 So, last time we discussed the compactification of the moduli space on a cylinder, y times r, and just to recap that briefly, first thing, so here I've just indexed the moduli spaces between a pair of critical points by the dimension. I should really keep track of the homotopy class of paths between them, but which is simplicity. So, if I look at a zero-dimensional moduli space between a and b, it's empty except if, sorry, except if a equals b, then there's just a constant trajectory between them, okay? If there's a one-dimensional moduli space between a and b, then mod translation, I get a zero-dimensional set, finite set of points by compactness, and the two-dimensional moduli space mod translation, if I compactify it, consists of, well, the top stratum is one-dimensional, and is the two-dimensional moduli space mod the one-dimensional action, the group of translations, and then its boundary is all possible breaks that give you one-dimensional moduli spaces between them, and then mod up by translation. For the three-dimensional moduli space, top stratum is a three-dimensional bit, then it can break into a one-dimensional and a two-dimensional bit on one side or in the opposite order, and then there's a bit that's where there are two breaks. So, this guy's two-dimensional, and it looks a bit like this, so there's some, you know, looks a bit like a two-manifold with a corner, and these things look a bit like manifolds with corners as you go down, then something interesting happens in this particular story in dimension eight. Dimension eight, well, we have the same, the pieces that we've seen before, but then there's a new phenomenon, which is that a bubble can happen, and a bubble leaves behind, in this case, a moduli space of dimension. It's an 800 number, which means they're probably telling me that my social security number has been compromised, and you know, I'm going to go to jail or something like that. Anyway, so the next, so the new phenomenon that we're seeing is that there's this potential for bubbling, but you notice that what it leaves behind is a zero-dimensional moduli space in this case, so this only, there's only, there is a contribution here, but a has to be equal to b, and then there'll be more bubble contributions as you go down, but this is the first one, right, so here a equals b. Okay, and then I just want to recap why d squared is zero. The definition of d, the differential on our potential complex is that we count the number of trajectories in one-dimensional moduli spaces from a to b, sum over b, and that's our answer, and you notice that if I look at the matrix element for d squared from a to b, then we're just summing over all c that are allowed breaks, and that's the boundary of this compactification of the, boundary of the compactification of the two-dimensional m2 mod translation being the boundary, well at least it's zero m2, this is a one-manifold, it has two ends, in pairs of ends, even number of ends, okay, so that's, so d squared is zero, so this gives us our complex, and nice thing about, you know, as you learn about this stuff you have two choices, you can read lots of books on homological algebra, or you can listen very carefully to the whispers that the moduli spaces are trying to tell you, the latter is a much more productive activity, so let, so we can define maps between these chain complexes by looking at moduli spaces on, well instead of on a cylinder, let's put some interesting co-bordism between them and form this manifold of cylindrical ends, we're going to study instantons on here, behavior's the same, the only failure of compactness is bubbling, and then we can try and define a map which just counts, there's no longer a translation here, right, because this is just some form manifold we can't translate, so a sensible thing to try to do is just count the number of solutions between a given a and b, and then let's try and define a map, the image of a is, we look at all zero-dimensional moduli spaces on this manifold, count the number of solutions that end up at b, multiply by b, and that's a map, and then observation is that's actually a chain map, why, because if I look at, instead of moduli space, that's one-dimension higher, so I look on this four-manifold cylinder collend at a one-dimensional moduli space, what is the compactification of this guy look like, well there's only two interesting things that can lead to non-compactness, the dimensions too low for bubbling, right, so there's only two interesting phenomena, some energy can slide out to the right, your right, my right, depends, so that's somebody's right, anyway, left, or it can slide out to the right, that's stage, stage left, stage right, yeah, I don't know, I never remember stage left, stage right, is yours or mine, anyway, so if it slides out, dimension counting tells you the only thing that can happen is that it can slide, this is one-dimensional, so the dimensions add up, so either one-dimension goes out that way, it leaves zero here, or one goes out that way, it leaves zero here, so you get this kind of a, so the compactification of this one-dimensional moduli space is the moduli space mod translation sliding out that way, union zero-dimensional moduli space here and the opposite, and if you interpret this in terms of that map, it's just saying it's a chain map, right, and then, well, of course, now you should appreciate that we've made many choices in the construction, and we should check that these things are independent of the choices that we make, and then you can convince yourself that if you now look at, you know, you can look at moduli space, kind of parameterized moduli spaces and come up between chain homotopies, between these things, etc., so just kind of staring at the moduli spaces gives you a lot of nice stuff, okay, eraser, great, alright, so we basically got a floor homology group for a three-manifold, so we have, you know, we'd like to say the floor homology is just the homology of the chain complex, there's a bit of a lie, so because there might be reducible connections, reducible flat connections, so those are where the stabilizer of A is not, well, the center of the gauge group. Let's think about SU2, sorry, not equal to the center, centers, centers just plus or minus the identity in this case. If we have reducibles, then that kind of, if you think carefully through the story, it reeks some havoc, but there's a nice trick that we can do to avoid reducibles, alternatively we can work much harder. Now, the trick is, remember, we observed that on the two torus, this is T2 and P, there's a unique irreducible flat connection, you know, with SU2 connection up to conjugacy with holonomic conjugate to I around the point P, so we discussed that last time. Now, that's not quite a three-manifold, but, yeah, sorry, I'll do this a little bit out of order. So, let's look at, so this is the three-sphere, that's the hop flink, there's this blue point, let's consider, look at the hop flink in the three-sphere. If you think the complement of the hop flink retracts onto the torus that links either of these guys, it sits around either of these guys. So, I can think of S3 with the hop flink and this blue curve being my second Stiefel Whitney class is giving us, you know, a three-manifold whose representation spaces are identified with the representation space of the torus, so this guy has a unique irreducible flat connection. So, this S3H and this omega, this has a unique irreducible flat connection, flat SU2 connection up to conjugacy. So, we're going to connect some with this guy, so we take, so let's call this Y-sharp, so that's just to connect some of our three-manifold with this three-sphere and the hop flink, hop flink in this curve omega. Now, this has all the, all representations are irreducible, right? Because, since it's irreducible over here, well, this guy doesn't have any stabilizer, so whatever it has over here, the whole thing can't have a stabilizer, so that's great. But now, we've run into a slight problem. We have this hop flink and actually the holonomie, well, by construction, where's some other chalk? Did I say, I said, so the, the holonomie around here is minus the identity. So, the holonomie around here is minus the identity, but the holonomie around here is, is conjugate to, to the quaternion I, but as a, I mean that's the holonomie around one of these loops, right? The, the, remember this representation had holonomies say I around this loop and J around this loop, so it'll have holonomie I around this guy, J around this, this guy, those are, remember, conjugate, but okay, so this is not quite the story we have. This is now a three-manifold that has a knot in it. Okay, well, that's, maybe we should just figure out how to do that. Then we can do knot theory, too. So, Orbe folding. So, it turns out that a lot of the story for, for instanton fluoromology, etc., it, it works just as well for Orbe folds, not just ordinary manifolds. So, let's make our, but let's think about the simplest possible kind of Orbe fold that we could get. So, if we have a three-manifold with a knot, I'm going to construct an Orbe fold and it's going to be the kind of absolutely simplest Orbe fold. Namely, you know, there's my knot. Let's make the knot have a color. It's sitting innocently in the three-manifold and what I'm going to decide to do is, is think of constructing a manifold that has a cone angle of pi along the knot. So, kind of change the picture to think of it like this and here. So, in other words, I think, you know, to say another way, I could, I remove the tubular neighborhood of the knot, then I divide by a pi rotation to make an Orbe fold and then I stick in the tubular neighborhood again. That's just made the kind of, you know, the normal disk half as big. It's still a disk. The neighborhood that's boundary is still a torus. I can stick it back in and we get, you know, so that's the construction of the Orbe fold and we can sort of, you know, at this level of detail, we just repeat the construction of the instanton complex and, well, so there's one modification that happens. So, the main change, don't get something for nothing. So, by the way, when I put in this Orbe fold structure, I want to really exploit the fact that I have a knot and remember it in the definition. So, the way I'm going to do that is via what kind of connections I'm going to look at. I'm going to look at connections that are genuinely interesting Orbe fold connections so they should have non-trivial holonomy around this loop. So, if I have an Orbe fold, it's natural to think of its fundamental group as, you know, if I take a little loop that links the Orbe fold locus, well, going once around that loop should be an interesting curve. But in this case, if I go twice around it, that should be null homotopic. And, you know, that obviously generalizes to higher order kind of Orbe folds. So, I want the holonomy, so we're going to require the linking holonomy to be of order exactly 2. So, it's not trivial, it has to be order 2. Now, if, there's a little, this guy, I, of course, has ordered 4 in SU2. So, the way I'm going to think about this is really as, I'm going to think of it as an SO3 bundle away from this locus. I descends to an element of order 2 in SO3. So, it's fair to think about this kind of connection. So, I take my Hopflink in the story, construct an Orbe fold along the Hopflink, you know, look at interesting connections, think of, you know, interesting SO3 connections, but I'm only going to divide out by a kind of SU2 gauge group. Anyway, so, just sort of for the moment, believe that when you do this Orbe fold thing, you can construct an instanton of floor homologies, so that amongst the flack, if you computed the flack connections here, you just get the ones that we're looking at. The main difference in, in the, sorry, yes. No, if I, if I take, if I take, yeah, yeah, yeah. So, what I mean is that the limiting holonomy should be non-trivial. I mean, just to say, that is, I'm going to look at Orbe fold connections where in the Orbe fold chart, you know, an Orbe fold connection means that when I lift the Orbe fold chart, it's an honest connection, but it's invariant under the symmetry group, and I want the action on the bundle to be a particular non-trivial order 2 action. The connection has to respect that, which in kind of geometric terms passing down means that the holonomy as I link the loop, you know, shrink, have shrinking loops linking the thing, they have this order 2 holonomy. Okay, so the main difference is, is what the compactification looks like. And now, what can happen, well, so if I look at four manifold, I can do the same construction with a four manifold in a surface, like that. And now, there's two kinds of bubbling phenomena. You can bubble here, at a point interior, and the dimension drops by eight. And this story happens to be set up that, you know, so this has drop of eight. Now, you can also bubble at a point on the surface, and this has a dimension drop of four, turns out. So, the compactification's a little, you know, now it's a little more complicated, you need to keep track of bubbles on the four manifold bubbles on the surface, but the dimension drop is four, and that's still, if you go back and think about that compactification story that we looked at, the interesting thing, the interesting new thing that starts to happen when you try and look at the issue of defining d squared happens in the four dimensional moduli spaces between pairs of trajectories because you could have a bubble that leaves behind a trajectory between two equal connections, but only once it's four dimensional. So, you know, we can still get a differential on c star of y k, and d squared is zero. So, so we can define, so now that's great. Now, at least it makes sense, you know, if I do this procedure now using that definition, this instanton four homology makes sense for any three manifold, as long as I do this connect sum, and more generally it makes sense for knots in three manifolds. Yes? Oh, yeah, the model is, sorry I should have said that, I mean that if you look at instantons on r four, you could look at the instantons that are invariant under z two action that fixes a two plane and is minus one on the orthogonal two plane. Now, if you, you remember that, so, you know, the standard instanton is of course invariant under that, but the only conformal transformations which are invariant under that are conformal transformations of the fixed r two. So, the moduli space becomes, instead of s o five one mod s o five, it becomes s o two one mod s o, sorry, s o three one mod s o three, a hyperbolic, sorry, yeah, sorry, s o two one mod s o two, yeah, hyperbolic two space. And so, you should think of the frame moduli space. Anyway, I don't want to say more about that, just, I mean, I can say more about that later, but I want to kind of get through this story. No, no, you have to perturb. Yeah, I mean, you know, we're not talking about perturbations because there's not enough time, but everything has, you know, you have to perturb the equations to get everything to be more smell. Alternatively, well, of course, you could do more spot if you wanted, but think of doing more smell. So, some hidden perturbations that are swept under the rug. Anyway, so, we get, so, the compactification is a little more delicate, but there's still a differential, d squared is zero, and yeah, great. And so, more generally, we get an instant on floor homology for three manifold and a knot and, you know, the, just to draw that picture again. And so, here's why. Maybe there's some interesting knot in it. And there's the hot flank over here. There's three over here. Connect some. Right. Now, you know, we have to be a tiny, so, but also the co-bordism story works. Now, you can have a co-bordism of pairs, a knot and a three manifold, and there's a surface in a four manifold, which just comes out, another knot and another three manifold. You have to keep track of this bit of data, so you need to take, drag a base point from one three manifold through to the other side. But there's, so, we get this invariant. Lovely. Okay. We got functoriality. Great. All right. So, I want to sketch for you an important property. So, this floor homology group, you know, it's like any homology group, it turns out that there are exact triangles that help you compute it. And I want to kind of sketch the story for the exact triangle for knots. There's actually an exact triangle with, just for three manifolds, which involves surgery on the three manifold. But I'm going to describe the exact triangle for knots. So, the exact triangle involves these three knots. So, I imagine there's some knot in the three manifold, and I see inside some three ball, this picture, and I replace it by these two pictures. Okay. And let's call this knot k0, k1, k, sorry, k2, k1, k0. And so, of course, you remember from Jake's lecture, this looks like the Havanoff picture, but the crossing is different. So, that's a, I'll explain more. I mean, sorry. So, if I have a planar projection, then there's a difference between this crossing. I mean, you know, if I turn the picture, everything turns. Okay. What do you see from the back of the board? You see the same picture, actually. So, let me put this in a little bit more, you know, yeah. Just bear with me for one moment. Sorry. Why are there three anyway? Sorry, let me, I should have, okay, let me, so these are three tetrahedra, and let's say, sorry, I got it. I have to get the convention right. Sorry. A tiny bit hopeless. I didn't think about them. Sorry. I don't want to do that one. I just did that one. Let's do this one. What's left? These opposite edges, that opposite edge. There's a meaning here. So, the tetrahedra, why did I draw that one differently? Sorry. Anyway, the tetrahedra has three pairs of opposite edges. The really important thing is how I'm going to go between these guys. Okay. So, in the Havana story, you know, it's just algebra. You pick some convention, you just got to make sure you stick with it somehow, and life is okay. In this story, we need some geometry. We need a co-boardism to define a map. And what the co-boardism is, if I want to go from here to here, there's a natural, there's a natural co-boardism. Now, what it does, so, you know, I start, so, I take this edge to kind of describe it. So, this is actually going to be a co-boardism in four space. I'm going to describe its projection into three space. I take this edge, and I start rotating it till it meets, you know, take this red edge, rotate it so that it stays on, the ends stay on the tetrahedra until it gets to here. And then the boundary includes this bit, which is this blue curve. All right. So, that gives me a co-boardism from here to here. Then there's similar co-boardisms that go, you know, so maybe that's sigma 21, sigma 1-0, sigma 0-2, and, you know,