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0:01
All right, welcome everyone! I'm Norman Wildberger, and we're here at the University of New South Wales. This is a beginners' course of algebraic topology, and in this lecture, we're talking about the relationship between covering spaces and fundamental groups. So, we've been talking about these covering spaces. The basic setup is that we have a map from one space to another that sort of covers the second space, in the sense that we could think of the
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base space as being downstairs in the plane, and the first base being somehow on top of it in a kind of systematic way, something like that. For example, a three-to-one covering of the circle, so the helical covering of the circle. The basic property of a covering space is that if we take any point in the base, say "A," then the inverse image up here will be a finite set of copies of "B," and if we look at a little neighborhood of "B" in the base space, that interval there,
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then the inverse image looks like a bunch of copies of that interval. All right, so that's a covering space. And today, we want to investigate what the relationship there is in such a situation between the fundamental group of the base and the fundamental group of the upstairs space. So, to make that a little bit easier, I'm going to first talk about some other examples of covering spaces where the base space is just a little bit more complicated than
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just being sort of one-dimensional. So, let's start by thinking about coverings of a torus. So, if we have the base space being a torus, how could we get some coverings of that? Well, the idea is somehow that's very similar to this. In fact, what we could do is we could take exactly this picture here and dip both these spaces into a batter like how the way you make pretzels, making them instead
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of just one-dimensional things, making them into two-dimensional fat sort of surfaces, surfaces. And then we would immediately have a covering space of the torus by this sort of helical torus. Now, that's a little bit hard to draw if you're not too skilled, so I'm going to draw that kind of thing in perhaps a slightly different way. So, let's draw a couple of copies of a torus. Say, people just use two; that's the idea is already clear there. So, let's draw two toruses,
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and we're going to create a space "X" by suitably cutting and pasting these ones here. So, let's draw a circle on the torus, each one, and slice the torus along a circle. So, we're going to make a little circular cut on that torus and a little circular cutting on that for us there. So now, we've cut these two tori, and each of the sides of the cut is a circle. And let's label those
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different sides of the circle, say by "C1" on that side and say "C1'" on the other side. Okay, we cut the torus into little bands, and we're calling those two circles "C1" and "C1'", and over here, we've also cut the torus, and we have "C2" as one circle and "C2'" as another circle. And now, we're going to create the space "X" by identifying "C1" with "C2'" and "C2" with "C1'". So, in other
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words, if we have a path in the space, let's say a path that's coming in this direction here and it gets to the boundary, so the "C1" side of the boundary, then because of this identification, it means that we have to immediately jump up to the second copy and where we glue it like this. So, this would be a continuous path on space "X", and let's say if that path happened to continue and go
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around here and go in this direction, well, then it would cross that cut, which is the "C2", and so according to this identification, you would have to come out on "C1'", which would be over here. So, we continue on like that. And if we looked at the projection of that path down onto the base torus where the projection is just exactly going straight down, then the corresponding path would
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be, well, we're going here and then jumping from here to here. The image doesn't feel anything, so it just would just go around, and then we would just go around like this. So, the image would just be a continuous path down here. All right. So, this is one way of creating a covering space without having to try to draw something that's self-intersecting. And we can kind of pretty clearly see the covering aspect. There's quite another way of drawing a covering of a torus,
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which I also want to show you. So, an alternative is to work completely in the planar situation, right? So, we're now going to think of a torus not as being the usual donut in three dimensions but rather just as a planar square with appropriate edges identified. So, the base space now, kind of working in the plane, one-one. So, maybe this will be our base torus, all it be, and it has the property that this edge is supposed to be identified with that edge,
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and this edge is supposed to be identified with this edge. Okay, now to create another torus kind of covering, and I'm going to create a more sophisticated one, a little bit more sophisticated than this one. I'm going to look at a bigger torus with the same scale, one-to-one, but maybe I'll go over three here and up to here, and so we'll have a bigger rectangle. So, this is a three-by-two rectangle, and we can make that into a torus in the same way by identifying
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opposite edges. So, we're going to identify this edge on this side with this edge on this side, and this top edge with that top edge, making a bigger torus. Then we'll call that "X". All right, now how can we define a nice covering map, a map P, a projection from X to B? Well, what we're going to do is we're going to take a point (x,y) here and we're going to send it to the corresponding point down here. That'll be P of this point (x,y). Now,
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by definition, that's equal to (x,y) mod 1. In other words, we just ignore the integral part and just look at the part between zero and one. So, this one here, for example, might be in this picture 1.8 and 1.3, maybe 1.4, okay, 1.8, 1.3, and then that would get sent to, we just ignore the integral part, so you get sent to 0.8, 0.4, 0.8, 0.4. That's the map.
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That's a covering space because, for example, the inverse image of this point here, if we look at this point here, maybe even look at a little neighborhood of it, its inverse image is going to have six points in it. There will be not just that one but also one there and one there and one there and one there and one there. And the various neighborhoods lifted will be these little neighborhoods around each one of these points. Well, so we can see the covering property. The
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inverse image of a point down here is a discrete number of copies upstairs, and the inverse image of a little neighborhood is a discrete number of little neighborhoods, each of which is completely isomorphic to this one here under the projection. This sort of corresponds to one of these helical things where we're wrapping three times in one direction and twice in the other direction. It would be a little bit more challenging to draw it in this kind of fashion, but we could do it,
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but we'd have to slice it also sort of along the other circle, not just along that circle. So, this planar representation is quite a pleasant picture. Alright, let's just mention coverings of a two-hole torus. So, if we have a two-hole torus, how could we get a covering space for that? Well, we could do an analogue of this version here. In other words, we could make a couple of copies of that. I'll just use two for simplicity. And then, we could make some cuts in corresponding
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places. For example, we could cut along here and along here. And just in the same way as we did before, we could call these sides C₁ and C₁ Prime, and C₂ and C₂ Prime. And we can agree that C₁ is glued to these, glued to C₂ Prime, and C₂ is glued to C₁ Prime. What we get also, and perhaps cut on the on some other side, so we can also well make some other cut. For example, a lot here and along here, and call these sides D₁ and D₁ Prime, and E₁ and E₁ Prime. And then
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we can also make the identification that D₁ is identified with E₁ Prime, that's D₁ ≡ E₁ Prime, and D₂ with E₂ Prime. And then we would get some kind of covering of the two-hole torus.
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And a very natural question is, alright, well, so there's an obvious analog of this picture for the two whole tours. What about an analog of this picture for the two whole tourists? Is this would be there would be some way of kind of making a plainer representation of the two whole tours by cutting it and laying it open? Well, it turns out that there is, but that's a little bit more complicated involved some hyperbolic geometry and I
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might mention that in my next lecture. Alright, so now we've got some examples of covering spaces not just with one-dimensional things but also with some two-dimensional things, so now we can talk about the relationships between covering spaces and fundamental groups. So the kind of question that we're interested in is if P from X to B is a covering space, what is the relationship, what if any is the relationship between the fundamental groups PI of X and PI of B say.
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So that means we're talking about paths and we want to then connect paths in inve space to pass in the covering space and there are two important results here that that we have to establish. So the first is the lifting path lemma.
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Okay, so let's write this down. It's a bit long, but the content is, I hope, pretty easy to understand. So if P from X to B is a covering space, let's choose some base point in both spaces, with A in B and X little X in X, and let's suppose that X is above A, so with P(x)
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equals A. And if alpha is a path in the base space B from A starting at A, say to another point B. Then there is a unique
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path alpha tilde in X, starting at X with alpha tilde opposed P equals alpha.
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So let me draw the picture beside that.
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Alright, so the situation is we have, okay maybe I'll use the Taurus. We have a Taurus, there's a base space, and we have some covering space, and let me be rather ambiguous about it. I'll just draw it rather diagrammatically like this. Okay, those are kind of some kind of covering in a space, and we were saying we have a particular point A in B that's chosen and a particular point X in X which is lying above A, in other words, P(x) equals A. What else we have is a
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path from the point A to some, well, to some other point, say B. Let's make it a little bit longer.
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Alright, this path is named as alpha, it's a path starting at A in the base going to some other point B. Okay, what we want is we want to lift that path to the covering space and we want to lift it so that it starts at X. There might be other points lying about A and typically there will be other points lying above A but we've chosen that particular point lying above A. We want to lift this path alpha to a path in X which is right above it which starts at X. So the claim
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is there's a path and then we'll draw it like this, roughly diagrammatically. This is going to be the path alpha tilde which is a path in X starting at X and has the property that, well, if you compose the path which is actually really a function from the interval into the surface, okay,
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that's really what a path is, it's a function from the interval into the surface, either alpha or alpha tilde, then this diagram commutes. In other words, if you go to a point here under alpha tilde and then you go down and that's the same as if you took that initial point and go over there. So for example, this might be alpha tilde of maybe a point 2 because we're point 2 of the way along the
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curve and here is alpha of 0.2, 0.2 along the way, and we want that this points its reign above there which is the statement that alpha tilde composed with P equals alpha. This is the lifting property.
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Alright, so why is this true? Why can we lift such a map? Well, I can show you the idea right on this picture here. So what we do is we say, "Alright, we starting with this path alpha down here and it's beginning at A, and we want the corresponding path up here to begin at X. So let's take a little neighborhood of A." So we'll take a little neighborhood of A, okay, and look at the inverse
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image of that neighborhood which is containing X. So there's some little neighborhood of X which is exactly a copy of this little neighborhood of A. So maybe we'll call this neighborhood U or maybe U1 because there will be more of them, and this is U1 tilde. It's just a bijective copy of the neighborhood upstairs around X, and it's exactly the covering map property that ensures that such a
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thing exists. Alright, so locally, if we just look at U1 and U1 tilde upstairs, those are isomorphic, they're exactly the same. So the first part of the path, this part of the path that's in red, that's in U1, we can lift that up in exactly a unique way because this is an isomorphism. P is an isomorphism at the level of these two open sets, so there will be exactly a unique path in U1 tilde that lifts this path down here, we just go via P₁, P inverse, well P restricted to U inverse.
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So that's how we construct the first part of the path. Alright, so we've lifted the first part of the path and now instead of having starting from here now we're at some new point that one there and a point above it. Well, now what we do is we repeat the process using this point and the point above it. We take a new open neighborhood, call it U2, around that new point, and the inverse image around this point,
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a new neighborhood U2 tilde. These are going to be isomorphic and so we can lift the second part of the path to U2, and we just keep doing this, we just subdivide the path here into little pieces, each of which has an inverse image which is completely homomorphic to it, and we lift the path on each one of those individual pieces by the inverse of P on those pieces.
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So in other words, locally at any given time, locally, there is no possible choice of how to wear to lift because there's a complete isomorphism locally between these two things and so as we traverse here, we never have any choice about where the blue path has to go, it just has to follow its nose because of the isomorphism of P at every point. So that's the lifting property of paths in the covering space.
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And a lot of number of items head, sorry, yeah, because this is up, this is a finite path, you know, just a finite path from the interval into the space so it can be covered by a finite number of these little neighborhoods. Nowhere we don't work with any kind of perverse situations where we have things like this or our neighborhoods are all nice and reasonable and nothing ever gets infinitely small and so on.
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That's a reasonable question. Okay, there's a variant of this of this lemma called the lifting homotopic lemma which has a very similar form. It says if P from X to B is a covering space and let's say alpha and beta are homotopic paths in in the base space B from say A to B with a homotopy given explicitly by a map H then there is a unique homotopy H tilde from okay, well, let's say the lifts the lifts alpha tilde and beta tilde of alpha and beta to
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X starting at some point have some point X with P of x equals pain.
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Alright, so let me draw the diagram that'll stick with the torus B and now maybe I'll be a little bit more careful about this diagram here.
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Alright, so the base example will be a torus. Our covering space will be X where we've got an X by gluing two tori together in the same way that we talked about before. So we'll identify the side C1 of this cut with C2 Prime on that side, and we'll identify C2 on this side with C1 Prime on this side. It's a covering of the torus. And the other setup is that we have a point A down here and we have some point in X above it. We have two paths emanating from A, so maybe one will be
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this path to a point B, and that will be the curve alpha. And then we have another curve beta which is homotopic to it. Homotopic means that, well, we can continuously vary alpha to beta. There's our path beta in red. So there are two paths going from A to B downstairs and B and they're homotopic. Let me remind you what homotopic means. Okay, homotopic means that there's a homotopy and the homotopy is a map from a square to a square, so from 0 to 1 to 0 to 1, so that
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the base goes to alpha and the top goes to beta, and for every sort of intermediate value s we have an intermediate path, say alpha sub s, which is part of the movie going from alpha to beta.
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Alright, so the way we read a homotopy is this. If we go along the base here, then we get the path alpha in blue. As we go along these successive horizontal paths, we're getting intermediate paths that are somehow between the blue one and the red one. For example, maybe something like this. So that might be alpha at some point 5. If we take a movie, we can see this continuous family of paths, all of them starting and ending at A and B,
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that go continuously from alpha to beta over here in this space, base B. That's what a homotopy is, that's what it means for alpha and beta to be homotopic paths. So what we want to do is we say, "Alright, we've just established this lifting lemma property. The lifting lemma property says that we can lift alpha and we can lift beta to the covering space." Okay, so we can lift alpha.
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What's the lift of alpha going to look like? Well, I'll try to draw it reasonably faithfully. So it's rather continuous, smooth up here. Here's the place where the cut, the image of the cut is. So we're going around here until we get to that point there where the cut is, and then the lift will have to continue up along here. Okay, so this is the lift that was alpha, wasn't it, in blue? Okay, there's alpha tilde, the lift of alpha. Now we can also lift beta by following
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our noses. We just follow it up, starting here, following it, so we got to about there and then we had to come out and go something like this. That's the lift of beta which we call beta 2.
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So these two paths downstairs, alpha and beta, we have their lifts, alpha tilde and beta tilde upstairs. We now want to convince ourselves that alpha tilde and beta tilde up here are homotopic.
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But if we just look upstairs, alpha tilde + beta tilde are homotopic paths.
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So how do we do that? How do we get a homotopy from alpha tilde to beta tilde? Well, the proof is we just lift. We lift alpha sub s for each s in 0 to 1 to alpha s tilde in X. So in other words, in the homotopy here, each individual horizontal line gives an intermediate path. We take each one of those intermediate paths and lift them up. For example, alpha at point 5, this one here, one that
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I drew here, we lift that one up, we're going to get a path also from X. Well, it'll be some other path, whatever it was, so then why that would be alpha 0.5 tilde. So if we watch the movie of the alpha sub s's as they go from alpha to beta down here, then those the lifts will be a continuous movie of paths up here that go between alpha tilde and beta tilde. They'll all start and end at A and
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B respectively because at each point everything is continuous and alpha and beta start and end at A and B and are at A we're just maintaining that. So we end up with a family of paths up here giving us a motion-picture view between alpha tilde and beta tilde, so we are lifting this homotopy.
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That's essentially a definition of the lift of the homotopy.
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Alright, so those are nice useful theorems. Alright, so now let's apply the now let's apply these theorems to two fundamental groups. So our setup again we have E from X to B covering space when we're talking about fundamental groups we like to have base points so let's choose a base point in both spaces so let's let A be a point in B and X a point in X with P of X equals A. We know that whenever we have a map between spaces we get an induced homomorphism between fundamental
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groups that happens automatically whenever we have a continuous map between two spaces we get an induced homomorphism and out what's called P star from the fundamental group of this space say centered at X to the fundamental group of this space centered at the image of X which is A.
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So, very loosely, there's base space B, here's some covering space X, here's some point X and some point A below it. If we take any loop in this space starting and ending at X and we just look at its image under the map, then we will necessarily get a loop that begins and ends at A. So, we're just composing the map from the interval, say that's alpha, with P to get a map here. So, we're just pushing down this thing. P star of
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alpha or the class of alpha is the class of the composition alpha followed by P.
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This is a homomorphism of groups. That happens whatever we have a function from or continuous mapping from one space to another. We get a homomorphism of the fundamental groups. That happens whether it's a covering space or not. Okay, but in this covering space situation, something extra happens, something special happens with this map, and the theorem is that if P from X to B is a covering space, then P star from PI of X to PI of B, B is injective.
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That's the same as saying that it is one-to-one. And because it's a homomorphism, that's the same as saying that its kernel is the identity. The kernel of P star equals the identity.
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So, why should this be true? Well, let's suppose we have a path. Let's call it gamma in PI X, X. So, it's a path upstairs starting and ending at X which has the property that it's pushed down such that P star of gamma equals the identity, i.e., in PI B, B. So, I'm investigating this aspect of it, looking at the kernel, trying to prove that the kernel is the identity.
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So, maybe here is a picture. Here's our space, B. Here's our space, X. Base point X, our base point B. We're assuming that we have some path, gamma, up here whose image down here, whatever it is, is actually trivial. What does it mean to be trivial? It means that over here in this space, the image of this path that we have is homotopic to the identity. Homotopic to a constant map. Okay, so here's the path gamma that we started. Here's the image P star of gamma,
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which is just what you get by composing it. And we're assuming that P star of gamma is homotopic to the identity. Okay, so then P star of gamma is homotopic to the constant. To the constant map, a constant loop at B. Let H be this homotopy.
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In other words, down here, we know that we can make a movie between P star of gamma and the constant loop. The constant loop which doesn't just start and ends at B and doesn't move at all. So, down here, we're assuming that there's a homotopy. We can make a movie between P star of gamma and the constant. What are we going to do with this homotopy? We're going to lift it up to here. We're going to lift this homotopy between these two loops down here to upstairs now. So,
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now, lift this homotopy, lift this homotopy H to H tilde between what?
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Well, what is the lift of P star of Gamma? What is the curve up here which is directly above gamma down here? It has to be the original gamma that we started with because the original gamma starts at X and at every point is directly above the image downstairs. So, the lift of P star of gamma between the lift of P star of gamma, which is equal to actually gamma, and the constant map, the constant map at X which just stays at X, doesn't do anything,
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and the identity at X. In other words, the movie that you've created between the constant and the image of gamma down here lift to a movie between the constant here and the original curve gamma that you started with. And that means that the original curve that you started with is actually homotopic to a constant loop. So, therefore, gamma, the class of gamma, is actually the same as the class of the identity in PI of X at X.
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Okay, what's the consequence of this? It means that in the covering space situation, if you have an injection, it means that this group is actually isomorphic to a subgroup of this one here. It's actually really the same as its image. So, it means that covering spaces of B give you subgroups of the fundamental group of B. This covering space X of B gives us then a subgroup of the fundamental group of B.
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So, PI of X, X is isomorphic to, well, P star of PI X, X, which is a subgroup of PI B, B. In other words, covering spaces of B give us subgroups of PI of B.
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So, kind of a main fact is that this is a one-to-one correspondence. This correspondence goes both ways.
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In other words, if you have a subgroup of the fundamental group, there's essentially one covering space that it determines. So, we can classify covering spaces by classifying or studying subgroups of the fundamental group of the base. So, next time we'll have a look at this correspondence and it will turn out that the key to understanding that is to understand the covering space associated with the simplest subgroup, which is just the identity subgroup.
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The covering space that has essentially trivial fundamental group is called the universal cover, and once we have hold of the universal covering, then all the other covering spaces are somehow created from it in an interesting way. Alright, so we'll see you then.
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