler 3-dimensional manifolds. The purpose is to give you a way to see something of their large scale structure. Keep in mind in this regard that a neighborhood of each point in any one of these manifolds looks just like a neighborhood of the origin in 3-dimensional Euclidean space. Even so, none of these manifolds fits entirely into 3-dimensional space. For this reason, I depict these manifolds by cutting them into parts that do sit as subspaces in 3-dimensional Euclidean space. I then draw these parts and indicate how the parts fit togetther so as to make a single, 3-dimensional manifold.

The 3-dimensional sphere: Take two solid balls as depicted in the next figure. The boundary of each ball is a 2-dimensional sphere. Now glue the two boundaries together.

Figure 15

9 The result is depicted schematically in the next figure. The figure shows a path that exists the left most ball at the boundary 2-sphere but due to the gluing, the path then enters the right most sphere.

Figure 16

Likewise, a path the exits the right most solid ball through its boundary sphere must enter the left most solid ball. The manifold that is obtained by this gluing is called the 3- dimensional sphere. This depiction of the 3-dimensional sphere as two balls with their boundaries glued together is analogous to the depiction of the 2-dimensional sphere below in Figure 17 as the union of the upper hemisphere (a disk) and the lower hemisphere (another disk) with their boundary circles identified as the equatorial circle.

Figure 17

The manifold S1 × S2: This manifold is one of two direct 3-dimensional analogs of the two dimensional torus that is depicted in Figure 10. It can be viewed as the set of pairs of points with the first member of the pair being a point in the circle and the second member of the pair being a point in the 2-dimensional sphere. This manifold is depicted in the upcoming Figure 18. The inner and outer 2-dimensional spheres in the depiction are glued one to the other so that a path that leaves the region between the spheres by exiting through the outer sphere automatically re-enters the region through the inner sphere. This path is depicted in the figure. By the same token, a path that exists the region between the spheres through the inner sphere automatically re-enters through the outer sphere.

Figure 18

The analog to S1 × S1 can be seen by depicting the latter space as in Figure 19 as a cylinder with the two ends glued together. The analog in Figure 18 of the cylinder in Figure 19 is the region between the inner and outer spheres.

Figure 19

The manifold S1 × S1 × S1: This manifold consists of the set of triples of points with each point being a point in the circle. It is depicted in the next figure as the cube in Euclidean 3-space with the opposite sides identified. Granted this identification, then a path that leaves the interior of the cube by exiting the top side immediately re-enters the cube from the bottom. By the same token, a path the exists the cube through the right hand side immediately reenters through the left hand side; and a path that leaves through the front side immediately reenters through the back side. One of these paths is depicted in the figure.

Figure 20

11 An analogous depiction of the 2-dimensional torus S1 × S1 as the square in the plane with opposite sides identified is given on the left below. The same square is depicted in the right most figure as a subset of the torus with the torus depicted as a surface in 3- dimensional Euclidean space

Figure 21

Manifolds from knots: The description that follows explains how to construct a new 3-dimensional manifold from two old ones via a surgery. To start the construction, take a 3-dimensional sphere as depicted in Figures 15 and 16 as two balls with their boundaries identified. A solid knotted tube in one of the ball components of the 3-sphere is chosen. The two balls that comprise the 3-dimensional sphere and the solid knotted tube are depicted on the left in the figure that follows.

Figure 22

Note that the boundary of the solid knotted tube is a knotted realization in the 3- dimensional Euclidean space of the 2-dimensional torus that is depicted in Figure 10. Figure 23 shows the boundary of the knot with the two angular parameters that identify it with the standard 2-dimensional torus as depicted in Figure 10.

Figure 23

The next figure again shows the two balls that comprise the 3-dimensional sphere with another solid knotted tube depicted.

Figure 24

Take out the solid knotted tube in Figure 24. The boundary of the result is the 2- dimensional torus. Likewise take out the solid knotted tube in Figure 22. The result has a 2-dimensional torus for a boundary also. Identify these two torus boundaries and use the identification to glue them together. The result is a compact, connected, 3- dimensional manifold.

THE CASE OF MANIFOLDS OF DIMENSION 5 OR GREATER: As it turns out, the question of a list of sorts for manifolds of dimension strictly greater than 4 has also been sorted out. This was work done in the 1960’s by Smale, Stallings, Milnor, Kirby, Siebenmann, and others (see for example [M] and [KS]). I won’t say more about dimensions greater than 4 but for the remark that distinctions between manifolds involve certain subtle sorts of algebraic data.

THE CASE OF 4-DIMENSIONS: I left this case for last because 4 is the only dimension where the question of a list is as yet unsettled. It is known that the analog in

13 dimension 4 of the algebraic data that is needed for the cases of dimension greater than 4 is necessary, but it is also known that the algebraic data is by no means sufficient. There is no compelling conjecture as to what else is involved.

4. Visualizing 4-dimensional manifolds Manifolds of dimension 4 are hard to picture. This is so even though we live in a four dimensional universe. Yet, some people are extemely good at visualizing and hence exploiting the intrincally 4-dimensional aspects of our universe. Magicians in particular are adept at this. Many standard magic tricks are done by exploiting the fact that the universe is 4-dimensional. A simple example involves a common trick whereby the magician asks someone from the audience to pick a card from a shuffled deck. The card is then shown to the audience but not to the magician. After the audience sees the card, the magician inserts the card in the deck and then has another member of the audience shuffle the deck many times. The magician then takes the deck and ‘magically’ picks out the chosen card. The trick is done as follows: In the morning before the performance, the magician travels in the 4’th dimension forward in time at an accellerated rate so as to appear in the back of the performance hall a split second before the trick is performed before the audience. The incarnation of the magician in the back of the hall then sees the chosen card when it is shown to the audience. The incarnation of the magician in the back of the hall then travels backwards in time to when he started, and then travels forward in time at the normal rate knowing before the trick is even begun what the chosen card will be. If you are at a magic show where this trick is performed and if you listen closely just after the chosen card is shown to the audience, you will hear a popping sound coming from the back of the performance hall. This popping sound is the sound that the air makes when it refills the void that is left when the incarnation of the magician leaves what was for him the future to return to the ‘past’ with the knowledge of the chosen card. This popping sound is the give-away. If you are not specifically listening for it, you won’t notice it amongst the hub-bub and misdirection on the stage.

5. Smooth manifolds and topological manifolds To say what we know and what we don’t know about 4-dimensional manifolds, I have to digress momentarily to talk about the notion of a smooth manifold. A manifold is said to be smooth if there is a consistent definition across the whole manifold of which functions have derivatives and which do not. This is a distinguished set of manifolds. As it turns out, there are manifolds that lack a consistent notion of a derviative. Of course, near any given point, the manifold looks like Euclidean space and we know from multi-variable calculus what it means to take derivatives of functions in Euclidean space. The issue appears because neighborhoods of different points will have, in general,

14 different identifications with Euclidean spaces and the functions that appear differentiable with respect to one identification need not be differentiable with respect to the other (and vice-versa). This issue is not present in dimensions 1, 2 and 3. It is however present in dimensions 4 and greater. Two manifolds that are equivalent with no regard for differentiation are said to define the same topological manifold. If differentiation is of no interest, then the question of a ‘list’ for 4-dimensional manifolds has been determined for cases when the higher dimensional analog of what I called algebraic data is not too complicated. This was done by Michael Freedman in the early 1980’s (see [FQ]). Freedman proved that the algebraic data in these cases is all that is needed to determine a 4-dimensional manifold as a topological manifold. If you care about differentiation, then the story is far more complicated and, as I said previously, not at all sorted out. That this is so follows from the pioneering work of Simon Donaldson (see [DK]). I say more about this in what follows. The notion of a smooth 4-dimensional manifolds is in any event, very natural and this is one motivation for listing them. In particular, the notion of differentiation is central to much of modern physics. By way of an example, classical mechanics in the guise of Newton’s laws equates the second derivative with respect to time of the position of a particle with the force acting on it. By way of a second example, the time evolution of the wave function in quantum mechanics is determined by the Schrödinger equation which is an equation that involves both time and spatial derivatives. With the preceding understood, I henceforth restrict my attention to smooth 4- dimensional manifolds.

6. Knot surgery As I said, the list of smooth 4-manifolds is not sorted out. To see where the problems lie, I now describe a construction due to Ron Fintushel and Ron Stern that can, under certain circumstances, change a given smooth, 4-dimensional manifold into a different smooth manifold without changing the listing as a topological manifold. The Fintushel-Stern construction goes by the name of knot surgery [FS]. I use X in what follows to denote the given smooth 4-dimensional manifold. Knot surgery requires as input an incarnation in X of a 2-dimensional torus. To say somewhat more about this torus incarnation in X, I introduce as notation D to denote the disk of radius 1 about the origin in the 2-dimensional plane and T to denote the 2- dimensional torus. The torus in X must have a neighborhood in X that can be identified with the space of D × T, the space of pairs of points (x, p) with x being a point in D and p being a point T. This neighborhood of T in X is depicted schematically in Figure 25. The figure accurately depicts the disks in D × T but it depicts the torus T as a circle.

Figure 25

There are other conditions on the incarnation of T but these I won’t describe. The boundary of D × T is the product space S1 × T, the space of pairs of the form (t, p) with t here denoting a point in the circle (the boundary of D) and with p again denoting a point in T. This is depicted schematically in the next figure.

Figure 26

Since T is S1 × S1, the boundary of D × T is the space that is depicted in Figure 20. The plan now is to remove the incarnation of D × T from X and replace it with a space that has the same boundary. To find such a space, depict the 3-dimensional sphere as in Figures 15 and 16 as two balls in Euclidean space with their boundaries identified. Choose a knotted circle in one of the balls and thicken the knotted circle to get a knotted tube in the 3-dimensional sphere. Two versions of this are depicted in Figures 22 and 24. Take out this solid tube around the knot to get a 3-dimensional space with boundary. The boundary is an incarnation of T in the 3-dimensional sphere. Letting K denote the chosen knot, I use MK to denote the resulting 3-dimensional space with boundary T. The space 1 MK × S is 4-dimensional, it is the space of pairs (z, t) with z being a point in MK and t 1 1 1 1 being a point on the circle. The boundary of MK × S is T × S and since T is S × S , this boundary is the same as that of the complement in X of the incarnation in X of D × T2. Granted the preceding observation, remove the D × T incarnation from X and 1 replace it by MK × S by gluing along their common boundaries. The result of this gluing is the knot surgery space. I denote it by XK. If the boundary identifications for the gluing are chosen in an appropriate way and if the incarnation of T in X has certain special properties, then XK and X will be the same when viewed as topologica manifolds. As explained momentarily, the knot K can be chosen so that XK and X are definitively not the same when viewed as smooth manifolds.

7. 4-manifolds and the Alexander polynomial Two knots in Euclidean space are said to be equivalent if one can be obtained from the other via a continuous, 1-parameter family of deformations with each member of the family being a bonafide knot. Some examples of inequivalent knots are drawn below in Figure 27.

Figure 27

The left most knot in the preceding figure is called the unknot since it is just a standard round circle in space. The knots in the next figure are all equivalent to the unknot.

Figure 28

Each knot has an associated Alexander polynomial. This is a polynomial in the variables t and t-1. The Alexander polynomial depends only on the equivalence class of the knot. The Alexander polynomials of the knots in Figure 27 are as follows: The

Alexander polynomial of the unknot is the constant Δunknot = 1, that of the trefoil depicted -1 in Figure 27 has the polynoma Δtrefoil = t - 1 - t , and that of the figure 8 knot depicted in -1 Figure 27 has the polynomial Δfig.8 = - t + 3 - t . As explained in [K], a polynomial in the variables t and t-1 is the Alexander polynomial of some knot if and only if the sum of its coefficients is equal to either 1 or -1 and it is symmetric under the interchange of t with t-1. This implies that there is an infinite set of knots with no two having the same Alexander polynomial. Fintushel and Stern prove the following [FS]:

Suppose that T is a torus in X with the desired properties. If K is a knot, then the

knot surgery manifold XK is equivalent to X as a topological manifold; but it is not

17 smoothly equivalent to X if the Alexander polynomial of K is not the constant 1.

Moreover, if K and K´ are two knots, then XK is equivalent as a smooth manifold to XK´ if

K is equivalent to K´ as a knot, but XK is not equivalent to XK´ as smooth manifolds if K and K´ have different Alexander polynomials.

Fintushel and Stern also prove that XK with K being the unknot in Figure 27 is the same smooth manifold as the original manifold X.

8. A candidate for knot surgery 4 I use C to denote the space of 4-tuples of complex numbers and I denote a given point in this space by (z1, z2, z3, z4). Set Z to be the locus in the complement of the origin 4 in C where the coordinates obey the equation

4 4 4 4 z1 - z2 = z3 - z4 .

Note that if λ is a non-zero complex number, and if (z1, z2, z3, z4) is a point in Z, then the

4-tuple (λz1, λz2, λz3, λz4) is also in Z. This understood, given points (z1, z2, z3, z4) and

(z1´, z2´, z3´, z4´) in Z are said to be equivalent if (z1´, z2´, z3´, z4´) = (λz1, λz2, λz3, λz4) with λ being a non-zero complex number. As it turns out, the corresponding space of equivalence classes in Z is a smooth, compact connected 4-dimensional manifold. This manifold is called K3. A torus in K3 suitable for knot surgery is the space of equivalence 4 classes of the points in the complement of the origin in C where both

2 2 2 2 2 2 2 2 2 (z1 - z2 ) = z3 + z4 and z1 + z2 = 2 (z3 - z4 ) .

The locus where these equations are satisfied is contained in Z because the equation that 4 defines Z as a subset in C can be written as

2 2 2 2 2 2 2 2 (z1 + z2 )(z1 - z2 ) = (z3 + z4 )(z3 - z4 ) .

If K is a knot with non-constant Alexander polynomial, then knot surgery using the manifold K3 and the just described torus results in an X = K3 version of XK that is equivalent to K3 as a topological manifold but inequivalent to K3 as a smooth manifold.

9. What we know and what we don’t know

What we know is that XK is not the same smooth manifold as X if the Alexander polynomial of K is not the constant 1. But, we don’t know if X and XK are invariably the same as smooth manifolds if the Alexander polynomial of K is equal to the constant 1.

18 One of the simplest interesting knot with Alexander polynomial equal to 1 is the Kinoshita-Terasota knot pictured below.

Figure 29

Is XK the same smooth manifold as X in the case when K is this Kinoshita-Terasota knot and X is the K3 manifold? I don’t think anyone knows the answer to this question. This question sits on the ignorance side of the boundary in the year 2013 between what we know about 4-dimensional manifolds and what we don’t know.

References [DK] Simon K. Donaldson and Peter B. Kronheimer, The Geometry of 4-Manifolds, 1997. [FQ] Michael H. Freedman and Frank Quinn, Topology of 4-manifolds, Princeton University Press 1990. [FS] Ronald Fintushel and Ronald Stern, Knots, links and 4-manifolds, Inventiones Math. 134 (1998) 363-400. [K] Aiko Kawauchi, A Survey of Knot Theory, Birkhauser Press 1996. [KS] Robion Kirby and Larry Siebenmann, Foundational Essays on Topological Manifolds, Smoothings and Triangulations, Princeton University Press 1977. [M] John Milnor, Lectures on the h-Cobordism Theorem, Princeton University Press 1965. [MF] John Morgan and Frederick Tsz-Ho Fong, Ricci Flow and the Geometrization of 3-Manifolds, American Mathematical Society 2010. [P1] Grigori Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math/0211159. [P2] Grigori Perelman, Ricci flow with surgery on three-manifold, arXiv:math/0303109. [P3] Grigori Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math/0307245. [T] William P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Amer. Math. Soc. Bull. 6 (1982) 357-381.