 Before I forget, there is one point to make in terms of notation that I didn't do last time. So if x is a path-connected space which has a fundamental group pi 1 of x0, which is trivial, then we say that x is simply connected. And I guess the first question would be, is this a well-defined concept? Why is that? Right, so he's saying that this is indeed well-defined because it doesn't matter what the point, the base point it is that you pick any other point because the space is path-connected will give you an isomorphic group, and any group isomorphic to the trivial group, of course, is trivial. So this has to do with a space, not with the choice of a base point. So this is an important concept to know about. So let's say, give me a couple of spaces that are simply connected. Sorry? I can't hear. Rn, yes? Because it's convex, for example. What is another example? Yeah, any ball, cube, convex sets of Rn are simple examples of this. OK, then I mentioned the concept of retract when we proved at the end of the class last time that a function, this Broward theorem, of a fixed point. And so let me formalize that. Let's spend work. So we have a topological subspace A of a space X with an inclusion I. A map from X to A is called retract R. If A from A, the map R composed with I, is the identity. And so the claim, excuse me? Oh, let's see. The R is, is that better? Yeah, I'm running out of this ink of the other pen. See if I can keep using this one. So the claim is that I star, lower star, which is a map from pi 1 of x. Let's pick a base point x0 that belongs to A. So there's a map from pi 1 of A to pi 1 of x. It's injective. And the proof of this is quite simple. Once we have the functoriality of the construction of the fundamental group, meaning that not only we associate a group to a space, but we associate homomorphism between those groups. If we have pointed spaces, we're spaces together with a fixed base point and maps that take continuous maps between the spaces that take a base point to a base point. And functoriality also means that if you have a composition of maps, then the resulting maps at the group level are also the corresponding compositions. So that is to say, because r i is the identity, it follows that r lower star i lower star, which is r i lower star. This is one of the properties of functoriality. It's equal to the identity of A lower star. And this is the identity of pi 1. Because another property of functoriality is that the identity map goes to the identity map. Although we haven't proved exactly, but they're all fairly simple, so I won't do them. But once we know that r composed with i is the identity map in pi 1, this implies that i has to be injective, i star. And this, if you go back to what we were doing, we didn't spell it out. But that's exactly how we use the fact that we have a retract in that example. So for example, if pi 1 of x is trivial, sorry, if pi 1 of x is trivial, in other words, if it's simply connected and pi 1 of A is not, let's assume all the spaces are path connected to avoid trivialities, then what can we say about A and x? So A is some subspace of x. If this group is trivial, this group is not trivial by assumption, then we cannot have an injection between taking this group into that one. Just a big group doesn't fit into this mole. So that means that we couldn't have a retraction. So A is not a retract of x. And a concrete example of this would be if we take x to be the unit disk and A to be its boundary, the circle. And this is the way exactly we did it last time. There's no retract. I think that's what we did, anyway. There's no retract from D to A. So maybe a picture to have for a retract would be something like this. Say if you have a unit disk times the unit interval and you look at the map that projects down. So this is our x. And you take A to be D. Then the projection down, any point of the disk stays where it is. It just projects to itself. And everybody above it gets squashed down. So this is a retract. So multiplying by the unit interval doesn't change the fundamental group. And there's one other point I want to make sort of a general discussion. Another claim is the following, that if we have the Cartesian product of two spaces, x and y, and a base point, say, x0, y0, then in a natural way, this fundamental group is the product of the two fundamental groups. And let me just give you a sketch of what you need to do. It's fairly simple. Map, path, a loop is a map from i to x times y, starting and ending at the base point. And any such map has to be of the form x of s, y of s, with xs and ys loops into x and into y. So this gives you a way to go from, you attach to the homotopy class of gamma, the corresponding pair of homotopy classes of these paths x and y. And then it's a simple matter to verify that this works with homotopy classes. And that indeed, in that way, you get an isomorphism. So we try to build the knowledge of this fundamental group. And I think it's good to have the sort of theoretical things down, but also get an intuition for how this behaves in spaces that you're familiar with. So you should try to think of various spaces as we incorporate more facts about the fundamental group so that you know how to compute them. It's always a good thing to internalize the concepts to be able to compute any given concept in as many examples as you can. So let's discuss one example of this. For example, let's take x and y, both to be the unit circle. So the statement is that pi 1 of the circle cross the circle at some base point x0, y0 is isomorphic to z cross z. So that's a abelian group. So what is a more familiar way to describe s1 cross s1? It's the torus. So you can represent this pictorially like this. And implicit when you go through the proof of this theorem, you'll see that this isomorphism here is not a completely abstract thing. We know the pi 1 of s1 is a circle. At least we talked about it and we'll prove it. And so you can pick a generator here, which for z there's only two generators. You have either 1 or minus 1. So this will define up to an orientation. So let's take 1 for this copy and 1 for this copy. So we have two circles. And they are going to map over here and be the generators of say here 1, 0, 0, 1. So in this form of the space, we have a circle and a point and a point and a circle as the two generators of z cross z. And in here, let's say this is the base point. They become a loop that goes along one of the circles. And then the other circle is just the point. And then another circle that goes along the other circle and is a point with respect to the first circle. Is that clear? So these two, let's call them say a and b, generate. So if we write it in sort of an additive notation, this would be pi 1 of the torus at some point z, say. And so given this fact, how would a homotopy class of a path look like? Well, we've been drawing paths that are fairly simple. But a path could be doing all kinds of stuff. But it's going to be homotopy to something quite simple, something that goes a certain number of times along one of the circles and a certain number of times along the other. In case it will be how many multiples of a and how many multiples of b you have. So if you draw them, you can see some of the pictures in Hatch's book. They will look like these knots sort of tied around the torus. OK. Now maybe before we get into more nitty-gritty proves, I thought I'll show you a picture. And hopefully we'll get to discuss the mathematics. So maybe one of you would like to read it and could tell us. And this is a trick that apparently I didn't know this before, a trick that apparently Dirac came up with. So let me say it in words, and then I'll show you a picture of how it works. So you take a belt, OK, and you twist it twice. So fix this end, and you take one end, and you do it two twists. OK, now there's a way to untwist this without moving these two. So you're going to move the thing, but the end points are going to be fixed. So this is an example of sort of a homotopy, and we'll discuss the math that goes behind it. If you like, I'll give you the URL if you want to read. So this is called the Dirac belt trick. And it's actually something you can physically do. So if you want to impress your friends by home, you can try this trick. And maybe you can impress us if you want to do the trick and learn how to do it and come here and do it. So the first thing this is going to do is actually do the two twists. And then it's going to unravel it by this homotopy. So first is twisting the thing twice, OK? So that's the belt twisted twice. And now without moving the end points, it's going to untwist it, OK? So let me try again. So you could say, well, maybe it's an optical illusion. The computer is tricking us. But you should try this at home and see if you can actually do it with a belt. And he's already trying. So first, this gets twisted once. And now it gets untwisted without changing, without moving the end points, OK? Now, well, don't do it now, because otherwise you won't just pay any attention to what we do next. But so tell me what you think this means. Why is this? So maybe let me ask a different question. So suppose I twist the belt once, not twice. Can you unravel it? Can you do something like this moving around and without moving the end points of the belt sort of untwisted? No, you can't. And why not? OK, so what he's saying is that if you do it once, you wouldn't, because somehow you will get to the other side of the belt. But if you do it twice, you're coming back to the original side. So OK, can you guess a more mathematical or something in terms of the fundamental group, given what we're discussing? OK, well, it's not completely apparent. But what we'll see is that this has to do with this space, which has fundamental group z mod 2z, so a group of two elements. And so the fact that you do this twist twice means that you're taking a generator and multiplying it by 2. And in this group, an element by 2 is 0, because it's just a group, the classes of 0, 1, or 2. And so this thing, the logit-loot twisted, in fact, it isn't. It is up to homotopy is actually untwisted. So I'm not pretending that this is a full proof, but this is somewhat what's behind this. Well, in order to do this correctly, you have to understand exactly what is the space that is involved. But I don't want to get into that right now. So let's go back to this. So what I thought we could do now is do a bit more actual proofs. And so in particular, this thing that I've been talking about for some time, or the pi 1 of the circle. So there's a concept that we're going to discuss now that is very much tied with the idea of the fundamental group, which is the idea of covering spaces. And I think it probably will, if you haven't seen this before, it will come out as something fairly strange. You probably are happy with the fundamental group. You can live with the idea that, you know, paths, classes of paths, those things that you've seen before in calculus as I tried to illustrate at the beginning last time. But covering spaces is a bit more, a bit stranger. So I'll try to motivate it. But let me just say that it will be a concept that, ultimately, is even perhaps more important than the actual fundamental group. Or at the very least, it's very much tied to it. So let me give you an example of a covering map. And then we'll abstract from this example what is it that we expect this to be. So the example is what I was doing last time. So there's a map from R to the circle, which is to send theta to either the 2 pi i theta, or if you want in real coordinates, cosine 2 pi theta, sine 2 pi theta. And this will be an example of a covering map. And I'd like to abstract what are the features of this map that make it so, and how can we use it? And as I was doing last time, and maybe this today will make it a little more clear what I was trying to say, in order to visualize this map, I'm going to consider the helix formed by something in R3. So the first coordinates follow the circle. But the third is the angle itself. So visually what we have is a cylinder. This is sort of the theta direction. And it's sitting above the unit circle. So the map, so think of this vertical line as R. And the map takes R to the circle. But instead of just keeping track of the angle, we sort of think of theta as the height in the cylinder. But we also keep in track of where it looks down below by describing this helix. So one way to think of what this is is a way to describe the angle of any point in the plane. So if we have this point here, it has a certain angle. But the angle has an ambiguity. And we sort of, rather than making a choice, which is the typical thing, you maybe take between minus pi half and pi half, or 0 to 2 pi or something like this. Now we take all possible things. And so above this point, we're going to have all possible angles. And as we discussed before, all possible angles differ from any given one by an integer, or rather an integer times 2 pi. But I'll keep the fact that 2 pi out. So after dividing by 2 pi, the ambiguity will be an integer. OK, so I want to understand topologically the properties of this map. And hopefully this will make clear why we define covering the way we will. So the point is this. Here is a point and its pre-images. If we pick a little interval, a little open set around this point, a little neighborhood, and we pull it out, we can have these around each one of these pre-images little intervals that look exactly like the one below. So more precisely, this map P, let me write it like this, there exists a neighborhood U of any point downstairs, such that the inverse image of that open set. So here's our U down here. Let's say any point Z down below. So what is the pre-image consists of all these little intervals that are disjoint? So it's a disjoint union over some indexing set, which for us are going to be the integers, of open sets U alpha, and such that if we take the map P, so here's map P goes like this, and restricted to any one of these given subsets U alpha, it maps homeomorphically to U. So what this, we want to say this in words, it says that you can pick a little neighborhood of the point so that above any pre-image, you'll see exactly that same little open subset around each one of its pre-images. The same meaning that the map P takes from one to the other in a homeomorphic way. And I hope this is clear from this picture for the case of R and the circle. I mean, what you would do concretely, you would define the argument or the angle of the point on the circle. And if you take a small enough neighborhood of a point, the angle is a perfectly good invertible function. And so that is what this is. So if you have a point that is, let me see, so if we want to do this not with the helix, let's say, I take this point, this is my z. So here's R. So if I do it not in 3D, but just in 2D, so I will have, for example, so this is 0, 1, 2, et cetera, inside R. So this measured in the usual way has an angle of 2 pi over 8. Did I do that right? OK, so it means that one of the possible angles is there here at 1, 8. So if I take a little neighborhood, we'll have a little neighborhood about that. And then the similar one around one more, et cetera. So the pre-image of that little segment around this point is little segments that look exactly the same around 1, 8, 1, 8 plus 1, 1, 8 plus 2, 1, 8 minus 1, 1, 8 minus 2, et cetera. So this is the basic property of a covering. And it will have, as I said, although maybe not quite clear yet, a very close connection to the idea of the fundamental group. So let me give you some more examples of coverings. The one R to S1 we already discussed. Let's take the following map from S1 to S1. Take the map, take Z to Z squared. Think of Z as a complex number and square it. So if you like to think of it in terms of angles, double the angle. But it's just quicker to write it in terms of complex numbers. So I claim that this is a covering. So a covering map is a map from one space to another, y maps to x, with the property that any point of x has a neighborhood whose pre-image is a disjoint union of neighborhoods, which are in homomorphic relation via p to the one below. So quite a lot of things crammed into that definition. So let's look at this example. So here we have the circle, and the map is the square map. So if we think of the circle upstairs, if we move along the circle once, what happens below? You move twice as fast. You go twice around. So if I move below once, what happens upstairs? We only do half. So then we can say, let's try to find this decomposition into disjoint unions of the pre-image. So if I take a little neighborhood here, so let's say this is, let me do it in an angle so you can see it. So this is 1. This is S1 and S1. This is the point 1 in the sense of complex numbers. So what are the pre-images of 1? 1 and minus 1. So there's another pre-image here. So this little neighborhood will be here half its length, but it will also be here. So the pre-image of a little interval around 1 will become these two intervals, in this case, around 1 and around minus 1. And the same thing happens for all the other points. So maybe if I, to describe it this a little more graphically, like in this example, what we can say is that we have the circle and then the circle going around twice, but once you go to this, this should be the same as that. So it is the circle, but it's been, I sort of unravel it so that it looks like a little piece of that helix above, two loops of that helix. So if I look at 1, we would have 1 here and then minus 1 there and the little interval will look like that. So it's another, OK. So this is another example of a covering. And now we're going to prove certain things about coverings that will hopefully also convince you that this is a worthwhile definition. So this now will make the connection between this concept and the concept of fundamental group. So all throughout this discussion, we're going to have a covering map or simply a covering and giving a path gamma from the interval to i. So this is a path together with some y0 in y, which is in the preimage of our base point below. So this is a loop at x0, which is in x. So we have an x, an x0, a loop starting and finishing an x0. And we have now a covering map. And I want to do something about this. So let's go back to our basic example, which is to guide the intuition. Let me try to say it in words and by moving my hands. And then we'll do it. We'll prove it more formally. So let's just look at a very simple path down below. So for example, let's say we take the loop that goes counterclockwise twice. So that's a loop down in the x. So this is in our notation. This is x and this is y and this is our map P, the covering. So what we could do is pick any preimage of the base point and then follow upstairs the point. So when it gets to be once, it's right on top where it was, but it didn't close. Upstairs it didn't close. You do it again and you go two steps up. So what we did is this loop that went around twice, we unraveled it to make something that goes twice in this helix. So what we can say what was going on is that we lifted this path down below to a path in the covering. So we basically stretching out the loop below into something that is not going to be typically closed upstairs. And how would you prove this? Well, in this particular case, it's just a matter you basically do it step by step. So if you move a little bit downstairs, you move a little bit upstairs. Then you look up below again, you move a little bit and you keep going. So you can do it by little steps. And you cannot take the entire path and move it up because there's no inverse to this map, but there's locally an inverse. So each little piece can be locally lifted and in such a way that these little pieces get pasted together. And that's what we can now prove more precisely rather than by just hand-waving. So what's the statement? So this is a lifting property of coverings. So we have a loop below and we want to lift it to a loop upstairs. So the statement is there is a unique path, gamma twiddle. That is a path. It goes from I to this space Y with the property that is a lifting, which means that if you project it down, it looks like the path we started with. So think of this map as a shadow. You have the shadow of a path below and what we're doing is seeing where it came from. What is the shadow of? So this means that this is the actual path upstairs. When you project it down, what you see is the path you started with, gamma. And this will not completely pin down the path. You have to start somewhere. So we're going to declare that the beginning of the path is our favorite pre-image Y0. So the point is you have an X0 here. You pick a pre-image somewhere. And then from then on, you can lift it up or down. I mean, whatever it does. But you lift it starting from that point on. So the upstairs, we have a loop. Upstairs, we typically don't. This interplay is going to be fundamental to understanding things about fundamental groups. OK, so I will follow Fulton's description, which is, I think, there aren't that many variations on the theme. So Hatcher most likely does something similar. I just found it a little easier to read. So let's do this in a formal way. But if you like, keep the picture of the helix on the circle in mind to see what's going on. So consider, so let me use a name that Fulton uses. So call U a open in X as in the definition of covering. Call it evenly covered. I think that's what he says. Yeah. So the definition of covering says that each point has an evenly covered neighborhood. There's a little open set containing a point whose preimage is this joint union, et cetera, et cetera. So call those things evenly covered, so we don't have to repeat the description that many times. So consider the collection of all evenly covered subsets of X. Actually, let's do that. But let's bring back for each one the preimage by the path gamma. Gamma is a continuous function from i to x. So it brings back any open set to an open set of i. So this gives us an open set covering of the interval i. By the property of covering, every point of X has an evenly covered neighborhood. The map gamma takes i to some subset of X. So if you pull back the open sets that are in the image, all the others are going to give you the empty set, you're going to get a collection of open subsets of i. And they cover all of i because at some point of i has to go somewhere. And so in the image, there'll be an evenly covered subset that contains it. So this is a covering. Unfortunately, it's the same word. So it's a collection of open subsets of i covering, containing every point. Is this clear? Yeah? OK. So what then Fulton uses is something called the Begg's lemma, which is a simple fact of point set topology. So this is a key sort of technical tool for this purpose. Have you seen this lemma? So what does it say? Can you tell me what it says? Perfect. Well, let me repeat for the camera, for posterity. What's the statement? As he outlined it perfectly. So for any open cover of a compact metric space, so a collection of open sets whose union is the whole space in question, there is a epsilon, say, such that any set of any open subset of diameter less than epsilon, it's inside one of the sets of the cover. There is such an epsilon, small enough so that every one of these little sort of balls of radius epsilon, so to speak, in the metric space are inside one of the subsets of the open cover. So this will mean that we can get a subdivision, a finite subdivision, of the interval i. Let's say 0 is t0, so equal to t1, tn equals to 1, such that if we look at the image of the interval from ti minus 1 ti, it's inside one of the evenly covered sets. Say, call it ui, evenly covered. So to lift it, we start with the interval t0, t1. So this is i equals to 1, 2n. So t0 is 0. You define gamma 0 to be y0. That's what we want. We had to start somewhere, so we chosen where we start. So that's this second property that we have as a hypothesis. So now, from t0 to t1, the image of gamma, which is some little path in the space downstairs in x, is sitting exactly inside a open set, which is a evenly covered subset. So this little open subset has a pre-image, all these layers, all of which are bijectively, each one of them bijectly related by p to the one below. So I landed in exactly one of those. So now what I do is I take the corresponding inverse of p, this local inverse of p, to lift it up. Is that clear what I'm saying? So we ended up with a little, well, it starts here. This is x0. So let's say this is x1 is this. If you want to fix ideas, let's say gamma of ti is xi. So this is inside the interval u1, and this interval, sorry, this open, let me do it again, x0, x1. This is u1, is evenly covered. So I mean, this may be a little misleading, because it doesn't have to be sort of discreet, but I can't draw something this joint with a bigger cardinality very easily. So let's say y0 is here. This is my chosen pre-image of x0. So p maps each one of these little pancakes down below exactly in a homomorphic way. So once I decide which of the pancakes to consider, there is a bijection back. There's a homomorphism back. So I can lift it by just taking sort of p inverse, restricted to that particular slice. Is that clear? So by this local inverse, which is guaranteed by the definition of covering, I can take this path down here, gamma, a little piece of gamma, and lift it up exactly, well, it's a homomorphism. You can sort of draw it. It will maybe deform it a bit, but it has an exact copy of it in the corresponding slice upstairs. And now what do we do? So we lifted a little piece of the path, and now we repeat. We repeat the same, same argument, but now apply to x1 and the interval t1 to t2. The interval t1 to t2 is sent over by gamma to some other open subset, u2. And the same argument applies. And I start off with y1. And of course, you have to convince yourself, verify that this process is such that this gluing of these maps at the end of the day gives you something continuous. So now iterate. So we would have some, I think, this is, yeah. So I would have x1, u2, x2, and then a local inverse starting at y1 that will connect to y2. And so I keep doing this until I get to the end of the interval. So do until done. So what this then showed, what is it that we're trying to do? We were trying to show that, well, we showed the existence of a lift. So this covering condition precisely allows us in a very clean way. It sort of captures what is needed to make sure that if we have a path downstairs, we can lift it to some path upstairs. So a loop downstairs turns into some path upstairs once you fix an initial point. OK, now I'm claiming a little more. I'm saying that the path that lifts one below is unique. So once you declare where it begins, then the thing is completely determined. And there is something to prove because in the proof, we use the Lebeck Lemma to divide the interval into pieces. What if I had chosen 100 pieces instead of 10? What would I get? We would get the same path. So uniqueness does require a proof, but it's not hard. So let's check the uniqueness. I think what one sees the beauty of abstraction. All of these notions took a long time to get reduced to these essential things. I mean, if you try to read literature of topology or even before topology, it would be much, much harder to understand what they were trying to say. So for uniqueness, we can do a little bit more generally, which doesn't hurt. So we want to show that this path that we constructed upstairs, gamma-tittler, is actually a unique path given all these constraints. So let's say we have z mapping to x. And let's say z is a connected space. So for us, z is, at the moment, the interval, 0, 1. But it doesn't hurt to do it in general like this. And so instead of a loop, we just have a map, f. And suppose we have two lifts, f1, twiddle, and f2, twiddle. So these are maps that have the property that projected down. They look the same. So to speak, the shadows down in x look the same. They look like f, the one we started with. So they lift f. That's one way to describe it. And so this is one point. And the other point is let's say that f1, twiddle, and f2, twiddle agree at some point in z. So we have a map below from a space z to x. And then two lifts of that map to the covering. And these two maps that lift the one below agree at one point. Then the conclusion is that they agree everywhere. So then the conclusion is then f1, twiddle is actually equal to f2, twiddle. And this will, of course, prove the uniqueness of the other what we were really trying to prove if we apply this to the interval i. And if the function f was the loop, then this common value would be, say, f1 of 0, f1, yeah. That's what one of the conditions that we had is that we fix one of the initial points. So we have two such. This will be exactly the statement. Anyway, so the proof of this is along the lines of this kind of abstract stuff that you don't really know what you're doing, but somehow the thing kind of proves itself. So we're going to strongly use the fact that the z is connected. You easily can see that this will be completely wrong if you didn't have that. So to prove that the functions agree everywhere, we're going to look at this subset of z where they agree. And we're going to show that it is both open and closed. And then what? Sorry? Good. We need to verify that the subset is also non-empty. So we're going to use the fact that it's connected, as pointed out. If you have an open and closed set that is non-empty, then it's the whole thing. And it's non-empty because we are requiring it. So what we need to do then is verify that the subset is closed where they agree is both open and closed. So take an evenly covered, sorry, this pen. I'm going to run out of ink here. Take u to be an evenly covered neighborhood of f of z0. So f of z0 is some point in x. Every point in x has an evenly covered neighborhood by the hypotheses of coverings. So p inverse of u is a disjoint union of subsets u alpha. What are the properties that we know? Let's call y0 to be the common value of f1 twiddle in this point z0. OK? And now let's do a little picture so we get some of our bearings. This, they say, is sort of a picture of x. This is f of z0. Maybe we'll call it x0 if we need to, just to keep our mind focused. And then each one of these has these various slices upstairs. OK? And so actually what I'd like to do is the same for some x. So if I have an arbitrary x, which is equal to f of z for some z in the sense z, then there are these neighborhoods. And let's say this is the value of f twiddle of z down here. And this is the value of f2 of z. So they're sitting above x. So actually this is what I meant to do with an arbitrary z. OK? So we pick a z in the space z. Look at this value x. Look at a open neighborhood of f of z, which is x. OK? And upstairs f1 twiddle and f2 twiddle are going to be mapped to some preimage of x, which may or may not be the same. We're trying to prove that they are the same. But they're sitting inside their own little neighborhood that covers the neighborhood below u. So this is some, let's call it u1 and u2. OK? So let's look at f2 twiddle inverse of u2. That's some open neighborhood of z intersect with f1 twiddle inverse of u1, which is another open neighborhood of z. So this is an open neighborhood of z. So let's see. If f2z twiddle is not equal to f1 twiddle z, we call this v, then it means that u2 is not equal to u1. These two neighborhoods are neighbors of two different points. We can take them small enough so that that's the case. And f2 twiddle and f1 twiddle are different in all of v. Yeah, u1 and u2 are disjoint. And so if they contain different points, they are different. So either they are the same, but in which case x, y1 in the image of f2z and f1z twiddles should be the same. So if they are not the same, the subsets are different, the open neighborhoods. And the pullback of the intersection by each function will give us an open neighborhood v, which gets sent over by f2 twiddle and f1 twiddle to these two separate things. And so the functions are different in all of the subsets. So if they happen to be the same, then u2 is equal to u1. And f2 twiddle is equal to f1 twiddle on all of v. Because this is we're using that they're mapped to the same function. Anyway, I'm about to get myself confused. So I'll leave it at that. I think I'm saying the right thing. You should verify it. But the point is then that the subset where they're equal, every point of the, if you have a point where the functions agreed, you have a neighborhood where they agree. And if you have a point where they don't agree, you have a neighborhood where they don't agree. So that tells us that where they agree and where they don't agree are both open sets. And as we said before, that's the proof. OK? All right. So OK, so what have we proved? Let's go back and digest this. We proved that given any loop downstairs, when you have a covering, you can use the covering to lift up that loop. And it will give you, once you fix a starting point, some path that is typically not going to be closed is not going to come back to the same spot. And what we should think of this is sort of an unraveling of the path below. And so if we think, for example, of the circle and the helix, this is precisely this unwinding of the path below, the loop below, makes it into something that now sort of doesn't sort of go around itself. And this will be an extremely useful tool to understanding the fundamental group of a space. And now for this to be useful for the fundamental group, we not only want to lift paths, but we also would like to say that if we have two paths below the homotopic, when you lift them, which we know we can do uniquely, they're also homotopic upstairs. So this will allow us to deduce things about homotopic classes, not just about paths. So there is a corresponding statement then about lifting homotopies. So I think I'll state it and then leave it to you to prove. We'll give you a list of exercises I think next time so you can work on. And I'm sure I don't have to repeat it, but the only way to learn all this amount of stuff that you are seeing so quickly is to do a lot of exercises and you have to work through examples on your own. So I'll give you the exercises and I'll look at what you write the solutions. But this is not for me, this is for you. This is something you should do just to get a hang on what is being developed and make sure you internalize it in your own words in your own way. OK, so what's the statement? Lifting homotopies. And if you look in Hatcher, he does this all in one go, which is fine. And then the uses are the bigger result, the two things that we are doing today from the big one. I think it's a little easier to understand one by one, so that's how I'm doing it. So what we have is, again, a covering and you still have to be convinced as to why this concept is actually of some use. But hopefully you are beginning to see it. So this is a covering. And what we want to do is lift a homotopy. So we have a map F from, I'm going to try to keep my notation straight, i times i to x. And we think of what happens at time t equal to 0 as a path, gamma 0. So we have a function of the square into x. And time t equal to 0, I'm thinking of time going up like this, is what we have there. This is some path gamma 0. And we know how to lift that. So there is a unique lift, gamma 0, twiddle of gamma 0 through some y0. Make sure I give you the hypotheses. So this is from our previous construction. And so the claim is that there exists a unique lift F twiddle, which is a homotopy to y. So of course, I never say this, but all the maps that we consider are continuous unless mentioned otherwise, such that at time t equal to 0, it is the lift of our path, and it actually is a lift of F. Namely, if you project it down, you get back the F you started with. So what this will allow us to say is that if we have downstairs not just two paths, but two paths that are homotopic, then we can lift the initial path and it will give you a homotopy between lifts of each one that you have below. So you can lift the entire homotopic to y, not just the paths themselves. So you think now instead of having an initial point, which will be the case of a path below, you have an initial path. And once you have a lift of the initial path, then you have a unique way to sort of continue it to in a way that covers what the sort of homotopy that you did of the path below. Does that make sense? I don't think I want to explain it too much. I find myself out of training, and by this time, I'm too tired to think too clearly. So I think I'm going to leave it at this. But in words, what we're doing is not just lifting paths, but lifting homotopy between paths. And this will allow us to do various things with the fundamental group. So in particular, and this will likely be the last concept we'll discuss today. So in particular, if we have, say, so gamma 1 of s is the path at times 1. So we have these two paths, gamma 1 and gamma 0, below. So this is by hypothesis is equal to gamma 0. So these are two homotopic paths. And so gamma 1 twiddle of s and gamma 2 twiddle of 0 of s are lifts of these. And by the construction is such that they start at the same point, starting at y is 0. Because these lifts, so each one of these paths could be lifted, as we did before, starting if we start at a given point, there's a unique lift. That's what we did before. But because there were homotopic downstairs, and we just at least stated that we can also lift the corresponding homotopic upstairs, it means that these paths are homotopic in this usual sense of the word. So what this means, in particular, is that the value of these paths at 1 is the same. So let's think of what this means in terms of the circle and the helix. I've said before that, well, so let's try to now match the two things that we did, sort of this abstract of the structure of lifting a little bit at a time with Levec's lemma and so on with the property of the covering map as our basic tool. And then what we are more familiar with of the circle and the helix is things that we can do with just calculus. So what would be the meaning of this value in the case of the circle and the helix? Well, we have a path on the circle, and then we can lift it. And as we said before, if we start somewhere, is that some angle of the initial point. And when it ends, it has to be some angle, modulo 2 pi, so 2 pi times an integer. So the difference between the beginning and the end has to be some integer, right? So each path has associated to it a unique integer, which is the winding number of that path. So for the case x equals to s1, y equals to r, gamma 0 twiddle of 0 is some angle of x0. And in fact, so is gamma 0 of 1. So the difference is some integer. Well, if I do angles, it will be some integer times 2 pi. So what this says is now, if I take downstairs a path that is homotopic to gamma 0 and I lift it starting exactly the same place, so I pick as the initial angle the same one as I did before, then when I lift it, I'm going to end up exactly in the same spot. So what this says, so this is called the winding number of gamma 0. And it doesn't matter where you start, because everything will move the same amount if you choose a different starting point. So the winding number, so homotopic paths in loops in s1 have the same homotopic, homotopy, same winding number. Namely, take the path, get it lifted by starting anywhere you like, and look as to where it ends, whereas the difference between where it ends as to where it began. That's some integer. That integer is the winding number of the path. And no matter what path you take in the homotopic class, these statements that we're discussing show that it is the same number. So if you like, then, well, we can interpret our discussion of liftings as some sort of generalization of winding numbers. So if you have a space below and some covering space, then this procedure gives you a way to assign a quantity, which is where a path that you lift ends. And that point where it ends depends only on the homotopy class of the path below. And what is your starting point upstairs? OK, so that is a very crucial fact that we can manipulate. We can do this type of analysis of winding numbers without using differential forms, without using the real numbers. This is an abstract way of doing something that is the usual one in the case of R. OK? So what we would like to say, then, is that now it should be clear how we can formalize what we've been discussing about the circle. Say you take S1 and the point 1 as your base point, then we've been saying that this is isomorphic to Z. OK? So what should be the map? Well, the homotopy class of a loop gamma will be sent to what? To the winding number of gamma. Let's call this W of gamma 0. And it's around 0, but I'm going to ignore that for the moment. And the argument that we had before says that this is a well-defined number. It doesn't depend on the choice of path in the homotopy class. And I claim that this is an isomorphism, and we still have a few little things to do to check that. Let's see what we can do now. So for example, how do we know this map is subjective? What does that mean that the map is subjective? It means if I give you an integer, you should be able to produce a homotopy class of a loop that has that winding number. We just go around n times in the circle, just at a normal speed. That will have winding number n. I guess we would have to verify that, but it's pretty intuitively clear. And what we still would have to do is to show that this is injective, so that if you have something with winding number 0, it is 0, and then that is a homomorphism, so that when you multiply homotopy paths, the winding numbers add. But this is something I'll leave for next time. So I'll show you only a few pictures of this to be continued, in other words, to complete the proof of this. And you may be thinking, well, we did all this covering space. That looks like a lot of stuff for proving something we can probably do with our bare hands. But I hope you agree that this, in fact, is a powerful tool that allows to do a lot more things than just dealing with a circle, which is what we're going to do. So in some sense, the covering spaces are a lot, in some sense, a lot more interesting than the fundamental group itself. So I'm just going to show you just to give you a flavor of the richness of covering spaces, what Hatcher has a discussion will suggest you look at. So if we look at that space, the figure 8, which we'll spend some time discussing. So there's two paths here, the A and the B. And you cannot think either one of them. And at least we discussed last time, and I mentioned that. Basically, if any path consists of a word in A, A inverse and B and B inverse, you either go in A in its correct direction or in the opposite direction, or you do the same with B and B inverse. So a path would be sort of do A and then B, and then maybe do A inverse A five times. Any word in A, A inverse, B, B inverse would be a path. And in fact, any path is uniquely determined by that word. There are no relations other than the obvious one that A, A inverse is one and B, B inverse is one. In fact, the fundamental group is the free group in A and B. So by now, if you look at this, you can also think of this as a sort of a graph. You have one vertex in the middle, and then there's one edge that comes out that looks like A, and one edge that comes in that looks like A. So this portion of A and this portion of A. And then one edge that has a label B coming out and then label B that comes in. So let me show you a few of possible coverings of this space. So so far, we don't have a whole lot of examples of coverings. We have S1. We said we have R, or S1 is covered by S1 by this square map. And you can easily see that you can have a covering with the cubed map or the fourth map and so on. And that's basically all you can have for the circle. But for this space, which is just two circles put together, things become a lot more interesting. And you have a lot more covers. So these are some of the covers you can have. So each one of these spaces is actually a cover of the figure eight. And what is it that determines them? How would you construct them? Well, let's look at this one. So you have a number of vertices. And the thing to focus is that if you look around a vertex, you have an A coming in, a coming out, and coming in, and a B coming in and coming out. So any graph that has that shape is actually a covering of the figure eight. So coverings could have quite a lot of interesting things in it. So for example, we see these little thingies that look like snowflakes. So at each point, there are these four things. And you have to keep doing this. So they don't close back until it continues at infinity. And the more interesting one, so you can take this to each one of these that is here, has a fundamental group that is not trivial because there are little loops in them that you cannot shrink. But you can sort of go all the way and get a covering that doesn't have any loops that you cannot shrink. So you can have a covering that is simply connected. And that's what it is. So if you take the circle, it has a covering, namely R. And R is simply connected. And so we'll see that all most natural spaces have a covering that is simply connected. And that's going to be the mother-all coverings. Any cover will be in between the big one in the space you have. And so that's called the universal cover. And here's the universal cover of the figure eight knot. And for those of you who have seen Galois theory, I don't know how many of you have. Have you seen Galois theory? Some, some of you. So this actually will make this theory become very much like Galois theory. And this universal cover will play the role of the algebraic closure of a field. And so we'll see how groups that look like the Galois group come in. And the Galois group is replaced by the fundamental group. All right, so enough mystery. And we'll continue on Thursday, I believe.