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0:00
Here's a topology problem: How many essentially different continuous functions are there from one kind of sphere to another? The n-dimensional sphere, S^n, is just the set of unit vectors in (n+1)-dimensional space, and two functions are "essentially the same" here if one can be continuously deformed into the other, and they're "essentially different" otherwise. This seems like one of the simplest topology questions you could possibly ask,
0:25
since spheres are some of the most basic objects in topology. But this is actually an unsolved problem -- mathematicians don't know a complete answer. The question is usually asked as "What is the k-th homotopy group of the n-sphere (abbreviated as pi_k of S^n)?" So no one on earth knows, say, pi_100 of S^4. On the other hand, we do know a lot of the story. This is a table of some of the stuff we do know.
0:53
So for example, if you're looking at maps from a 10-sphere to a 2-sphere, there are essentially 15 possibilities. For maps from a 4-sphere to a 3-sphere, there are only 2 possibilities. If you're looking at maps from a 7-sphere to itself, there are infinitely many essentially different continuous functions -- one for each integer. There are tons of patterns you could notice in this table, and I think it's really cool how this simple problem has such a complicated answer.
1:23
My goal for this video is to give you some animations to help understand some of the entries in this table, especially this top-left corner where everything is low-dimensional, and then I'll point out some more interesting patterns going on here. Instead of including difficult proofs, I'm going to focus more on building geometric intuition to help understand why some of these answers are believable.
1:44
I especially want to point out some ideas that should be accessible and interesting to someone curious about topology, but which you don't always get to hear about in a first or second topology course. Homotopy groups are sometimes studied without much geometry or visualization, so it should be interesting to compile some of the abstract theory down into nice concrete animations. Let's start with the first homotopy group pi_1 -- it's also called the fundamental group.
2:09
If you have some shape "X," then pi_1(X) is supposed to describe the different continuous functions from a circle into X (which you might call "loops"). Start with a circular racetrack for X. I'm actually thinking of a thickened-up annulus instead of an infinitely-thin circle, but this won't end up mattering. If you want to walk around the circle, starting and ending at one distinguished point, then you might just walk around once. But you could also go around twice…
2:40
or three times… or any number of times that you want. You could also go around backwards once… or twice… or you could go backwards three or more times. You could even just stay still, keeping your position constant. If you do 3 laps around and then go 2 more laps around, then in total you've gone 5 laps around. So we have this sort of "number system" of all of the different ways to walk around a circle: doing one kind of loop after
3:22
another just adds the corresponding integers. So we say that pi_1 of a circle is the group of integers under addition. We get a different "kind" of loop for each integer, so this is an example of the sort of thing that pi_1 should describe. You might think, "What about a little loop like this that doesn't go all the way around?" Remember that we only want to count "essentially different" loops. But we can continuously deform the little loop to squish it down the basepoint,
3:52
so we say that the little loop is homotopic to the constant loop. It corresponds to 0, not some new number. I like to think about going for a little walk one day, then a slightly shorter walk the next day, then making the walk shorter and shorter until you don't walk at all. Each new day, you didn't change the walk by that much, but in the end you've transformed the starting loop to a constant. Equivalently, you can think of the whole loop continuously deforming
4:20
like a rope being hauled back in. In the same way, if you have some very squiggly loop that nonetheless goes around just once in total, then you can continuously deform it by gradually straightening it out, and keep straightening it out until you're left with a basic circle that goes around at an even pace. The squiggly path still just corresponds to 1 in this number system.
4:54
More generally, the number of times a loop goes around is called the winding number, and the winding number is the complete invariant here, which just means that two loops are homotopic if and only if they have the same winding number. As another example, if we take a loop with winding number 1 and then a loop with winding number -1, then the result should have winding number 0. And sure enough, we can gradually deform this doubled-up loop back down to a constant.
5:26
Notice that the hole in the middle of the circle is significant -- it's what makes it so that we have any interesting loops at all. Without the hole, any loop you want can just be shrunk uniformly down to the basepoint, so there's really only one kind of loop. So pi_1 of a shape somehow detects its "holes," whatever that means. Having this circle example in mind, let's make this into math and write down some definitions of what we're thinking about.
5:55
I'm going to assume that our world X is a metric space, so just any sort of shape with a good notion of distance. That's not traditional, but I do think it's helpful for understanding. Let's also pick some basepoint in our space, which we'll call x_0. Then the set of all loops in X based at x_0 is denoted as Omega(X, x_0).
6:16
It's just the set of every possible function from the interval [0,1] into X with the constraint that the curve starts and ends at the basepoint. This set of loops almost always has uncountably many elements. You could also think of this as the set of basepoint-preserving maps from a circle with some basepoint into your space. If we view a circle as an interval with its ends glued together, it's the quotient topology that lets us go back and forth here.
6:44
The quotient topology lets us take the property of an interval-based loop just happening to start and end at a point and exchange that incidental property for some topological structure in the domain. Now the remarkable thing is that this set of loops comes with a nice metric -- it's just the uniform metric. The distance between two loops is just the maximum distance between corresponding points. So two loops are close together exactly when
7:09
all of their corresponding points are close together. Instead of just thinking about a set, we often get more out of a piece of math if we look for a nice topology (or metric) to put on the set, so that's what we're doing here. Now it's worth noting that this loopspace is infinite-dimensional in some sense -- there are infinitely many independent little directions in which you could nudge a loop around.
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So there's not really a nice way of visualizing it as a set of points in space; you really have to think about it as just the set of all loops where some loops are close to others. It's more like a space of functions that you might encounter in analysis. Okay, so we know what loops are; now we want to define which loops count as "essentially the same." We say loops gamma_0 and gamma_1 are "homotopic," or write this tilde-equals symbol between them,
7:56
exactly when there is a path between them in loopspace. This means there's a continuous function from [0,1] to the loopspace of X which is gamma_0 at 0 and gamma_1 at 1. This sounds sort of abstract, but this is exactly the data of one of the animations we've been looking at: each day we only take a slightly different walk, which corresponds to the continuity of the homotopy h; plugging in nearby values
8:20
into h should return (uniformly) nearby loops. Now the thing we wanted to define was pi_1, which will just be the set of homotopy classes of loops, or equivalence classes based on which loops are connected by homotopies. Sometimes people don't write the basepoint if it doesn't matter, and just write pi_1(X), but properly it's always pi_1(X, x_0). We can then define a concatenation operation as a piecewise function for when we do one loop "and then" another.
8:50
In the concatenation of two loops, we go through the first loop during the first half of the time, and the second loop during the second half of the time. This concatenation plays nicely with homotopies, so we can concatenate two homotopy classes of loops by just concatenating loops in those homotopy classes. This whole definition is a lot to digest, so feel free to pause and try to make sense of it all, especially focusing on our circle example
9:14
where pi_1 should be isomorphic to the integers. I will also note that if you've seen the fundamental group before, the definitions here are probably different from what you're used to, but they are ultimately equivalent. The definition here is nice because there's supposed to be this intuition that a homotopy is a path between loops, but that's literally what I wrote down: it's a path from one loop to another in loopspace.
9:37
A keyword is "left homotopy" versus "right homotopy." If you want to work with topological spaces instead of metric spaces, you can do the same thing, but you need the "compact-open topology," and I don't want to get into that, so I think sticking to metric spaces will make some of these things easier to think about.
9:55
There are a couple facts about these definitions that we're supposed to check, like: homotopy respects concatenations, and we have inverses, and identities, and associativity all up to homotopy, but the punchline is that pi_1 is a group. So far we've thought about how pi_1 of the circle is the integers, and pi_1 of a filled in disk with no holes is the trivial group where there's only one element. For another example, consider this figure
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eight sort of picture with two holes. What are our homotopy classes of loops? We could go around A, forwards or backwards. Or B, forwards or backwards. Or go around multiple times. Or we can mix and match -- get AB or BA. In total, we can get any string of "A"s, "B"s, inverted "A"s, and inverted "B"s.
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But if we ever do A and then immediately do A inverse, then that loop is homotopic to a constant loop, just like we saw with the circle, so that's not anything new. Also, if an inverse pair shows up consecutively in a longer string, we can cancel just that inverse pair. So the essentially different loops here are the so-called "reduced words," where nothing adjacent can cancel. This system of reduced strings here is called the free group on two generators,
11:17
and that's our pi_1 of a figure 8. I want to point out here that order matters; A and then B is not the same as B and then A -- the homotopy class of a loop still knows some of its history, and it keeps track of which part of the shape it essentially came through most recently. You can't wiggle something that ends with B to get it to end with A. So pi_1 is not always commutative.
11:45
Next let's think about pi_1 of a sphere S^2 -- that's just the outer surface of a ball. Given a loop on the surface of a sphere, based at the south pole, as long as the loop doesn't touch the north pole, we can make our loop uniformly melt away back down south to a constant map. Any sort of loop that you can think of, just push it south, away from the north pole and toward the south pole until it's just a point. There is one little wrinkle:
12:23
what if the curve hits the north pole exactly? But this isn't a real issue, since any piece of the curve that hits the north pole can just be nudged away to not hit the north pole, and then pushed back down to the south pole. Even if you had some wild space-filling curve on a sphere, you could still do the same thing, working one small piece at a time to miss the north pole, and then push everything uniformly south.
12:46
Since everything here is homotopic to a constant, pi_1(S^2) is trivial. Before we move on, here's a quick summary about pi_1. We defined pi_1 as loops up to homotopy, and saw that pi_1 of a disk was trivial, pi_1 of a circle was Z, and pi_1 of a figure eight was a free group on two generators. There is a pattern here: a shape in the plane with two holes gets a free group on two generators as its pi_1; a shape with one hole gave us Z which is a free group on one generator,
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and zero holes gave us the trivial group, which is the free group on zero generators. More complicated shapes can actually make pi_1 be any group you want, not just free groups, but at least in these cases you can see the sort of thing that people might mean when they say that pi_1 "detects holes." One funny thing here is that a disk and the surface of a sphere both have the same pi_1 -- they're both trivial. And it sort of makes sense
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to say that a sphere has no holes, in the sense that it has no circular holes, but there is something else that makes a sphere different: a sphere has a hole in the same way that a block of swiss cheese can have holes: there's a "cavern" or "void" on the inside of the sphere that pi_1 doesn't detect. So trying to detect "caverns" like this is an inspiration for finding higher-dimension versions of pi_1, like pi_2 which we'll see next.
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Now is a good time to point out that it doesn't matter if we "thicken up" our space: for the sake of homotopy groups, a thick annulus and an infinitely thin 1-dimensional circle behave exactly the same way. Likewise with a figure eight. We could even add an extra "altitude" dimension to a sphere or another surface, and it wouldn't make a difference to homotopy groups. This kind of "thickening" relationship is the premier example of a homotopy equivalence,
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and today we'll think of homotopy equivalent spaces as being "the same," since they'll have all of the same homotopy groups. So far in this table, we've looked at pi_1(S^1), which was Z, and pi_1(S^2), which was trivial. The same idea will work for pi_1 of higher spheres: we uniformly push a loop away from the north pole and toward the south pole, after maybe nudging small pieces away to miss the north pole.
15:00
So pi_1 of higher spheres is also trivial, and that finishes the first column of the table. Now we're ready to look at column 2. The second homotopy group is going to look at maps from a 2-sphere into a space, but for a lot of purposes I think it's nicer to think of "loops of loops." So before we define pi_2, I want to walk through what pi_2 "feels like." If pi_1(S^1) was "essentially different ways of walking around a circle," then pi_2(S^2) will be "essentially different
15:32
ways of swinging a loop around a sphere." Instead of just the surface of the sphere, It'll be easier to see if we thicken things up in altitude so our space will be all of 3-dimensional space with the exception of a ball at the center that we can't move through. We can put the basepoint just floating off in space somewhere. We'll think of an element of pi_2(S^2) as a way of swinging a loop around in here,
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like a lasso or a jump rope. So an element of pi_2(S^2) is not just a loop; it's an animation of a loop moving through space. And we'll assume that throughout the animation, the ends of the loop stay pinned to the basepoint; you need to keep a hold on both ends of the jump rope. And we'll also assume that the animation starts and ends at the constant loop, where the loop is bunched up at the basepoint. You could even do two swings around,
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one after another. Or any number of swings that you'd like. We can concatenate two animations by just playing one after the other. Two swings around and then three more swings makes five swings. As usual, we only want to count the "essentially different kinds" of these animations. So for example, swinging a wiggly loop around is homotopic to swinging a standard loop around, since we can gradually iron out the wiggles: if on the first day your loop is very wiggly,
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then on the next day you can make it slightly less wiggly, then keep ironing out the wiggles day after day until you're left with the standard loop.
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In a similar way, if we have an animation where the loop doesn't swing past the ball, then we can gradually shrink that animation back to the basepoint day by day, gradually replacing the animation with shrunken down copies. Ultimately, the homotopy classes of these animations exactly correspond to the integers, so pi_2(S^2) is Z. So there's some sort of higher-dimensional version of the winding number that counts "how many times
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did I swing around the ball?" You might say that the number of times you swing a jump rope around is some well-defined integer. Or maybe think of the number of times a basketball goes through a hoop, as long we're playing a strange version of basketball where the hoop is infinitely bendy and the ball hovers in place. In this goofy game of basketball, if the ball goes through the hoop you get a point, but then if the ball goes back
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out of the hoop then you can lose that point. That's because you can gradually cancel out a forward swing with a backward swing by shrinking the maximum swing amount until you're left just staying at the basepoint. There are probably many more imperfect analogies you could use here, and maybe you can think about your own, but what I want to communicate is that it's possible to get this hands-on visceral feeling of pi_2 based on experiences swinging loops around in real life.
18:19
Holding onto that feeling, we're ready for a definition. The animations we've been thinking about are loops of loops, so let's try writing that down. I'll still assume X is a metric space for convenience, with some basepoint x_0. Then remember we had the loopspace Omega(X, x_0), which was itself a metric space. So let's do it again; take the loopspace of the loopspace. So that'll be our definition of the double loopspace, written as Omega^2.
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This would be very abstract to look at with no lead-up, but this is really just translating from English into math symbols exactly the sort of animation that we've been looking at: a loop of loops is a continuous function gamma from an interval to the loopspace, which takes in a "time" parameter and returns the loop displayed by the animation at that time. We require gamma(0) and gamma(1) to be the constant loop,
19:09
since that's where our animation is supposed to start and end. The fact that each frame of an animation has the loop based at the basepoint (and not just freely floating around) corresponds to the fact that gamma is a function into the loopspace, which already only contains based loops. The continuity of gamma corresponds to moving the loop around without teleporting, so nearby times should have nearby loops.
19:32
With this double loopspace defined, defining pi_2 is just like defining pi_1: we quotient out by homotopy, so we count each essential kind of animation only once. There's another concatenation operation here, where we can do one swing of a loop and then another swing, and this operation (just playing one animation after the other) again makes pi_2 into a group. We don't even need to check associativity and all that,
19:57
since pi_1 is pi_1 of the loopspace. This is just saying that homotopy classes of loops-of-loops are the same as homotopy-classes-of-loops of loops. This "animations" view of pi_2 is nice to get intuition, but isn't pi_2 supposed to be about maps from S^2? Let's try to see how that works. Imagine some loop of loops in some random space X. What's the general shape that the loop of loops sweeps out?
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It seems to be a sphere. The data of a loop-of-loops lets us pick one blue loop from the loop-of-loops, and then pick out one green point from that loop, all in a continuous fashion. So these two parameters, "which loop" and "how far along the loop," form a sort of coordinate system for the sphere. It's sort of like latitude and longitude, except the international date line is collapsed to a point. We could even build a flat
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map based on this coordinate system, unpeeling the sphere into a square. When reading the map, we have to remember that the edges of the map all get glued together to just one point in the sphere. The top and bottom edges are glued to the basepoint because every blue loop starts and ends at the basepoint, and the left and right edges are glued to the basepoint because the animation starts and ends at the constant loop.
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In short: A SPHERE IS A LOOP OF LOOPS. Maybe you'd want to specify that we mean a basepointed loop of basepointed loops (since a free-floating circle of circles would be a torus), but regardless, this makes for a nice snappy slogan. Let's write this down more precisely, again using metric spaces for convenience. We already defined the double loopspace as literally the loopspace of the loopspace,
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but there is an alternate definition: we look at maps from a unit square into X, where those four red edges we saw all get sent to the basepoint. Similar to what we saw for pi_1, instead of thinking of maps from a square that just happen to send the boundary to the basepoint, we could also encode that information into the topology of the domain by taking a quotient of the square, wrapping it up into a sphere,
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and the quotient topology lets us say that these two sets of maps correspond exactly. Of course, the double-loopspace should still be a metric space -- let's just use the uniform metric again. Then as you might hope, the two definitions are equivalent. A loop of loops really is just a sphere. Not only are the two metric spaces homeomorphic; they're actually isometric -- there's a distance-preserving correspondence between them.
22:32
Remember that in the blue definition, you have a loop-of-loops gamma, and when you plug in a time t you get a loop, and then you plug in a loop-progress s and get a point, whereas in the green definition, you plug in just one time/loop-progress pair (t,s) into alpha all at once, and then you get a point. These are really two formats of the same data -- computer scientists call this relationship "currying," after Haskell Curry,
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so this is a continuous version of currying, or more properly a basepointed continuous version of currying. You can check that the correspondence is well-defined and continuous in both directions, and it's not too hard to show that this preserves distances. It boils down to the fact that the maximum of a whole bunch of maximums is just the overall maximum. This also means that if we quotient out by homotopy,
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we get two equivalent definitions of pi_2: either loops-of-loops up to homotopy, or spheres up to homotopy. Let's see how this adds to the intuition about pi_2(S^2). If we swing a loop around our 3D-space-with-a-forbidden-ball, and pay attention to the sphere that it sweeps out, it's sort of like swinging a butterfly net around, instead of just a single jump rope. And because we've "caught" the missing ball in our butterfly net,
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this particular swing cannot be deformed down to a constant swing. So pi_2 successfully detects that this space has a swiss-cheese sort of hole, or "cavern" in it, unlike pi_1 did. To make it easier to see, here is just what the bottom half of the butterfly net is doing; the top half is just the mirror image. So swinging around the ball caught the ball. But if we miss the ball entirely, then as we saw before,
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we can continuously shrink down to a point. If we swing the loop forward and then backward around the ball, we sweep out this funny doubled-up sphere, but we can peel the doubled-up loop back down to a constant as we did before. This one is much easier to understand for me if we just look at the bottom half again. We'll talk about this more later, but it's not a coincidence that this "forward plus backward equals zero" animation, when restricted to the xy-plane,
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is exactly the same as the "forward plus backward equals zero" animation for pi_1(S^1);
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the midpoint of that loop does that lower-dimensional animation in the xy-plane, and the rest of the loop just sort of gets carried along for the ride. This is called the "suspension" of the function, but we'll come back to that later. Another thing that I want to point out here is that if we do two swings of our loop around, then the surface that it swings out definitely has to pass through itself somewhere, and that is allowed. Again, here's just
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the bottom half so to see better. So our butterfly net is magic in such a way that it gets caught on the missing ball, but it doesn't get caught on itself. When swinging a loop around a sphere, there's an extra sort of freedom we have when compared to walking around a circle: we have this extra dimension that we can rotate through. So the standard one-swing-around is homotopic to a rotated swing around. So is there even any way of telling apart a
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forward swing from a reversed swing? How is 1 different from -1?
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Let's add a green color to the top end of the loop, or wherever the loop starts when you plug in numbers close to 0. Then we can see that gradually rotating the whole animation around puts the green at the bottom. This whole time, there's a sort of right-hand rule: if you take your right hand and curl your fingers from the green start of the loop toward the blue rest of the loop, then the loop is moving in the direction of your thumb. Contrast this to the basic
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time-reversed swing animation, where the loop is swinging in the left-hand direction. So forward and backward really are essentially different swings. Another way you might think about it is to say that a fabric net can be continuously twisted around, but you won't ever be able to turn inside out. Or you can still define which side of a basketball hoop gets you the points, even if the hoop gets turned around.
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Since we have all of this freedom to twist around, it's sort of subtle to precisely define our higher-dimensional winding number, which is usually called the degree. And I'm not going to prove that degree is well-defined. But hopefully some analogy here works for you, whether that's jumprope, or weird basketball, or butterfly nets, and hopefully that convinces you that such a number should exist.
26:46
So the most important pi_2 example is pi_2 of a 2-sphere, so we've covered that -- we got the integers. What if we generalize a little bit and have two forbidden balls that we can't pass through? So we have two swiss-cheese sort of holes. If you thinned everything down, you could think about living on two copies of a 2-sphere, butted up against each other. So you should think of this as the higher-dimensional version of the figure eight.
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But for visualization it's nice to have this extra 3D space to move through. So here you could swing your loop around the yellow ball forwards or backwards; you can swing around the red ball forwards or backwards. And you can mix and match and do one combined swing like Yellow-then-Red. If we do Yellow and then Red, this new swing "Yellow-Red" doesn't actually have to stay cinched down in the middle.
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We can sort of "loosen the belt," and deform until our loop sweeps around both balls at once. But now something really interesting can happen. We can rotate this whole animation. So at the start, the loop swings around Yellow-Then-Red, but at the end, the loop will swing around Red-Then-Yellow. You can even cinch the middle back down. It looks like the red and yellow parts got individually rotated upside down,
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but that won't matter -- they still have the same right-handed swings, so we could always individually rotate them back right side up. So in fact, yellow-then-red is homotopic to red-then-yellow. It might be more enlightening to see the surfaces that the loops swing out. I'll make the yellow ball part of the swing yellow and the red ball part of the swing red. So we loosen the belt and grow to one big inflated balloon,
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then rotate the big balloon, then cinch it back down, so now the yellow part is on the red ball and vice versa. Here it is again with partially see-through surfaces. This pi_2 turned out to be commutative: A times B is equal to B times A. So since we group together all of the red swings and all of the yellow swings, we really only need to keep track of those two numbers: pi_2 here is isomorphic to the integers times the integers.
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The little vee symbol between the "S^2"s means to glue the two spheres together at one point. I animated this as being thickened up into 3D space, but remember that that doesn't affect the homotopy groups. This Z times Z is kind of surprising, because if you remember back to the the figure eight (which was S^1 wedge S^1), in pi_1 there was AB which was NOT the same as BA, since a curve can remember the essential history of where it's been.
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But with pi_2, we have this extra dimension of wiggle room which allows A and B to slide past each other. So this 1-dimensional story went from Z to a free group on two letters, while the 2-dimensional story went from Z to Z^2 (also called the free abelian group) on two letters. You might even call S^2 wedge S^2 the "universal example," in the sense that the exact same sort of twisting can make any two swings of loops slide past each other,
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no matter what kind of holes are sitting around. So actually pi_2 of any space is always commutative. pi_1 was just any old group, but pi_2 is specifically an abelian group. You can imagine taking any map from a square and pre-composing with a rotation like this to switch the two pieces around. This rotation argument that pi_2 is commutative is pretty visual, but there's a different version of the proof, called the Eckmann-Hilton argument, that I want to advertise.
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It's so nice that you deserve to work it out for yourself, so I'll leave this on the screen for a few seconds, and you can pause and try it out.
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The solution to part (A) here is really a completely symbolic version of "sliding things past each other," sort of like a fifteen puzzle, and I think it's fun how weak the assumptions are compared to the conclusions. So pi_2 was designed to detect these swiss-cheese-cavern-style holes. Let's see how it acts a dimension lower. If we look at pi_2 of a circle S^1, it's like we're finding ways of shoving a loop of pipecleaner down into a circular pipe and then reeling it back in.
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So if we have one way of shoving, maybe the next day we do the same thing but just shove it in a little less. And then a little less the next day, and keep going, gradually shrinking back to the basepoint. So any loop of loops is homotopic to a constant, and so pi_2(S^1) is the trivial group. Maybe you can buy this fact as is, but the usual framework that topology has for thinking about this is to notice that
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shoving stuff down a circular pipe is a lot like shoving stuff along an infinite straight pipe. And that's because when you track the movement of a point along the circle, you can keep track of how many times it's gone around, so it's just like the winding number, except it's not necessarily an integer. Then instead of thinking about the loop as living in the circle, it can be helpful to instead think about it as living in this "winding space"
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of all real numbers as winding amounts. Pushing things around the circle exactly corresponds to pushing things around the infinite line. This fact is called the "unique homotopy-lifting property," and the line gets called a "covering space" for the circle. So once you buy into this idea of looking at the covering space, it's easier to see how to homotope a loop of loops down to a point: just uniformly scale things down toward zero in this winding space.
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So pi_2 of a circle is trivial really because pi_2 of the real line is trivial. And the same idea will actually show that taking a covering space always gives you the same homotopy groups, except pi_1, which will be different. So pi_2(S^1) was trivial and pi_2(S^2) was Z. Now let's look one dimension higher, at pi_2(S^3). The first hurdle in trying to visualize this is how do we even think about S^3 -- it usually exists as a subset of 4D space, which is hard to think about.
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Well let's start by thinking harder about some of the lower-dimensional stuff. If we wanted to visualize a circle S^1 using only 1-dimensional intuition, we could think of the top and bottom semicircles separately, remembering that the left endpoints of the two segments are really the same point, and same with the right endpoints. If we want to visualize a sphere S^2 using only 2-dimensional intuition, we could think of the top and
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bottom hemispheres separately, remembering that the two circular boundaries really represent the same points, and ought to be zipped together, but they can be artificially separated to fit in 2D space. The same thing works for the 3-sphere S^3: take two solid balls next to each other, and remember that corresponding points on the two surfaces are really the same point.
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So for example, any time you run through the top of the left ball, you come out the top of the right ball. I'll put a white basepoint in the left ball, and I'll mark the center of the right ball in yellow. You can think about the two balls as being the higher dimensional versions of the southern and northern hemispheres, and the south and north poles are these two marked points. Remember our strategy for pi_1 was to push everything away from the
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yellow north pole and back down towards the white south pole. So for any loop, we can always nudge it to make sure it doesn't touch the north pole, and then uniformly shrink it down to the south pole basepoint. The same thing will work for pi_2. Any time we sweep out a surface, we can always nudge it to make sure it doesn't touch the north pole, and then again uniformly shrink it back down to the south pole basepoint. So pi_1(S^3) and pi_2(S^3) are both trivial.
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This is a special case of what's called the cellular approximation theorem, which very roughly says that if you have a continuous map, the 2-dimensional parts of the input space can be homotoped to stay at most 2-dimensional in the output space. So a map S^2 to S^3 doesn't need to touch the 3-dimensional entirety of S^3; it can be deformed to miss some point (like our yellow point), and then pushed away from that point back down to the basepoint.
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More generally, the cellular approximation theorem gives that pi_k(S^nn) is trivial whenever k < n, since then any map S^k to S^n can just be pushed down to the basepoint. So we can go ahead and fill out the bottom half of our table: any time we're "going up in dimensions," there's only one homotopy class because everything is homotopic to a constant. To summarize this chapter on pi_2, we defined the double loopspace as loops of loops in X, but this can also be
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thought of as maps from a square into X where the outside of the square gets sent to the basepoint, or as maps from a 2-sphere where the basepoint gets sent to the basepoint. When we then declare that homotopic things should count as the same, we get pi_2, and we saw that this was always an Abelian group, so everything commutes. We saw pi_2(S^1) was trivial with covering spaces, pi_2(S^2) was Z, and pi_2(larger spheres) was also trivial,
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so pi_2 seems like it might be doing a good job of detecting these "swiss-cheese-style holes," or spheres that aren't filled in. With two copies of the sphere, we get two copies of Z, so that all still makes sense. But since pi_2 couldn't detect whatever was going on with S^3, we'll need to move on to pi_3. The definition of pi_3 is going to involve the triple loopspace, so we're thinking about loops of loops of loops in X,
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everything based at the basepoint. Since a loop of loops is a sphere though, we could also think about this as loops of spheres in X, or as spheres of loops in X. And by very similar "uncurrying" or "basepointed topological uncurrying," we can translate any of these into maps from a cube into X where the outside of the cube gets sent to the basepoint. The quotient topology on the cube modulo the boundary then makes this equivalent to basepointed maps from S^3 into our space.
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So all of these are equivalent definitions of the triple loopspace. For any topologial space X, all of these will be homeomorpic to each other, and if X is a metric space they'll even be isometric. We then want to say a pair of triple loops counts as the same in pi_3 if the triple loops are homotopic, which as usual just means there's a continuous path from one to the other in triple loopspace. By associating stuff around, we can see that pi_3 is
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just pi_2 of the loopspace. And since pi_2 is always abelian, so is pi_3. In other words, in two dimensions there was enough room for things to slide past each other, and that's still going to be true in three dimensions. Okay now what's the best way to get some geometric feeling for the triple loopspace and pi_3? Of all the variations here, I find loops of spheres to be the easiest to think about. We want to think about based loops of based 2-spheres,
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so I'm thinking about ways to take a balloon and swing it through the space X, while keeping the little knot of the of the balloon pinned at the basepoint, and while starting and ending with the whole balloon scrunched up on the basepoint. It feels like it might be hard to get this to "catch" on anything, until you remember that the purpose of pi_3 is to detect "holes" that look like S^3. So if we look at S^3, just as before thinking of two balls with
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corresponding points on the surfaces identified, we can go ahead and swing our balloon around in here, first growing it out to the right 'til it makes an appearance in the second ball, then sliding it across both balls, then shrinking it back to the basepoint from the left. Of course, you can play this animation any number of times in a row, and you can play it backwards. There's a different class of animations for each integer,
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and since adding the integers corresponds to concatenating the animations, we get pi_3(S^3) is Z, the group of integers. So this says for example that one playthrough of the animation cannot be continuously deformed into the constant animation where the balloon stays at the basepoint and does zero sweeps around, since 1 is not equal to 0. This requires some mathematical machinery to prove rigorously, but one way to think about it
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is to count the number of times the yellow dot pierces through the surface, counting with signs so that piercing through backwards counts as negative. Counting with signs makes the number not change under homotopy, so for example if you gradually add little folds to the balloon in this animation, the yellow dot might pierce forward-backward-forward through the balloon, but it'll still have one net forward pierce. There are some hard corner cases making a definition like this rigorous,
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like what if the balloon scrunches up around the yellow dot for a while, but this method of computing the degree with "signed pierces" can be made rigorous using something called local degree. But without going into that here, hopefully you find it plausible that triple loops like this on S^3 are homotopic exactly when they have the same value for this degree integer. Just like pi_2(S^2) came from suspending pi_1(S^1),
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that sort of "carrying loops along for the ride," pi_3(S^3) also comes from suspending pi_2(S^2).
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That'll be easier to see if we put the basepoint on the sphere shared by the two balls. Then you can see that you're swinging loops around a sphere, and the surface bounded by the loop in either ball just comes along for the ride. Okay, so we saw pi_3(S^3) was Z, maybe as expected. For pi_3(S^1), another covering space augment will show that this is trivial -- shoving balloons down a circular pipe is a lot like shoving
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balloons down an infinite straight pipe. And pi_3 of any higher sphere is trivial since we can always push stuff away from some north pole and back down towards the basepoint. If you only saw what we have so far in the table, maybe you'd just predict "Z"s down the diagonal, trivial groups everywhere else. That would be a nice view of the world, where spheres only have holes in exactly the dimensions where you expect. There's a different "theory of holes" called homology,
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where that actually is the case, but homotopy groups are somehow more mysterious than homology. For some bizarre reason, pi_3(S^2) is NOT trivial -- it's Z. Let's try and see what's going on here. We've already met S^2 by thinking about all of 3D space with a ball removed. And we're just getting to know pi_3 as sweeping balloons around, but now it's time for these two characters to meet. It seems like most balloon swings you can think of doing here will just be homotopic to a constant.
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But there is a sneaky way you can make an essentially new balloon swing. First, you'll want to approach the ball, and I'll go ahead and make half of the balloon this yellow wireframe so we can see better. If you don't cover up the whole ball, then you could just push back away from any point left uncovered, and that won't be interesting, so we'll want to engulf the whole ball in balloon. When we do that, the balloon gets two layers -- just like would happen if we
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did this in real life, there's an outer layer that we can see and then an inner layer closer to the forbidden ball. The two layers meet pinched together up above the north pole. Now what? We want to do something nontrivial, and we've already done some up/down sort of movement, so let's try another degree of freedom that we have: spinning. Spinning everything won't really help because that's just spinning our whole world and not really changing anything. But we have these two layers,
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so let's try spinning just one of the layers, say the inner layer. When we give that a twist, if we were using a real rubber balloon, this would put a twist in the little neck up at the north pole. But our magic balloon is allowed to pass through itself, so the twist isn't actually a concern, and we can have exactly the same balloon configuration before and after the spin. Our only requirement is continuity -- that nearby points get mapped to nearby points -- and that's still true here.
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So then we can just do the start of the animation in reverse to retreat back to the basepoint. Altogether, we engulf, then twist, then retreat. Of course we can play this animation multiple times in a row, and we can play it backwards, and maybe it starts to become believable that pi_3(S^2) is Z. Actually, instead of engulf-twist-retreat-engulf-twist-retreat multiple times in a row, a retreat then an engulf is really undoing something and then re-doing it,
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so we can gradually cancel those out to get just one engulf-twist-twist-twist-retreat. So the integer we're counting for pi_3(S^2) is really how many of these inner-layer twists we do. We just have to engulf the ball to be in a position to be able to do a twist like that. Instead of "degree" or "winding number," the integer we count is called the "Hopf invariant," since it was introduced by Heinz Hopf in a paper from 1931. By the way,
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the fact that the animation relies on some kind of spinning is an instance of something called the J-homomorphism, but I won't say much more about that here. Anyway, one engulf-twist-retreat animation like this gives us a map from S^3 to S^2, but it is more traditional to look at a slightly different but homotopic map called the hopf fibration, denoted with this greek letter eta. The easiest way to write this down is to think of S^3 as being normalized pairs of complex
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numbers rather than quadruples of reals, but this doesn't change the shape at all. And we think of S^2 as being the Riemann sphere, which is just the complex plane but where we have one extra point at infinity. Then the hopf fibration is then just taking your two complex numbers and dividing them, where division by zero results in infinity. Something interesting about eta here is that multiplying both of your complex numbers by, say, i, does not change the result,
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since you can just cancel the "i" on the top and bottom of the fraction. In fact, you could do this with any complex number, though we do need to restrict to unit complex numbers if we want to stay in S^3. But by choosing different complex numbers u on the unit circle, we get a whole circle of different points in S^3 that all map to the same point in S^2, and in fact this is all of the collisions that we have, so the preimage under eta of any point in S2 is just a circle like this.
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Niles Johnson has this great animation where you can see exactly which circles of points in S^3 correspond to which points in S^2. And there's actually an interesting way of defining the Hopf invariant using knot theory by counting the linking number of any pair of these circles. But I definitely recommend checking out Professor Johnson's video. It'll be linked below. I want to show what happens if you animate a loop of spheres
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by literally typing in "divide these complex numbers" like this. There are a few extra steps here to make this a viable animation, but the heart of it really is just dividing complex numbers. The resulting animation looks more like this, where instead of engulf, then twist, then retreat, the twisting is smeared out over the whole animation instead of just happening during the middle portion. So the hopf fibration is this really this
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nice smooth, very uniform thing, but before I had just separated two layers and split the animation up into multiple steps to make it easier to explain. To be honest, my main idea for even making this video came from coding up this animation without knowing what it would look like, and then trying to understand what was going on with the result. With that, that's already pi_3 of all the spheres; pi_3(S^3) and pi_3(S^2) were both Z,
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and everything else was trivial. If you want to say homotopy groups measure holes, it's pretty strange that S^2 has this hole in the wrong dimension. You might've noticed the pattern, but to be explicit, we can define the kth homotopy group pi_k using the k-fold loopspace. The kth loopspace is just iterating the loopspace construction k times, which we can write recursively like this. But you might think about it as loops of (k-1)-spheres
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or as maps from a k-cube that send the boundary to the basepoint, or as basepointed maps from a k-sphere. Then pi_k is the k-fold loopspace modulo homotopy, so k-fold loops connected by a path are declared to be the same. pi_k will always be a group, but it'll be an abelian group if k is 2 or greater. Even if you haven't heard of homology, it's interesting to compare two different theories of "holes" provided by homotopy groups (pi_1, pi_2, pi_3, ...) versus
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homology groups (H_1, H_2, H_3, ...). People sometimes say that the homotopy groups are easier to define but hard to compute, while homology is hard to define but easy to compute. So I'm going to skip defining homology here, but you can try to imagine what it might mean. My favorite example of the difference between the two theories is to look at a torus, where pi_2 is trivial but H2 is nontrivial. pi_2 is trivial because of coverings spaces again,
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so shoving loops around on the surface of a donut is just like shoving loops around on an infinite flat plane. But H_2 is nontrivial, and the way I think about this is that pi_2 can detect a sphere-shaped cavern, while H2 can detect caverns bounded by other surfaces, like a donut-shaped hole that a torus has. So a torus has nontrivial homology but trivial homotopy dimension 2. On the other hand, in dimension 3, S^2 has trivial homology, but we saw nontrivial homotopy.
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So this is saying we have an interesting map from S^3 to S^2, but if we were allowed to slice it up into pieces and let the pieces "cancel" with each other somehow, then everything would cancel out. So homotopy groups don't allow slicing stuff into parts, which is maybe what makes them so hard to calculate. A few more examples to think about where homotopy and homology are different: There's the real projective plane RP^2, which has an interesting map from S^2 --
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actually it's a covering space map -- but homology lets you slice that map up into two opposing halves then cancel them out, so H_2 is trivial. Another difference is that pi_1 can be non-commutative like we saw, but H1 is always abelian. In fact H_1 can be described as the abelianization of pi_1. Finally, one last tricky example is a sphere glued to a circle, where homology is just telling you that you have a sphere and you have a circle,
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but pi_2 and pi_3 have infinitely many different copies of Z, which you can think about as coming from another covering space argument. The final thing I want to do in this video is to quickly run through some theorems about homotopy groups for you to chew on. First of all, one reason to care about homotopy groups (other than that they're cool), is Whitehead's theorem. Whitehead basically says that if you can find a continuous function that makes all
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of the homotopy groups of one space line up with the homotopy groups of another, then that's not a coincidence; actually those two spaces are homotopy equivalent So while other invariants like homology can be used to tell you two spaces are not homotopy equivalent if you get different invariants, homotopy groups are unique because they can sometimes affirmatively say that spaces are homotopy equivalent. I should probably point out that being individually isomorphic isn't enough;
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we actually need one continuous function f that makes all of the "pi_k"s line up at the same time. But it's crazy the problem of homotopy equivalence is sort of solved in some sense, if only we knew how to compute with homotopy groups. Next, we have the Hurewicz theorem, which says that the homology groups and homotopy groups are the same, but only once. I mean that the first nontrivial homotopy group is the same as the first nontrivial homology group.
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To make that literally true you do need to assume pi_1 is trivial, or else make some assumptions about pi_1, but that's the basic idea. So there's at least one point of agreement between homology and homotopy groups, and they can become wildly different after that, and that's pretty much all we can hope for. Next, we have the Freudenthal Suspension Theorem, which says that if we keep doing the "suspension" thing where we boost the
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input and output spheres both up a dimension and we just take a loop along for the ride, then in some circumstances this doesn't change the group. On the table, this manifests as having this border with a slope of -1/2, and everything below the border is constant on each diagonal stripe. Actually a lot of homotopy theory tries to just work out these so-called "stable" homotopy groups that the different diagonals approach -- starting with the main diagonal
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we have (Z, 2, 2, 24, 1, 1, …). The numbers here mean the cyclic group of that order so the stable groups are Z (the group of integers), then the cyclic group of order 2, cyclic group of order 2, cyclic group of order order 24, trivial, trivial, and there keep on being more groups after that. We know about 90 of these stable homotopy groups of spheres. And interestingly, we can pretty much study them one prime factor at a time, and we even know enough of what's going on with primes
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7 and bigger that most of the ongoing work relates to the primes 2, 3, and 5. The stable groups can also be used to say thing something about what's called "framed cobordism," so there are nice connections with other parts of topology and geometry. It's much harder to calculate what happens on each diagonal "before" stability -- these are called the "unstable" groups, occurring above this border with slope -1/2.
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We know the stable group for the first 90 diagonals, but peeking back to earlier in those diagonals, we only know the first 20 or so diagonals completely. For some more patterns, we know where all of the "Z"s in this table are: there's the main diagonal, and then this double-knight's move line of "Z"s just above the stable range. Everything else in the table is a finite abelian group. Another pattern to notice is that the rows for S^2 and S^3 seem to match up exactly except for pi_2,
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and this continues to be true to the right. You could think about this as being a consequence of pi_3(S^2) being Z, so S^2 has this secret S^3-shaped hole sort of thing, and so S^2 and S^3 are a lot alike somehow. It also turns out that there are no more trivial groups in these two rows, and this was proved as recently as 2015 by Ivanov, Mikhailov, and Wu. Some more patterns appear if you start looking at prime factors,
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like how the first 2 to appear in this table is in pi_4 of S^2 and S^3. The first multiple of 3 to appear is in pi_6. And The first multiple of 5 to appear is in pi_10. And it'll keep being true that the first time a prime p appears in this table is in pi_2p of S^2 and S^3. It actually even "survives to stability" from pi_2p(S^3),
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meaning everything that follows after it the diagonal will also have that prime p hiding in it. A lot of this stuff was proved by Jean-Pierre Serre in the 1950s. For another pattern, just like how the Hopf fibration was based on dividing complex numbers and it made S^2 and S^3 have similar homotopy groups, there's a variation where dividing quaternions turns out to make S^4 have homotopy groups based on the groups for S^7 and the previous group for S^3, which seems pretty wild.
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You might wonder why something in this table can even have torsion, like why are there all these finite numbers instead of just being "Z"s? One example to think about might be pi_4(S^3), which is the cyclic group of order 2, and it comes from the J-homomorphism: it comes from some sort of "spinning" just like pi_3(S^2) did, but this time we would get to spin all of 3D space instead of just spinning around one axis,
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and the space of all orientations of 3D space has this weird thing where two full rotations cancel out to be trivial, but only one rotation is somehow nontrivial. Physicists would want me to say "spinors" or "spin one-half" or "plate trick" or "belt trick," but however you want to think about it, pi_4(S^3) comes from pi_1 of the 3D rotation group via the J homomorphism. What I just said might have been incomprehensible, but this video is already getting too long,
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so I might make a follow-up video about pi_4(S^3) specifically.
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There are so many weird little patterns to study in this table, and a lot of hard math is needed to do the computations rigorously, but I think it's cool how one of the most basic questions in topology turns out to have this very intricate answer that we still don't even know all of.
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