1/6 CAT(o) Cube Complexes
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WürfelCW-KomplexRechter WinkelSchlussregelMaß <Mathematik>Komplex <Algebra>Produkt <Mathematik>GrenzwertberechnungKugelFormation <Mathematik>Euklidischer RaumVerschlingungKnotenmengeKoordinatenSimplexverfahrenVorlesung/Konferenz
03:46
WürfelRechter WinkelSimplexverfahrenVerschlingungArithmetisches MittelBijektionFahne <Mathematik>GraphSimplizialer KomplexMultigraphKomplex <Algebra>KnotenmengeLoopPaarvergleichGanze FunktionVorlesung/Konferenz
08:02
WürfelGraphVerschlingungKugelSchlussregelDickeFahne <Mathematik>KnotenmengeEbeneAbgeschlossene MengeTeilbarkeitKomplex <Algebra>Simplizialer KomplexFlächentheorieHomöomorphismusCW-KomplexDreiecksfreier GraphMinimumRechter WinkelDimension 2KurveQuadratzahlVorzeichen <Mathematik>DimensionsanalyseMengenlehreGanze FunktionInjektivitätPunktKombinatorikProdukt <Mathematik>ÄhnlichkeitsgeometrieMultigraphSimplexverfahrenKonditionszahlGammafunktionEinbettung <Mathematik>Projektive EbeneDreiHomomorphismusLoopTorusObjekt <Kategorie>IdentifizierbarkeitDifferenteVorlesung/Konferenz
17:52
Fächer <Mathematik>Uniformer RaumMinimumGraphProdukt <Mathematik>Euklidischer RaumMinkowski-MetrikQuadratzahlWürfelRechter WinkelFormation <Mathematik>OktaederKomplex <Algebra>Objekt <Kategorie>MannigfaltigkeitMultiplikationsoperatorUngleichungVollständigkeitDreiecksfreier GraphMultigraphKnotenmengeVerschlingungPunktEindeutigkeitGeodätische LinieAlgebraische StrukturEbeneGruppenoperationDimensionsanalyseGeradeBipartiter GraphKrümmungsmaßLastImmersion <Topologie>MathematikTermMomentenproblemOrdnung <Mathematik>KurveVorzeichen <Mathematik>Vollständiger VerbandVorlesung/Konferenz
27:43
Minkowski-MetrikGammafunktionCoxeter-GruppeRelativitätstheorieFundamentalgruppeRechter WinkelGraphWinkelverteilungProdukt <Mathematik>VertauschungsrelationEinfach zusammenhängender RaumEndlich erzeugte GruppeMultiplikationsoperatorKnotenmengeVorzeichen <Mathematik>VerschlingungGruppenoperationDickePunktQuadratzahlKurveKette <Mathematik>Bipartiter GraphGüte der AnpassungÜberlagerung <Mathematik>Figurierte ZahlVollständigkeitMinimalgradKomplex <Algebra>TopologieWürfelSortierte LogikKreisflächeModulformStandardabweichungDreiecksfreier GraphObjekt <Kategorie>MannigfaltigkeitMultigraphKnoten <Statik>Vorlesung/Konferenz
37:33
WürfelKomplex <Algebra>MengenlehreMultiplikationsoperatorEinsDreiecksfreier GraphIdealer PunktKurveOrientierung <Mathematik>Vorzeichen <Mathematik>VerschlingungGammafunktionProjektive EbeneNumerische MathematikClique <Graphentheorie>Vollständiger GraphVertauschungsrelationRechter WinkelFamilie <Mathematik>KnotenmengeModulformZahlensystemEbeneGraphRandwertPaarvergleichGeradeUnendlichkeitLokales MinimumMomentenproblemQuadratzahlNachbarschaft <Mathematik>GraphfärbungVorlesung/Konferenz
47:23
PunktPaarvergleichFundamentalgruppeDreieckVerschlingungAbstandWürfelMinkowski-MetrikHomotopieKomplex <Algebra>OrtsoperatorRechter WinkelArithmetisches MittelMetrischer RaumProjektive EbeneDreiecksfreier GraphGeometrieEuklidische GeometrieRandwertGeodätische LinieDickeMetrisches SystemKnotenmengeÜberlagerung <Mathematik>SummierbarkeitMereologieLokales MinimumKubischer GraphSchnitt <Mathematik>BeweistheorieQuadratzahlEbeneZusammenhängender GraphKnoten <Statik>TheoremEinbettung <Mathematik>PrimidealKreisflächeGrenzschichtablösungEuklidischer RaumMannigfaltigkeitKurveVorzeichen <Mathematik>MinimumDehn, MaxHyperebeneDifferenzkernUngleichungVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:16
An n-cube is just a copy of product of n intervals.
00:30
So here's a 3-cube over here. And of course, it has sub-cubes.
00:42
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
01:01
often. And a cube complex is a cell complex, or CW complex, built by gluing cubes along sub-cubes.
01:32
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.
01:47
So here's an example. Make it a little bit more interesting.
02:02
Maybe a little 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.
02:24
Now, the link of a 0-cube is what
02:44
I like to call a simplex complex, or the complex formed by gluing simplices together instead of cubes. It corresponds to the epsilon sphere at X.
03:08
If you were imagining that all of these were Euclidean cubes, unit side lengths, and you pretended that they were a metric, then you probably know what I mean. So let's look at that 0-cube right over there
03:23
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.
03:45
And first you see ends of 1-cubes. Those are 0 simplices. And then whenever there's a 2-cube, like over here, for instance, that's a 2-cube, you
04:01
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 will give you a 2-simplex.
04:21
I'll fill it in, even though you know that if you see it, it's there. So that's the link. So the link has an n-1-simplex for each corner of n-cubes
04:52
at x and glued together in the appropriate fashion. And the cube complex X is non-positively curved.
05:11
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.
05:40
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
06:15
an n-simplex if and only if they are pairwise adjacent,
06:34
pairwise connected by edges. So actually, this orange link over here is a flag simplex.
06:46
And I should say, so flag complexes are in one-to-one correspondence
07:03
with simplicial graphs, meaning graphs without loops or bygones. How so? Well, whenever you have a flag complex,
07:22
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,
07:45
well, you guys are probably looking at it and you're imagining, you're probably imagining the entire flag complex because you can't help but complete it and add
08:01
all of the missing simplices. So that makes flag complexes an especially easy type of simplicial complex. OK, so you would like to see a non-non-non-non-non-non
08:20
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.
08:44
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.
09:00
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
09:26
is not non-positively curved. All right? Thank you for that question. So some examples of any graph, not just a simplicial graph,
09:56
but any graph is a non-positively curved cube complex.
10:04
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
10:23
complex, a graph. And well, in the exercises, I ask you to show that any closed surface except the two
10:41
sphere and the projected plane is homeomorphic to a non-positively curved cube complex, a two-dimensional non-positively curved cube complex. So we'll often call those non-positively curved square complexes.
11:01
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
11:21
am suspending the rule that if you think you see a cube, there's a cube. So it really is a graph. And 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
11:43
cube at the bottom. Can you all see the difference between white and orange? 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 flat complexes. So what is your role for the attaching
12:01
map of the CW complex? Here, when you have this loop, the map is not injective. It seems that you didn't specify exactly the rules of attaching things. I never told you that my cubes are
12:22
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, what I suggested instead is that you
12:40
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 long sub-cubes, so in the language of CW complexes, we would call these a type of combinatorial CW complex.
13:06
Don't make it difficult. You guys, a lot of you, are trying to make everything difficult when you're learning things. Really, just like when you take a bunch of edges and glue them together along sub-cubes, glue them together along vertices, you get a graph.
13:22
Right? Likewise, you take a bunch of cubes and then look at certain sub-cubes in 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, well, it doesn't really matter, but not some crazy gluing.
13:41
You should imagine, because I know there are 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 at that homeomorphism on a single face, but not on the entire boundary at once. Yes.
14:00
Yes. The entire attaching map is not, it's not a homeomorphism on the attaching map. We could, we should move forward. Is there anybody else who willfully wants to not understand this?
14:21
Huh? Let's go. I will get to the point where I'll be guilty. I'm not guilty yet. OK, so I'll leave this surface to you. It's in our collection of problems. But of course you're familiar with taking a torus,
14:49
two-dimensional torus, and identifying opposite sides. What you're doing is you're taking one 2-cube and identifying these two 1-cubes, and identifying these two 1-cubes. And of course certain 0-cubes are also going to get identified.
15:01
And 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. OK, well, here's a, just to, I find
15:27
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,
15:53
is non-positively curved if each link of each 0-cube
16:03
is a graph of girth greater than or equal to 4. So the girth of a graph, girth of a graph gamma,
16:21
is equal to the infemal length of closed cycle. Well, I think that, for me, I feel I probably
16:45
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. OK, so maybe one more example.
17:11
The product of non-positively curved cube complexes
17:25
is a non-positively curved cube complex. So there's a bit to do over here, right? 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 in the factors. And then you have to convince yourself that,
17:41
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 0-cubes? And so the reason is going to be that the links of the 0-cubes of the product
18:04
are joins of links. You can look it up on Wikipedia and see what that means. And I didn't put this in the exercises, but this is a good exercise to think about as well.
18:22
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-positively curved.
18:41
Well, each link, and in particular, the link of this vertex over here, the product of these two 0-cubes, is going to be a complete bipartite graph. That's just an example of the join.
19:07
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.
19:20
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
19:42
as 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? In three dimensions, the link of a 0-cube is an octahedron when n is 3. And an octahedron is a beautiful fly complex,
20:02
isn't it? OK. So perhaps now a meteor example.
20:20
All right. OK, so let's see. I want to use this one. I'm going to keep this guy up. Let's see if it can be done. Probably not.
20:46
Maybe a somewhat less artificial example, which leads to many types of examples, a graph of spaces
21:01
where all vertex spaces and edge spaces are graphs,
21:21
and all attaching maps come in a tutorial emergence. So this is a little bit packed,
21:41
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.
22:08
And I like to label things in order to be able to describe what the maps are. And then there's an edge space.
22:32
Well, we like to take the product of this edge space with an interval.
22:51
And you see from the labeling how to glue the left and right sides of this object
23:00
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 this a little bit more interesting.
23:37
And I'll add another edge space.
24:03
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.
24:21
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
24:40
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
25:02
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. So 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.
25:22
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.
25:43
Indira, did you get the exercises? Are they legible? Can you guys read my handwriting so far?
26:03
Is everybody here under 40? You'll be OK. You'll be OK. So I'll give one more example, which I think is interesting, that maybe you'll
26:24
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,
26:44
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.
27:06
Bear with me for a moment.
27:25
Here's the attaching maps in orange. This is going to be a non-positively curved square complex.
27:43
Six squares over here. It's nice because it's a covering map. The two attaching maps are covering maps. Yes? You've got two double-arrowed edges coming from the same vertex. Yeah, I made a mistake. Tell me which vertex. This one over here. Well, OK, this one and also that one.
28:01
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?
28:21
So you can all figure out what the other five squares are. Excuse me? Can you explain a remark about this being a covering map? OK. So the attaching maps, 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.
28:41
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, I'm gluing just 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.
29:01
Those two orange maps are covering maps. They're covering maps of degree two, two-shaded covering maps. Is that what you wanted to hear? You're good. OK. So it turns out this is a very, very strange example.
29:22
It's a 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.
29:41
So right-angled artin groups. The right-angled artin group, these are affectionately known as RAGs, G of gamma, associated to a
30:16
simplicial graph, gamma, has a presentation of a very
30:32
simple form. There's a generator for each vertex, and two generators
30:49
commute if and only if, so this is just a commutator of u and v is a relator. Two generators commute, u v equals v u, if and only if
31:04
the pair u comma v is an edge. So this is a presentation over here. Let me make this a little bit bigger.
31:22
So for example, the right-angled artin group
31:49
associated to that simplicial graph, there are three generators, and any two of them commute.
32:06
And well, the right-angled artin group associated to this simplicial graph, the complete bipartite graph on 2 and 3, there are two generators on the left, and
32:27
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
32:43
exercise, if you like, if you like knots and links, the right-angled artin group of that graph over there, I suppose we could write out the presentation, right? a, b, c, d, a commutes with b, b commutes with c, c
33:09
commutes with d, it actually turns out to be the fundamental group of the 3-manifold that you get from
33:21
the 3-sphere by removing a chain of four circles, which
33:47
is a foreboding example. All right, now the connection between, we're going to hear about artin groups, I am the warm-up talk for Ruth's
34:02
talk on artin groups. So close your eyes, relax, and sleep during the next couple of hours or an hour or so, so that you can be alert for artin groups in general. Why do we care about writing artin groups in this context?
34:28
Well, first of all, when the girth of gamma is greater than or equal to four, then the standard two-complex
34:50
associated, let's call it R of gamma, associated the
35:08
standard two-complex of the presentation for g of gamma is a non-positively curved square complex.
35:26
And that's, I suppose, an exercise you can, well, for instance, this example that I just mentioned right over here, this writing of artin group right over here, it's,
35:47
maybe I'll redraw it, but I'll draw it in a suggestive way where you, instead of thinking of it as a
36:01
presentation written with letters, I'll think of it as a presentation written with a bouquet of circles, summarizing the generator A, B, C, and D. Those are three one-cubes that you're adding, one for each generator, and the one zero-cube that we have over here. And I would put the three relators.
36:23
I might summarize them like this. A commutes with B, B commutes with C, and C
36:43
commutes with D. I always run out of space for my close right angle. So that's the standard two-complex. There's a zero-cube, a one-cube for each generator. Here, the relators are squares. The relators are four-cycles. The relators are words of length four.
37:00
So we can think of each two-cell as being a two-cube. And we glue it all up together. And you'll find that the link, there's only one zero-cube. You'll find that the link looks like this.
37:39
Oh, I've been drawing my links in orange, haven't I?
37:49
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
38:01
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.
38:21
Likewise for the other two squares. Because back then, I was trying to convey where things came from.
38:41
So I'll do that. You're too close to the board. In general, G of gamma, our rag, is equal to pi 1 of a
39:17
non-positively curved cube complex that I'll call R, or
39:27
non-positively curved cube complex, called the Salvetti complex.
39:44
Ruth, who first noticed this? Do you know? For the Rags? I don't know. OK, so it's been known for a long time. It's in my paper with Micah on pi 1. Ruth has it in her paper, but is not willing to take credit for it. OK?
40:07
And the idea is add an n cube for each n-clique in gamma.
40:29
So if you kind of imagine what's happening over here, there was a 0 cube for the one 0-clique, a set of 0
40:42
pairwise-adjacent vertices. There's a 1 cube for each 1-clique. There's a 1 cube for each 1-clique. There's a 2 cube for the three 2-cliques. A clique is a complete graph with a certain number of
41:02
vertices and so forth. If there had been a 3-clique, if there had been a 3-clique, I'll add it like this. If there had been a 3-clique for ABC, then as you will do in the exercises, you will be adding a 3-clique.
41:26
And that 3-clique, 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
41:46
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
42:02
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.
42:25
Perhaps I'm not using this blackboard correctly. Let's see.
43:02
The question is, is that a cell phone, sir? OK, so you can delete that, whoever's recording this, yeah? You have a question?
43:22
So in the example you gave before of just, like, the ABCD with, like, 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
43:47
rethink 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?
44:03
Right? But there are no cycles. So the shortest cycle, the cycles make the cycles as big as you want. So you're good now. Yeah? Great. So the Dane complex, X, of a link projection.
44:23
Or if you'd like a not, if you don't, same thing. Not or 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,
44:49
a very simple not projection. It has two 0-cubes.
45:01
It has a 1-cube for each region. It has a 2-cube for each crossing. OK, so let's talk about that very briefly.
45:23
So you can imagine there's a 0-cube. You're kind of imagining that this is sitting inside of, this not sitting right alongside the plane. There's a 0-cube on top of the plane and there's a 0-cube below the plane. Right? And there are these regions, right? We often like to give it a checkerboard coloring
45:46
so you can see the regions. There's three dark regions and two white regions over here, right? Because this I'm calling white and that I'm calling dark. Right, some of you accepted that. And there's going to be a 1-cube that,
46:03
the 1-cubes are going to travel from the top 0-cube to the bottom 0-cube. Maybe I'll orient them in a moment.
46:30
There's also a region containing the point at infinity. And well, let's decide how to orient these.
46:43
Let's orient the ones that are shaded. Let's orient the 1-cubes that are going through shaded regions downwards and let's orient those passing through non-shaded regions upwards.
47:02
And maybe we will label these. So this is X. This is A, B, C. And maybe this will be X, Y.
47:21
And this is a complex that was introduced by Max Dehn 100 years ago. This complex is going to embed inside of the 3-sphere, or R3 if you like, minus this knot.
47:40
And the fundamental group of the complement is going to be the fundamental group of this Dehn complex that I'm describing. I've only described its 0-cubes and its 1-cubes. I told you that there's a 2-cube for each crossing. And let's think a little bit about how you might imagine
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a crossing. So here's a crossing over here. And maybe I'll call the four regions P, Q, R, S. And, I don't know, let's imagine that this was
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shaded, that these two were shaded over here. And so what I want you to imagine is that there's a path from the 0-cube on top to the 0-cube on the bottom. There's going to be a path that travels through P,
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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.
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Everybody sees this little orange square that I've drawn? And you can all imagine grabbing a hold of these two vertices over here
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and pulling them towards the 0-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 2-cell whose boundary is this 4-cycle. 1, 2, 3, 4.
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And that's certainly a null homotopic path over there. So there'll be such a 2-cell for each of these three crossings. And you can all see that if you do a, y, b, x, for example,
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that's null homotopic. a, y, b, x was one of them. There's two more.
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For this crossing over here, there's c, x, b, y. And which one didn't I do?
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Was it this one or that one? I think it's this one. I'm sorry, you want to do c, y, a, x? Oh, whatever. c, y, a, x.
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And so it turns out this is a non-positively curved 2-complex. The Dane complex is a non-positively curved square complex whenever the link, whenever the notch or link projection
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is prime and alternating. So alternating means that as you're traveling along in this knot,
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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.
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No circle cuts into, cuts the projection in two interesting, meaning not arcs, pieces.
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Where you say a link, you mean like you have multiple components, so you're not. 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 guy, this link or maybe a knot,
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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.
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These dame complexes are natural, showed up a hundred years ago, and you'll hopefully catch a sketch of a proof in the exercises of why this is non-positively curved. I just have a few more minutes.
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Let's talk a bit about geometry. Let me ask you, were cat0 spaces defined yet?
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Okay, so then I'll do that. A geodesic metric space X is a cat0 space if its geodesic triangles are at least as thin
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as comparison triangles in the Euclidean plane.
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So what that means is, so a 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 cat0 means that anytime you choose three points
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and you choose geodesics between them, you then, this is inside of your space, maybe I'll call it X tilde now,
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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 three points, P bar, Q bar, and R bar.
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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 length as the triangle delta. And it's an exercise that this can always be done.
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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,
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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
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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.
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Okay? So, well, non-positively curved cube complexes were introduced by Gromov as examples,
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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
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is called a cat-zero cube complex. And, in fact, it's a theorem.
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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. X tilde has a cat-zero metric. So it's entitled to be called a cat-zero cube complex
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because it has a cat-zero metric. And not only that, where each n-cube is isometric to the Euclidean n-cube.
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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.
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But for cube complexes, things work out really very nicely. The geodesic joining two points, well, you could look at all possible, you consider all possible piecewise parts of cubes, paths.
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There are so many. And you find the one whose sum of lengths in these Euclidean metrics is minimal.
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And that's the geodesic. So geodesics between two points P and Q in this cat-zero metric is the piecewise cubicle, if you allow me, piecewise cubicle.
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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.
01:00:02
and are unique. And it's an interesting exercise. A consequence of this cat-zero inequality is that geodesics between points are unique.
01:00:24
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,
01:00:45
in this sense, this is an idea from Riemannian manifolds 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
01:01:05
and will involve other things called hyperplanes that we're going to talk about soon. So I'd like to stop now.