 Okay, so I guess we'll go ahead and get started. So quick intro. Just want to welcome people to the Graduate Student Seminar Series for doing this summer. I want to say a couple of notes just to prep this thing, right, just despite having a graduate student in the title of the seminar series attendance is pretty open to everyone and people's backgrounds can vary a lot so please just be respectful to your friends and colleagues. With that being said, please do also ask questions. This is, you know, the primary goal of this seminar is for it to be a positive learning experience for everybody in the spirit of a usual seminar if you have a question or you would like clarification or anything. It's maybe just most preferable to unmute yourself and politely ask out loud. It's maybe the easiest way to go about things. You can also just type a question in chat and I will keep an eye on it and I'll relate to the speaker. And also there will be a little bit of time at the end of the talk. If you have questions that might be longer or more involved. All right. So, just double check. We're all set. Okay, so with that being said, I'm very pleasure to have Mike Meyer, you tell us about your splitting and trisections. Yeah, so it's excellent. I'm Nick Meyer. I am a grad student at the University of Nebraska Lincoln. I'm advised by Alex a zoo pan. I just finished my 30 years and went in my fourth year. I kind of study low dimensional geometry topology, especially like combinatorial decompositions. So these are kind of my bread and butter. So today I'm going to be talking about hair splitting and trisections, which are my two primary tools of understanding three and four. So a little bit about this title slide. So this is what's called a trisection diagram. And we'll learn a lot about them later on in this talk. So this is a special one, and this is a trisection of the force fear. And hopefully by the end of this talk, we'll be able to realize how to see this. Okay. Got us growing this way. So some background for the sake of this talk, all manifolds are going to be assumed to be compact orientable. We won't assume them to be closed. These smooth and all maps are going to be proper, which just means preimage of compact subsets of the compact. So we're going to define an n dimensional K handle body to be a copy of the boundary connect some of G copies of SK cross the n minus cable. We're going to define that Sigma G with a little underline below it to be the genius G surface. So that's a zero is the two sphere. So we draw what an n dimensional K handle body is. So the page. And the n equals three case so a three dimensional one handle body looks like a solid version of the genius G surface. So for example a three dimensional one handle body with three handles. So I have three handles looks like the way I like to think about it is I have a sphere, and then I attach handles to it, and everything is solid. So that's what a three dimensional one handle body looks like is it looks like a sphere, and we attach some solid pool noodles to it. And because I use three dimensional one handle bodies and four dimensional one handle bodies so much. We have special locations for them. So each upper G will be the genius G three dimensional one handle body, and Z upper G will be the genius, the dimension for genius one handle body, genius G one handle body, lots of numbers and indices going on here. So I'm going to try to draw what Z sub G looks like the way to think about it is look at this picture and scream to yourself, this should be four dimensional. That's really the only way I can visualize four dimensional spaces by thinking it as three dimensions and convincing myself that it should be what it is in four dimensions. Yeah, go for it. What is the difference between the boundary connects some and I guess like the the usual connects some. Yeah, great question. Okay, so I'm going to let M and N be compact. I'm going to say N manifolds with one boundary component. If they have more than one boundary component each, you run into an issue of well which boundary component, are you working with. So for sake of this definition might help one. I'm going to say M boundary connects some and is the end manifold obtained by the following procedure. Okay, so let be and be prime be and balls in boundary and boundary and that what I want. So we're going to take spheres in the boundary. So if you want to think of this in the three dimensional case. So take a three dimensional one handle body to solid or picking out a circle in its boundary. Really, this is a disk. Really, we're looking at a boundary parallel desk. And we're looking at another thing. So maybe we have a genius to three dimensional handle body, and I pick out a circle slash this here. And then what I do is I just identify these two discs. So you can think of it as I'm attaching a to. So we're doing is we're gluing in. We're gluing in a B and minus one cross I identifying B and one cross plus or minus one with this or really the balls. We don't want these to be animals. The idea here is that we're, it looks like the connection in the boundary. So the takeaway here is that the boundary of M connects some and is equal to boundary of connect some boundary. And then away from the boundary, it looks somewhat like a disjoint. The interior. Naturally is contained in the interior. Sorry, this should go the other way. The idea is is on the level of boundaries it's just a connect some, but you're gluing things with a solid tube instead of a hollow tube. It's kind of the manifold with boundary version of a connection. So here are some more three dimensional handle bodies. Again four dimensions hard. Imagine the 3D picture and think for really loudly at yourself. This is something my advisor told me to do and it's what I do. And it mostly works, except when it doesn't. So in general, three manifolds and four manifolds are hard to picture, because we can't imagine them with our tiny three dimensional brains. We're trying to imagine what a lens space looks like or what is the Poincare dodecahedron sphere to look like. That's, that's hard. So in 1898 will he guard introduce this notion of a guard splittings, which is a nicer way of thinking of the three manifold by reducing it to handle bodies and surfaces, because surfaces we can understand for the most part. So here are the ingredients. So M is going to be a closed orientable three manifold. You can also do this for non orientable three manifolds. But it's a pain in the butt. And you can also do it for non closed three manifolds so three manifolds with boundary, but then instead of using handle bodies you have to use something called a compression body, which I just don't want to get into in this talk. Each one of them is a decomposition of M into two handle bodies, each one in each two, such as the following two conditions hold. So first, each one of each two are both copies of the genus G three dimensional one handle body. And notably, it's the same G for each, each one and each two. One is that these two handle bodies intersect only at their boundary, and their boundary is the entirety of this genus G surface. So if we write M is equal to each one union over Sigma H two. Then we say that M has a genius G Hey guards. And this little notation means that. We can write this as color. We can write this as M is each one union each two, and each one intersect each two is equal to Sigma. So that's what that little notation means in this talk. Normally of course it means there's an adjunction. And that's technically true is just a trivial adjunction, because we're identifying everything with the identity map. So I'm going to give a pictorial definition of what a hangar splitting looks like before I give actual examples. So, if this is all of them, right, ideally this would be without boundary but yeah. And everything here is going to be one dimension down. So a surface will turn into a line segment, and three dimensional pieces will turn into two dimensional pieces. So I'm going to have some surface in the middle, and then everything above it will be one of my handle bodies and everything below it will be the other handle body. And this is kind of the plural definition of what a Hagar decomposition looks like is we just have some surface that divides M into two bodies that meet nicely. So one example of this is, is that we can look at the topology of Sigma, and look at how each two and each one interact, and that tells us pretty much everything we would want to know about that. If everything was nice in the world. So let's look at some examples. The start is always the end sphere, where it's kind of our trivial object in the category of closed for manifolds or closed and manifolds here and equals three. So we can view as three as the union of two four dimensional one handle bodies, three dimensional one handle bodies. So these are both copies of B three, and we're uniting them across a copy of S2. And how do we see this. Well, we can view S3 in as the one point compactification of our three. And we'll let Sigma zero BS to which is right naturally living in three space. Okay, so each zero for each one here will just be the regular unit ball in three space, and each two will be the complement of each one. So here's a way I like to think of it. So here's three space, then there's the point at infinity. I look at my unit ball. And then this splits everything into outside and inside. Right this is the Alexander theorem that every sphere in three space splits. All of space. All of space is my each two and my red stuff is my each one. We can do this in coordinates, and we'll return to this example later. So this is the idea of a Morse function, and we'll come back to this in a little bit. We'll take a look at another example. So now we can look at a genius one hangar splitting of S3. And we can do this in the following way. So here's your Taurus in three space, preferably it would be unnommed, which means that the complement of your Taurus will be a handle body. So just a regular standardly embedded Taurus, such as the picture. And on the inside of the Taurus, try to draw this. We have a disk, right this whole thing is filled up. So that's the handle body on the inside. So my red stuff is each one. My blue stuff, which is everything outside. Why that disk is there. My blue stuff which is outside, I have the single disk generator out here, and then it kind of goes out to infinity. And this kind of looks like, if you've ever seen the hop vibration. These are kind of your two orbit types here, right you have your external orbits which go transverse, and then you have your stable orbits, which are kind of orthogonal to those other orbits. So the inside part here is a one handle body, and the outside, while harder to see is also a one handle body, and here is kind of the handle that goes through. And then everything else is right a three ball. So that's a three dimensional picture. This has been known since the 1890s. The four dimensional picture wasn't as clear until recently. So in 2012 or 2015, depending on if you want to use archive dates or publication dates. They and Kirby introduced a, what's called trisections of four manifold. I'm also including references to the Meyer Schirmer and Zupan paper, because they kind of expanded on the definition a little bit. And I'll kind of point out their contributions to this theory as well. So here, X is going to be a closed orientable format of full. So within the past three years or so there have been results for both non closed and non orientable for manifolds as well, stating that they exist. So a GK1 K2 K3 trisection of X is a decomposition of X into three pieces, hence the name trisection. This is X1 X2 and X3, which will usually color red, blue, and green, respectively, such that each Xi is a four dimensional one handle body of genius K I. And looking mon three, the pair wise intersections are three dimensional one handle bodies. And the triple intersection here is a central surface of genius G. So what Meyer Schirmer and Zupan did is they looked at these unbalanced trisections and gain Kirby's original paper only defined balance trisections. So that's the case when all the case are the same implicit in the game Kirby paper was this idea of an unbalanced trisection, but they never really quite called that. So a picture of what this looks like is. This is a three dimensional one handle body of genius reform manifold. And now everything's going to be down two dimensions. So a surface is going to look like a point three dimensional pieces are going to look like lines four dimensional pieces are going to look like spatial regions. So we have some central surface. Sigma in the middle. And then we have three pieces, which are conventionally red. Green. So this red guy, this is x three intersect x one. The blue guy this is x to intersect x three. And this green guy up here is x one intersect x two. And we usually label them. I apologize. The green guy, this is x to intersect x three. And the blue guy is x one seconds to. So that this is x one. This is x two. And this guy appears x three. And this is kind of the standard schematic of a trisection is this disk with a Y in it. So it's cool and I'll talk about this later if we have time is that just like how we can understand a three manifold using a Morse function, which goes to a line, we can understand a trisection by looking at a Morse function that goes to a disk. And this Y picture is going to come up a bunch. So here's some examples. So we can look at S four, which is the five coordinates in our five that sum squared to one. So we're going to let X one be the piece that has argument of the last two coordinates between zero and two pi over three. And then X two and X three are going to be defined similarly. So we're just dividing up the last two coordinates, based on a unit circle, and it's not too terrible to check that each Xi is affordable. The idea here is, is that if you know what the argument of these two pieces is, and you know their magnitude that gives you three degrees of freedom. So if you have exactly four degrees of freedom, and these three plus one pieces have to some score it to one. That's a really bad description of what's going on, but it's a formal. If you stare at the point set definition. Moreover, you can see that each double intersection is a three ball. And the other is, if you look at X one intersect X two, you're fixing the ray that X four and X five have to lie on. Because for example, X one intersect X three, you're fixing it to have exactly angle to pi over three. So you can lie anywhere on that ray, and then these other three coordinates are free. So you have four degrees of freedom that sum to one. So the triple intersection is it has to the triple intersection is X two, because we're lying on this X four is equal to zero piece. That should not say X equals they should say X four X five is equal to zero zero is what it should say in all three of these pieces. Two coordinates are both zero, and the other three have to sum to one. And then of course that's a two sphere. You know, in our five. So this is a tri-section. Right, so we take you say what the, there were these, this like G and the K occurring in the definition, what are those end up being here. So here, G, K one, K two, and K three are all zero. And the theorem that S four is the only four manifold that I care about that has a genus zero tri-section. Okay, it's the only thing that you can split using three spheres. Okay, the genus is the center, the triple intersection. It says G. Gotcha. Okay. Yeah, that's usually why when I read a tri-section, I separate G with a semicolon because it's the most important parameter. And you can kind of cycle these K one K two and K three, just by rotating kind of how you're viewing it. Yeah, that's a great question. So here we have a zero zero zero zero tri-section, which is boring. Unfortunately, it's a little bit harder to picture other tri-sections. Here's one way to do it. And this is kind of where Morse two functions come in. So if you S four as living in our five as before. And this pie is going to be what's called a Morse two function. This is slightly different from my picture before. It depends on as you ask Rob Kirby or Dave Gay, which way to label pieces. Dave Gay is going to tell you one way to label them Rob Kirby is going to tell you the other. It's great. Mostly Rob Kirby has one out and this is the way people label them now. But we can view this exactly the same way as before. We can just define pie from S four to R two by pie of X. This is equal to the argument of X four plus X five. And then continued to zero. So this is continuous function. And then continued to the disk. Actually, you don't even have to take the argument function. This is literally just prediction on the last two coordinates. I apologize. And then the pullback is really what you care about. So you care about the pre images of these sectors here. So the X one X two and X three sectors. Now the disk is going to give you exactly your tri section of your format. So here was just a little bit more exposition on the double intersections and triple intersections. I actually gave us talk as part of an oral comp. So there were much more places in here for questions. But I think it's a fun little expository talk. So the question is how do we build tri sections and hangar explains. And here is where the talk can diverge into two places. I can either talk about cut systems and other pretty pictures, or I can take a little bit of time to talk about projective spaces, and how to try select and bisect projective spaces. So the question, if you, if you go back up to the, as a Morse to function, do you just have some kind of similar condition on my critical points or something like that, regular values. Yeah, let me, I can talk about that a little bit. I'm going to talk about the Morse to function picture. So a Morse. I'm going to say a trisecting Morse to function. And this is following a Kirby 15. And Dave gay has this really good survey paper, porting 3D 40 something. If you just Google David gay porting 3D to 40, it's like the first archive result. And this is 18. So Morse to function is a function from a four manifold to our to such fat at each. And this is in X, when I get this name, F, either F, either grad F at X does not equal zero, or if grad F equals zero, then special form pictorially describe what the special form soccer, because doing them in is not fun. So everything here has to be done on local coordinates right you're in a manifold mapping to our to so pretend you're in suitably nice local coordinates. So the first thing is it, I guess I should use a different letter. So a it's either definite, which just means locally. I'm really bad at this drawing part. It kind of looks like that you have this critical here, where it's indefinite. I'm being very loose here. The definitions in game Kirby and gay are very much more technical, where it kind of looks like that. We have this calm point. But you can perturb it so that the company is transfers. If we're following this path here, as we go along, right, the tangent space intersects transversely the time and space. That bit, I guess that was it. It's something happening here like the tangent space splitting into like a positive definite and a negative definite part. Yeah, it's, it's very technical. But the key thing is that all of these oopsies are convex. So none of these angles are bigger than 90 degrees, which helps preserve some nice geometry that goes on. But that's just kind of aside, I do not claim to understand the most two function picture as well. I just use them. And this is really useful because it allows us to look at CP to for example. How do we view CP to so there's one way that's just look at homogenous coordinates and pray. Right, try to play around kind of like we did with the four sphere and hope that we can find some nice embedded tour I were higher genus surfaces that just happened to split your happen to split CP to into the three components nicely. We can use what's called the action map. So what it's called action. Action map or potential map, but the idea here is right I have z1 z2 z3 in CP to, and I want to define a function that goes to R2. So what I do this is I can do. Oh boy I'm not remembering it exactly, but it's something like z1 squared plus z2 squared plus z3 squared. Yep, this is what it is. So we're going to send it to a plane inside R3, which might as well be an honest to God R2 after some affine transformation, there should be norms around everything. So here is we're mapping to a good old regular simplex in R3. So we have this map from CP to, and everything actually gets mapped into this unit simplex here, a one simple, two simplex. So everything gets mapped into this triangle. And then we can just finally transform this triangle to live in R2. And this is called the action or potential map. And it's really useful in complex geometry, which I know nothing about. The idea here is it comes from looking at it as a, you can view CP2 as a there's a nice torus action on CP2 is the way Dave Gay explained it to me one time, and this comes naturally from that torus action. And then we can use the exact same tools that we did before. So I can find a central point. So just like in regular Morse function, the pre image of a point pre image of a regular point. Well, it's a code of mention to sub manifold. So it's going to be a surface. And then I can just divide up. Oh, I don't want to do it there. I want to do it here. And then I just divide up my manifold like that. And it turns out that this will give you a genus one trisection, because the pre image of one of these points is going to be a torus. And this will give you a 1111 trisection of CP2. And it actually turns out that the same map will give you a trisection of CP2 bar, just by reversing the orientation on CP2. However, it'll turn out that these two trisections are different. So let's go back to some easier pictures and surfaces because we, because we like surfaces and pictures on that. So this is going to be the notion of a cut system, which is going to kind of underline all of the stuff that I'm talking about. I accidentally muted myself. So a cut system is a system of curves on a surface. This is introduced by Hatcher and Thurston in a really, really, really legible paper for Hatcher and Thurston in 1980, where they looked at the modular space of curves and surfaces, and how that interacts with the mapping class group. So this is going to be some genus G surface, and alpha is going to be a collection of alpha curves. Each alpha I is going to be a disjoint simple closed curve. Then we say alpha is a cut system, right, really for Sigma. Then we say cut along alpha, which we're going to define to be Sigma, and then we're going to remove open tubular neighborhoods of the pieces. The idea here is where you can view a curve on your surface, and then you literally just snip it out like it was made out of paper. And then it's a cut system if this cut along thing is a planar surface. In one of the two ways, it's either a sphere with some even number of boundary components, or it's a disk living in the plane with some number of disks, sub disks removed. And in particular, we're looking for one that has two G components. And if you're more algebraically inclined, you can think of a cut system in the following way. Capital A be the equivalent things in homology. Then you want capital A to be a linearly independent subset of your first homology of your surface, such that the intersection form is trivial on a intersection form. I just mean the natural pairing on each upper one tensor each upper one down to a lower. Oh boy, down to just regular old Z, given by cupping the Poincare duals and evaluating at the class. Right, the regular old thing that you would want to do with even dimensional stuff. Okay. The algebraic picture isn't super useful, because this just says everything is disjoint homologically disjoint. So let's think of some cut systems, and just kind of explore what they are. So I'm asked to any alpha works, particularly the empty set works right algebraically. Each one is zero. So, you know, pick your favorite element of zero. Wow that's a linearly independent subset. Or, right, pick a simple closed curve on the sphere. Okay now try to pick a different one that's homotopically disjoint from it. Okay, you can't, you can't do that. What about on the tourists. So what are possible cut systems on the tourists. Well, here's one right I can cut it along the wrong way to cut a baggle. And I can think of this and I'm wrapping and that gets a tube, and then I thicken it up and that gives me a surface, a sphere with two punctures. And it works in every genius. So I'm genius to just cut the baggle wrong twice, unwrap it, and I get a pair of pants. But it actually turns out that this piece, this piece, and another empty piece that looks like this but upside down, are the only pieces you need to build any cut system up to some notion of one. So lemon on the standard course is there any, is there any choice of curve which doesn't work like if you choose something homologically trivial or something. Um, yeah, anything that's homologically trivial won't work. So if I chose, for example, right, this disk here, instead of the red curve, and I cut it out. Well, that's not going to separate my surface into a planar surface. Does that make sense. Yeah, what did you need for a planar surface and this is a sphere minus points or just minus points. I couldn't quite hear you. What was the planar surface condition. One of them I think was like punctured disk right. Yeah. Yeah, planar surface punctured disk or sphere with some boundary disks. They're all equivalent notions. Okay. And the kicker is we need this planar surface condition. So let's, let's prove that cut systems exist, because a priori, maybe on some crazy hygienist surface, you can't find a complete collection of curves. The problem is that they exist and they have exactly g curves. We're going to do this by induction. Well, I already showed the g equals zero case it's trivial right pick the empty collection, and the genius one will teach you itself isn't a punctured sphere, but cut along along this red curve works. There's two or more sigma g breaks down into pieces of the following form. There's the piece a, which I call left macaroni. There's piece B, which is a topologist t shirt. And then there's peace a prime which is right macaroni. And if you're willing to believe me, then sigma g breaks down into a left macaroni g minus one copies of a t shirt, and one copy of right macaroni. We just blew everything like Lego bricks. And we can describe a cut system on here as follows. So on a, we're just going to cut the macaroni noodle in half. And on a, we're going to cut along the mid section of this t shirt thing, or if you want to think of it as weird pants, we're putting the pants in half. And on a prime, we're not going to put any curves. So, if I do everything like this, I get a cut system that looks like this. And this is on sigma sub G, or sigma super G. The colors mean nothing here. They're just to know what piece it came from. So we have a red cut here, and then a bunch of blue cuts. But how do we see that this gives us a planar surface. Well, we can take our scissors and cut all the way along here and cut out this entire thing. And then we kind of unbend it. And then we have this thing that looks like a flute. We have two open end bits, bounded by the red curves, and then we have 2g minus two finger holes. Okay. And this of course is a punctured sphere with 2g holes. Right if I really wanted to I could blow up this piece and kind of make it more spherical. And then you would maybe see a little bit more clearly that this is a sphere. So if we wanted to call it to be the red system union the blue system, this gives us a cut system. So how do we see that we can't have more or less than g curves. Well, if we had more than g curves, cutting along it, we'd have more than 2g boundary components. And if we had less than g curves, we wouldn't have enough boundary components either. So the definition that is definitely worth talking about is equivalence of cut systems. So when are two cut systems determined to be the same. So definition, the standard genus g cut system is this one. It's just you snip all the bagels like this. So the two cut systems are diffeomorphic. If there is an orientation preserving diffeomorphism on the base surface. And a permutation in the symmetric group on g things, such that fee of alpha I is beta of Sigma spy for some beta j's and beta. A diffeomorphism that sends one thing to the other. So for example on the tourists, these two cut systems are diffeomorphic. And it's orientation preserving because we can do it in C2. And this map, the Z1 Z2 to Z2 Z1 is orientation preserving by counting transpositions. Right. So this thing moves to spaces, this moves to spaces that moves to spaces that moves to spaces. Everything is even. So that's what it means for two cut systems to be diffeomorphic. There's also a slightly weaker notion of equivalence called slide diffeomorphism, which is technically what everyone uses in real life. But it's a little bit more complicated because you need to slide curves over other curves and it's, it's a bit of a picture nightmare. The Higger diagram is a triple Sigma alpha beta, such that alpha and beta cut systems for Sigma. And conventionally alpha curves are always drawn in red beta curves are always drawn in blue. Moreover, these cut systems have to satisfy a standard nest condition. Right, each pair Sigma Alpha and Sigma beta have to be diffeomorphic to the standard genus of Sigma cut system. So what is something that needs to look like? Well, on the sphere, here's a great Higger diagram. It's the empty Higger diagram. This is a Higger diagram for S3 because we can view these as telling us how to glue in handles. The red curves are going to tell you how to glue in one handles, and the blue curves are going to tell you how to glue in two handles. I'm not going to go over handle calculus today because I have like five, six minutes left. And that's definitely not enough time. But here are three genus one examples. Now I'll label these by the three variables that these come with. So this guy here, this is S1 process two. This guy is still in S3. And this guy is going to be a lens space LPQ. For some values of P and Q. Where PQ is the representation in the natural basis of H1 of your blue curve. Where P is your Lambda coordinate and Q is your mu coordinate. Right here, mu is that curve and Lambda is that curve. And where I remember is Lambda is longitude, that goes the long way around. Mu is the meridian, it goes the meaty way. I like to think of matrices as thicker around the midsection. So as I kind of mentioned Higger diagrams you'll take your splittings. And the idea is that the cut systems tell us how to build the three mental one handle bodies by specifying what curves found desks. And then this was a picture that kind of showed how the genus one thing was built. Okay. So now we're going to go up to tri-section one. And we see that the standard GK Higger diagram is a triple sigma alpha beta of the following form. So the alpha curves are all these standard pairs. K of them are parallel to alpha curves. And G minus K of them intersect alpha curves in exactly one spot. When we get to the definition of a tri-section diagram, this will hopefully become more useful. For example, a three one standard Higger diagram is this picture. So one of our blue curves is parallel to a red curve. And then the other two intersect transversely in exactly one point. So the idea is that alpha curves tell us how to build each one, the beta curves tell us how to build each two. And the standardness condition is required so that we can build the union in a nice, smoothly defined way. And a theorem, and this is pretty classical. So we're going to go ahead and omit a genus one Higger splitting, then M is defumorphic to one of the following three families. It's either S3, S1 cross S2, or some LPQ for co-prime integers P and Q. I'm going to skip over handle slides because we just do not have time. I might get to the tri-second picture if this ever lets. Okay. This is a GK1, K2 tri-section diagram, or more succinctly Higger curvy diagram. And this is following Meyer, Schirmer, and Zupan, their 2018 paper. Is it quadruple sigma alpha beta gamma, such that sigma is a genus G surface, alpha beta and gamma are cut systems on sigma, such that each triple is slide defumorphic to a number GK Higger diagram. So pairwise, these things bound three manifolds, right? So on each boundary three manifold, we want to have a Higger split. And so here is an example of a Higger diagram. This is a 1100 tri-section diagram. This tells us the number of intersections of parallel curves, I should say, in X1 intersect X3. And this is X1 intersect X2. And this tells you how many parallel curves you have to intersect X3. The notation is a little bit cumbersome because the genus and reverse. The notation is a little bit cumbersome because you're tracking parallel curves rather than curve that intersect. So here you see, when we look at red-green, so we ignore blue, we have one family of parallel curves. Whereas we don't have anything in the other two families. And this is actually a tri-section diagram for S4. It's called the one-fold stabilization. So this is a one-stabilized tri-section. So lemma is that the tri-section diagrams tell you how to build tri-sections. And the idea is you can build each of the three-dimensional pieces exactly like we did in the three-dimensional case. And then we can use this super-celebrated theorem of loading back and cornering, which says that if you know essentially the zero, one and two handles of a four-manifold, there's a unique way to complete that and cap it off using three and four handles. And it's unique up to diffeomorphism. It's an incredible theorem. I don't claim to fully grasp everything I can do. But it's one of these results that just revolutionized the field, because now we can use all these geometric techniques. And this was back in like the 70s, I think. And so theorem, and this is from Jeff Meyer, Trent Schirmer, and Alex Nupin. If X admits a genus one tri-section, then X is either S4, CP2, CP2 bar, or S1 cross S3. And all three of the Ks are equal, or X is S4, and the triple of Ks is some permutation on one zero zero. And that's it. That's the idea of, hey, are there some tri-sections. I could probably give another two-hour talk on more details, but I just kind of wanted to give an overview of what the heck these things were. So thanks. Oh, yeah. Thank you so much. Yeah, let's go ahead and thank Nick for his wonderful talk. Thank you. Thanks, Nick. Does anybody have any questions? One thing I was kind of wondering from earlier, it's not a huge deal. I think it was in the proof of existence of these cut systems. You had these sort of the pieces, like the two different pool noodles and the t-shirt, right? Are these just coming from, like, the elementary, cowardism sort of decomposition when you attach a Morse function and like, I forget how good, like you take sub-level sets or whatever? So that is one way that I thought about it. The easiest way is to honestly just think combinatorially of, how would I want to build such a thing? So I like to say I study Lego brick topology. I like to think of myself as I have a standard toolbox of pieces, what are all the different ways I can glue them together and give me something interesting. I had your thirst and prove this. Well, actually in their paper, they didn't prove that they exist. They just said, well, clearly they exist. And I think it was giving, worth giving a cool little proof for why these things have to happen. And what's actually even cooler is that if you're allowing non-orientable stuff, similar cut systems can still exist, but they might cut up to give you punctured Mobius bands, rather than punctured annuali. And that gives a whole bunch more difficulty as well. So these pieces really just came from thinking about what would have to happen and what is the easiest thing I can do to guarantee a cut system. Because right, I could have chosen just crazy curves on my Bs or my A's. So for example, if on my B piece, I wanted to choose this curve. If I pressed along these curves, right, I cut these out. That's going to separate, those are separating curves. And that's, that's not really what we want to do. We don't want to separate. We want to keep everything intact. I see these are kind of coming more from casework on kind of like all the different cut systems you could dream of. And that's kind of the thing I like about this way of approaching topology rather than approaching it from the more speary land or approaching it from thinking about surface bundles and crazy highfalutin differential geometry, is we really get a handle on exactly what you need to build a manifold and how much structure and information you need to specify a manifold up to diffeomorphism. And it turns out that cut systems from a combinatorial standpoint are pretty much the bare minimum. Do you happen to know how or how or if like the curvy calculus stuff ties into this or is this like the same coin or something or. Yes, yes, yes, I do. Okay. The idea for the curvy calculus goes through the Morse three land. So by knowing a trisection diagram, or a haggard splitting diagram, you're able to get a handle decomposition on your manifold. And that handle decomposition works in the following way. So let me scroll all the way down. So right here is kind of your standard picture of a four manifold. So here's X four. So Obama of Morse or probably Milner says that you can build it with a unique zero handle and a unique. So your zero handles and four handles are uniquely determined. And by construction, we know that X one is a four dimensional. One handle body. And X one into this picture. So here's up to my one handles, and this piece down here is X one. So X one is my zero handles union. And if I kind of turn my head upside down, I can do the same thing with X three. So X three is three handles and four handles. And the tricky part is how do we fill in the middle bit is how do we make these things match up. And this was the real innovation from the game curvy paper is that. Okay, well let's ignore this stuff up here. Let's not even pretend that this stuff up here exists. So I'm just trying to build a handle structure. Well, I can specify how I'm attaching my two handles. So I have X two. And I can tell it how to attach here under the one handles. And this gives me a link with a framing. And this frame link is exactly what you need for a curvy calculus picture. So I take X two and I go in here. And then I do the same thing up here. And I just kind of flow along a little bit. Keep doing X two. And then I see what link I have to glue in here. And what game curvy showed is that it's the same link with the same framing. More or less. So essentially what we've done is this. And then I have X two. And then this stuff here is just whatever we need to glue in with loading back and butter to cap off the picture. And nine times out of 10, it's just a four ball. So this one's I've looked at the original game curvy paper to fully understand this construction. But this is pretty much this kind of picture is that you get three layers and then you kind of fill it in. And really the picture most people look at this one, where you kind of let X three flow a little bit. And it also hits X one. And then this thing here, this triple point is your central surface. That's kind of the more scary picture. And by adjusting this to be two dimensional you can get a more to function. Well, yeah, curvy calculus appears naturally by gluing in these things. Any other any other questions. All right, so let's go ahead and make one more time. Thanks Nick. Awesome. Thank you everybody. And then stop recording here.