 An n-cube is just a copy of product of n-intervals. So here's a 3-cube over here. And of course, it has sub-cubes. You just restrict some of the coordinates. So there's six 2-cubes that are sub-cubes of this 3-cube. And there's a 1-cube, which is a sub-cube, and a 0-cube. Of course, we call 0-cubes and 1-cubes vertices and edges often. And a cube complex is a cell complex, CW complex, built by gluing cubes along sub-cubes. It's a very combinatorial thing. And you should be imagining that all of the gluing maps are modeled by isometries, if you'd like. So here's an example. Make it a little bit more interesting. Something hanging off. The rules over here for this talk. Here's a cube complex, X. The rules are that if you think you see a cube, there's a cube. Now the link of a 0-cube is what I like to call a simplex complex, the complex formed by gluing simplicies together instead of cubes. It corresponds to the epsilon sphere at X, if you were imagining that all of these were Euclidean cubes unit side lengths and you pretended that there were a metric, then you probably know what I mean. So let's look at that 0-cube right over there and try to make some sense out of this. So you imagine the epsilon sphere over here. And what you would first be seeing are put your eye at that 0-cube and look out. And first you see ends of 1-cubes. Those are 0-simplicies. And then whenever there's a 2-cube, like over here, for instance, that's a 2-cube, you will see its corner, which is going to be a 1-simplex. Here's another. And when you see a 3-cube, when a 3-cube whose corner is at that 0-cube, we'll give you a 2-simplex. I'll fill it in even though you know that if you see it, it's there. So that's the link. The link has an n minus 1-simplex for each corner of n-cube at x and glued together in the appropriate fashion. And the cube-complex x is non-positively curved. We'll always be writing non-positively curved. The cube-complex x is non-positively curved if each link of each 0-cube is a flag-complex. For each 0-cube, its link is a flag-complex. So what is a flag-complex? It is a simplicial complex such that n plus 1 vertices span an n-simplex if and only if they are pairwise adjacent, pairwise connected by edges. So actually, this orange link over here is a flag-complex. And I should say, so flag-complexes are in one-to-one correspondence with simplicial graphs, meaning graphs without loops or bygones. How so? Well, whenever you have a flag-complex, you could just look at its one-skeleton and you'll get a simplicial graph. But more interestingly, whenever you have a simplicial graph, you just see a simplex whenever you see its one-skeleton. That's the flag-complex that's associated to it. So if I were to just draw some simplicial graph, well, you guys are probably looking at it and you're imagining the entire flag-complex because you can't help but complete it and add all of the missing simplices. So that makes flag-complexes a specially easy type of simplicial complex. So can you show a non-positivity curve? You would like to see a non-positively curved cube complex. So let's unravel the definitions over here. We want to find a cube complex that fails to satisfy the condition that each of its links, each of the links of its zero-cubes, is a flag-complex. So I told you the rule. If you think you see a cube, there's a cube there. But let's break the rule right now. And now we only see the two-skeleton of this three-cube. So it's homeomorphic to a sphere. What does the link of this zero-cube right up over here in the front look like? Well, it looks like this. But there's no two-simplex there because there's no corner of a three-cube there. So this link over here of this sphere broken up into squares is not non-positively curved. All right? Thank you for that question. So some examples of any graph, not just a simplicial graph, but any graph is a non-positively curved cube complex. You're all getting used to an NPCCC, non-positively curved cube complex. Because what do the links look like? Well, they're not very interesting. They're a disjoint set of vertices that aren't with no edges. When you look at the link of a zero-cube, of a one-dimensional cube complex of graph. And now I'll, well, in the exercises, I ask you to show that any closed surface, except the two-sphere and the projective plane, is homeomorphic to a non-positively curved cube complex, a two-dimensional non-positively curved cube complex. We'll often call those non-positively curved square complexes. Excuse me. Is there any graph or maybe any tree? Because like a fourth cycle is not an. So any graph, OK, so let me, yes, I am suspending the rule that if you think you see a cube, there's a cube. So it really is a graph. And let's suspend. It's not even a simplicial graph. And your concerns, you've already set them aside, but I'm just going to draw the link of that zero-cube at the bottom. Can you all see the difference between white and orange? Yeah? So right in there, you're looking at it, and you're saying, yeah, all of the links are edgeless graphs. So they're all going to be flag complexes. OK? So what is your role for the attaching map of this CW complex here when you have this loop? The map is not injective. It seems that you were not, didn't specify exactly that the rules of attaching things. I never told you that my cubes are going to get attached, that the attaching maps of cubes were embeddings. And I really didn't mean that. And there are rules of attaching maps. Yes, it's true. And what I suggested instead to avoid making a huge mess, OK, what I suggested instead is that you imagine taking a collection of cubes and gluing them together along sub-cubes. And I think that's clear enough. So on every cube in the boundary, the map is injective? The attaching map is injective? Right, because that's correct. When I glue together along sub-cubes, so in the language of CW complexes, we would call these a type of combinatorial CW complex. The don't make it difficult, you guys, a lot of you are trying to make everything difficult when you're learning things. Really, we take a bunch of edges and glue them together along sub-cubes, glue them together along vertices. You get a graph. Likewise, you take a bunch of cubes and then look at certain sub-cubes and these cubes and identify sub-cubes. Of course, those are sub-cubes of the same dimension and start gluing them together. All those gluings are going to be not some crazy gluing. You should imagine, because I know they're uncountably many different ways to glue two cubes together by a homeomorphism. But I mean that one, OK? So shall I continue? I think the confusion is it's injective is that homeomorphism on a single face, but not on the entire boundary at once. Yes, yes. The entire attaching map is not a homeomorphism on the attaching map. We could, we should move forward. Is there anybody else who willfully wants to not understand this? Let's go. I will get to the point where I'll be guilty. I'm not guilty yet. So I'll leave this surface to you. It's in our collection of problems. But of course, you're familiar with taking a two-dimensional torus and identifying opposite sides. What you're doing is you're taking one two cube and identifying these two one cubes and identifying these two one cubes. And of course, certain zero cubes are also going to get identified. That's an example of a cube complex. And it's non-possibly curved. So you'll do something similar for arbitrary closed surfaces. Maybe you'll subdivide. Maybe you'll subdivide. So all right, well, I find that it's very useful to stick to two-dimensional examples, because it's already a rich enough world, a two-dimensional cube complex, or square complex, is non-positably curved if each link of each zero cube is a graph of girth greater than or equal to 4. So the girth of a graph gamma is equal to the infimol length of closed cycle. Well, I think that for me, I feel I probably could have spent the last 20 years just thinking about two-dimensional square complexes. And that would have been enough to keep me busy. I'm still thinking about them, actually. So maybe one more example. The product of non-positably curved cube complexes is a non-positably curved cube complex. So there's a bit to do over here, because you have to think about the product of two cube complexes. What are the cubes going to be in the product? They're going to be products of cubes and factors. And then you have to convince yourself that, yes, if you could form it by gluing cubes together along sub-cubes, that's OK. So the key is going to be, what are the links of the zero cubes? And so the reason is going to be that the links of the zero cubes of the product are joints of links. You can look it up on Wikipedia, see what that means. And I didn't put this in the exercises, but this is a good exercise to think about as well. So you might think, in particular, about the product of two graphs. So here's a three-valent vertex in a graph. And here is a four-valent vertex in a graph. And the product of two graphs is non-positably curved. Well, each link, and in particular, the link of this vertex over here, the product of these two zero cubes, is going to be a complete bipartite graph. That's just an example of the join. So that's the link of these. And you can kind of see that these three vertices over here, the link of this, correspond to these three points over there and likewise for the other. All right. Maybe, well, I guess you can push this a little further. And you know this example very well. Euclidean n-space, it's a product of copies of Euclidean 1-space. And Euclidean 1-space is just a graph. And we would think of Euclidean 1-space is just a subdivided line, of course. So this is a very nice cube complex over here. It's fun to think, what are the links, what are they even called in higher dimensions, right? In three dimensions, the link of a zero cube is an octahedron. One n is three. And an octahedron is a beautiful fly complex, isn't it? OK, so perhaps now a meteor example. All right, OK, so let's see. I want to use this one. I want to keep this guy up. Let's see if it can be done. Maybe a somewhat less artificial example, which leads to many types of examples, a graph of spaces where all vertex spaces and edge spaces are graphs and all attaching maps combinatorial immersions. So this is a little bit packed, especially if you haven't heard the term graph of spaces, but you catch on very quickly. So what's happening over here is that there's a kind of graphical structure to a certain space that I'm about to depict. It has some vertex spaces, maybe just two. And I like to label things in order to be able to describe what the maps are. And then there's an edge space. Well, we like to take the product of this edge space with an interval, and you see from the labeling how to glue the left and right sides of this object right over here. And well, if you take a product of this four cycle with an interval, then it's broken up into four squares right over there. Perhaps I'll make it this a little bit more interesting, and I'll add another edge space. And let me decide how I'm going to map it like that. And here you have, well, there's also four squares hidden over here. This is a graph with four edges crossed with an interval. Now that's four squares. And I've glued its top and the bottom using the way the labeling tells you what the map is. And so the map is just saying which one cubes are going to get glued to which one cubes. And well, all in all, when you form this object, which is kind of eight squares, but it's been organized so that we can kind of take it all in very quickly, this object, it turns out to be a non-positively curved cube complex. So what's critical over here is it uses that the attaching maps, which are these maps from the edge spaces to the vertex spaces, it uses that the attaching maps are combinatorial immersions. And if that's the case, you get a non-positively curved cube complex over here. Why is it called a graph of spaces? So you're looking at this and you kind of see a graphical structure. What graph is this looking like? Well, I prefer to draw it right underneath, but I'm not going to draw it alongside. It looks like there's two vertex spaces. There's an edge, and there's another edge. And if you've heard of the notion of graph of groups, then this is an allied notion. In the exercises, I will ask you to consider an example of this type. Indira, did you get the exercises? Are they legible? Can you guys read my handwriting so far? Is everybody here under 40? You'll be OK. So I'll give one more example, which I think is interesting, that maybe you'll spend some time thinking about. And one nice thing about this subject is that there are loads and loads of examples that are accessible. But there are also, of course, many, many examples, as we'll see near the end, where you're just going to have to grope in the dark to figure out what's going on. That's the way math is. Bear with me for a moment. Here's a, which are the attaching maps in orange. This is going to be a non-positively-curved square complex, six squares over here. It's nice because it's a covering map. The two attaching maps are covering maps. Yes? You've got to do a little error in just coming from the super. I made a mistake. Tell me which vertex. This one over here. So OK, this one and also that one. So thank you. Here, let me just draw just to make sure everybody knows where I am. So this square over here, let me draw it right over here. Everybody agree to that? So you can all figure out what the other five squares are. OK? Excuse me? Yeah, can you just play the extended remark about this being a covering map? So the attaching maps, so the edge space is kind of the cross-section of this object here. So it's just this. This graph with six edges and two vertices. And notice that the map from this edge space to this vertex space, which is the way in which I'm gluing the edge space to the vertex space, right? I'm gluing just sort of this end of the edge space to the vertex space, and I'm gluing this end of the edge space to the vertex space by identifying every point with its image. Those two orange maps are covering maps. They're two-sheeted. They're covering maps of degree two. Two-sheeted covering maps. Is that what you wanted to hear? You're good. OK, so it turns out this is a very, very strange example. It's universal cover. When you think about it, you'll find that it's the product of two trees. It's isomorphic to the product of two trees. I need to give. And I left something for you to think about over there, or maybe really something for you to admire over there in the exercises. So right-angled artan groups, the right-angled artan group, these are affectionately known as rags, g of gamma, associated to a simplicial graph, gamma, has a presentation of a very simple form. There's a generator for each vertex, and two generators commute if and only if. So this is just the commutator of U and V is a relator. Two generators commute U V equals V U if and only if the pair U comma V is an edge. So this is a presentation over here. Let me make this a little bit bigger. So for example, the right-angled artan group associated to that simplicial graph, there are three generators, and any two of them commute. And well, the right-angled artan group associated to this simplicial graph, the complete bipartite graph on two and three, there are two generators on the left and three generators on the right. And every generator on the left commutes with every generator on the right. So it's F2 times F3. And then a favorite example, which you can take as an exercise if you like, if you like knots and links, the right-angled artan group of that graph over there, I suppose we could write out the presentation. A, B, C, D, A commutes with B, B commutes with C, C commutes with D. It actually turns out to be the fundamental group of the three manifold that you get from the three sphere by removing a chain of four circles, which is a foreboding example. All right. Now the connection between we're going to hear about artan groups. I am the warm-up talk for Ruth's talk on artan groups. So sort of close your eyes and relax and sleep during the next couple of hours or an hour and so on so that you can be alert for artan groups in general. Why do we care about writing artan groups in this context? Well, first of all, when the girth of gamma is greater than or equal to 4, then the standard 2 complex associated, let's call it R of gamma, associated the standard 2 complex of the presentation for g of gamma is a non-positively curved square complex. And that's, I suppose, an exercise you can, well, for instance, this example that I just mentioned right over here, this writing of artan group right over here, it's maybe I'll redraw it, but I'll draw it in a suggestive way where you, instead of thinking of it as a presentation written with letters, I'll think of it as a presentation written with a bouquet of circles, summarizing the A, the generator, A, B, C, and D. Those are 3 1 cubes that you're adding, 1 for each generator, and the 1 0 cube that we have over here. And I would put the three relators. I might summarize them like this. A commutes with B. B commutes with C. And C commutes with D. I always run out of space from my close right angle. So that's the standard 2 complex. There's a 0 cube, a 1 cube for each generator. Here the relators are squares. The relators are 4 cycles. The relators are words of length 4. So we can think of each 2 cell as being a 2 cube. And we glue it all up together. And you'll find that the link is only 1 0 cube. You'll find that the link looks like this. Oh, I've been drawing my links in orange, haven't I? And you'll find from the first square, you'll get this corner right over here. I like to use the notation outgoing A, outgoing B. That was this right over there. You'll have incoming A to incoming B right over there. And then you'll also have these corresponding to the two sides. And likewise for the other two. Likewise for the other two squares. Why is the 1 nothing? Because back then I was trying to convey where things came from. So I'll do that. You're too close to the board. OK, so in general, G of gamma, our rag, is equal to pi 1 of a non-positively curved cube complex that I'll call R, or R of gamma over here, where R is a non-positively curved cube complex called the Salvedi complex. Ruth, who first noticed this? Do you know? For the road rags, I don't know. It's well known for a long time. It's in my paper with Mike on pi 9k pi 1. Ruth has it in her paper, but is now willing to take credit for it. OK? Yeah. Yeah, so OK. And the idea is add an n cube for each n-click. In gamma. So if you imagine what's happening over here, there was a 0 cube for the 1 0-click, a set of 0 pairwise adjacent vertices. There's a 1 cube for each 1-click. There's a 1 cube for each 1-click. There's a 2 cube for the 3 2-clicks. A click is a complete graph with a certain number of vertices. And so forth. If there had been a 3-cube, if there had been a 3-click, I'll add it like this. If there had been a 3-click for A, B, C, then as you will do in the exercises, you will be adding a 3-cube. And that 3-cube, because you've made two more, there'd be another 2-cube, of course, A and C commute. And then you'd be adding a 3-cube in the form of a 3-torus, et cetera. And what you're doing is you're taking that 3-cube, and you're identifying its 2-cubes, which are on its boundary. You're identifying them with the 2-cubes that were already there. That's all. And it turns out this ends up being non-positively curved. It's not very difficult to see. You're going to think about this in the exercises, I hope. And compute the link and kind of get a sense of what's going on there. Let me describe one more example. Perhaps I'm not using this blackboard correctly. Let's see which version it is. I've got a cell phone, sir. So you can delete that whoever's recording this, yeah? You have a question? Yeah. So in the example you gave before of just like the ABCD with Speak up. Before you threw those dotted lines, the girth of that graph was less than 4, right? The girth of this graph over here, that's the graph that you're referring to, is infinity. Because the scariest word here, I have to stop and think what's going on every time I use it, the infimum. And it's the infimum of the empty set which geniuses decided was infinity, yeah? OK, and if you're confused about it, yeah? But there's no cycles. So the shortest cycle, you make the cycles as big as you want. So you're good now, yeah? Great. So the Dane complex X of a link projection, if you'd like, a not, if you don't, same thing, not our link projection. So I'm giving you one more family of examples of non-positively curved cube complexes. So here is some link, a very simple link, very simple not projection. It has two zero cubes. It has a one cube for each region. It has a two cube for each crossing. So let's talk about that very briefly. So you can imagine there's a zero cube. You're kind of imagining that this is sitting right alongside the plane. There's a zero cube on top of the plane, and there's a zero cube below the plane. And there are these regions. We often like to give it a checkerboard colorings. You can see the regions. There's three dark regions and two white regions over here. Because this I'm calling white and that I'm calling dark. Some of you accepted that. And there's going to be a one cube that, the one cubes are going to travel from the top zero cube to the bottom zero cube. Maybe I'll orient them in a moment. There's also a region containing the point at infinity. And well, let's decide how to orient these. Let's orient the ones that are shaded. Let's orient the one cubes that are going through shaded regions downwards. And let's orient those passing through non-shaded regions upwards. And maybe we will label these. So this is x. This is a, b, c. And maybe this will be x, y. OK. And this is a complex that was introduced by Max Dain 100 years ago. And this complex is going to embed inside of the three sphere, or R3 if you'd like, minus this knot. And the fundamental group of the complement is going to be the fundamental group of this Dain complex that I'm describing now. I've only described it's zero cubes and it's one cubes. I told you that there is a two cube for each crossing. And let's think a little bit about how you might imagine a crossing. So here's a crossing over here. And maybe I'll call it the four regions P, Q, R, S. And I don't know, let's imagine that these two were shaded over here. And so what I want you to imagine is that there's a path from the zero cube on top to the zero cube on the bottom. There's going to be a path that travels through P. Then it goes through Q. Then it goes through R. And then it goes through S. Those are the names of these four regions over here. Actually, let me do it like this. Everybody sees this little orange square that I've drawn. And you can all imagine grabbing a hold of these two vertices over here and pulling them towards the zero cube on the top. And your friend on the other side of the blackboard grabs these two vertices over here that you can't see and pulls them towards the bottom vertex. And in that way, you'll get a two cell whose boundary is this four cycle, 1, 2, 3, 4. And that's certainly a null homotopic path over there. So there'll be such a two cell for each of these three crossings. And you can all see that if you do A, A, Y, B, X, for example, that's null homotopic. A, Y, B, X was one of them. There's two more. For this crossing over here, there's C, X, B, Y. And which one didn't I do? Was it this one or that one? I think it's this one. A, I'm sorry. You want to do C, Y, A, X. Oh, whatever. C, Y, A, X. And so it turns out this is a non-positively curved two complex. The Dane complex is a non-positively curved square complex whenever the link, whenever the knot or link projection is prime and alternating. So alternating means that you go, as you're traveling along in this knot, you go under, over, under, over, under, over. That's what alternating means. And prime means that you can't find some prime means that there is no interesting way of cutting it. No circle cuts into, cuts the projection in two interesting, meaning not arcs, pieces. Where you say a link, you mean like you have multiple components to your knot. That's right. A link means an embedding of several circles. And there's a lot that's floating around in the air over here. I mean, what are these projections? You have this link or maybe a knot. And you're picking a plane, and you're imagining pressing it down on the plane. But we're leaving these little gaps over here to show you where it was going over, where it was going under. And yeah, you can always project it some way. And these turn out to be kind of fascinating, also foreboding examples to what will be for three manifolds. These Dane complexes are natural, showed up 100 years ago. And you'll hopefully catch a sketch of a proof in the exercises of why this is non-positively curved. So I just have a few more minutes. Let's talk a bit about geometry. Let me ask you, were cat zero spaces defined yet? OK, so now I'll do that. A geodesic metric space x is a cat zero space if its geodesic triangles are at least as thin as comparison triangles in the Euclidean plane. So what that means is whenever, so geodesic metric space means that for any two points, there is a geodesic in the space, an isometrically embedded interval in the space that connects those two points. To be cat zero means that any time you choose three points and you choose geodesics between them, this is inside of your space. Maybe I'll call it x tilde now, x tilde. What you do is you can find, it's always the case that you can find what's called a comparison triangle inside the Euclidean plane. And a comparison triangle is a triangle, let's call it's three points, p bar, q bar, and r bar. And the distance between p and q and p bar and q bar are the same. So it's a triangle whose side lengths are exactly the same. The comparison triangle, delta bar, has the same side lengths as the triangle delta. And it's an exercise that this can always be done. And to say that this triangle is less than or equal to, it's at least as thin as that triangle means that whenever you choose two points over here, maybe a and b, and you look at the comparison points, a bar and b bar, meaning points in the exact same positions on the sides, then what you want is that this distance over here between a and b, the distance in x tilde between a and b, you want that to be less than or equal to the distance between the comparison points. So what we want is that the distance in x tilde between a and b is less than or equal to the distance in the Euclidean plane between the comparison points. And to be a cat zero space means that this holds for any geodesic triangle in your space. OK? So well, non-positively curved cube complexes were introduced by Gromov as examples, examples of examples that allow us to produce cat zero spaces. So let me say like this, a non-positively curved, a simply connected non-positively curved cube complex is called a cat zero cube complex. And in fact, it's a theorem. Let's call it x tilde because it's simply connected. Probably it was the universal cover of a non-positively curved cube complex. It's probably how you got it. Maybe not though, we'll see. A x tilde has a cat zero metric. So it's entitled to be called a cat zero cube complex because it has a cat zero metric. And not only that, where each n cube is isometric to the Euclidean n cube. And the metric is not so terribly bad. It's actually extremely simple. And this is really part of a much larger subject. These are just examples of cat zero complexes in general. But for cube complexes, things work out really very nicely. The geodesic joining two points, well, you consider all possible piecewise parts of cubes, paths. There are so many. And you find the one whose sum of lengths in these Euclidean metrics is minimal. And that's the geodesic. So geodesics between two points p and q in this cat zero metric is a piecewise cubicle, if you allow me. Piecewise cubicle. So this is a little part of a little geodesic in a cube. Piecewise cubicle paths of minimal length. And they exist. They exist and are unique. And it's an interesting exercise. A consequence of this cat zero inequality is that geodesics between points are unique. OK, so the truth is that even though these are called cat zero cube complexes, and I'm talking about cat zero cube complexes, which have a metric of non-positive curvature in this sense. This is an idea from Romani and Manifold, really. The truth is that this metric is not going to be so important to us. And the viewpoint that I'm going to adopt and that I will sort of try to transmit to you is much more combinatorial. And will involve other things called hyperplanes that we're going to talk about soon. So let's stop now.