 The last lecture. OK, so in lecture 1 to 3, we have described how the algebraic, so we've introduced the category of stable maps, a bundle category over it, Cauchy-Riemann section functor. We have described the algebraic relationships, and we have put smooth structures on this. And sort of the packaging is essentially of when you want to resolve the transversality questions for SFT. So you have this diagrams of covering of, actually, now smooth covering, SC smooth covering functors. Some other relationships formalized by having some functors which behave in some compatible way is the Cauchy-Riemann operator. So now the goal then is, so now we have a complete setup to perturb the subcategory of sort of J-holomorphic objects into some other sub-weighted subcategory which actually will be smooth, SC smooth, so that will be enough to extract data from it. And we would perturb it into a functor which has certain properties. That is the algebraic thing, so that we discussed in length. And of course, geometrically, you want that this is a pair in general position as far as the Fritam series. So there's a Fritam functor as far as the Fritam series is concerned. Now, if you think of this here locally as a finite number of sections, so you get a finite number of equations, and you want that each of them is in general position, which can be achieved for this. It cannot be achieved when you homotope from one perturbation to the other. You cannot in general achieve general position because of all the structures which you have here. But for not doing homotopes, but when you just want to perturb, that's still possible. So now, so we have to perturb. And of course, we make choices to do so. So if you have two different perturbations, we want to make sure that in each connected component of the orbit space of our categories, the solution spaces are still compact. And we want to make sure that we can guarantee that we can connect this sort of as generic as possible, these two perturbations. And we also have compactness doing that way. So now, in order to actually make sure that this is a case, you ideally distinguish something like a convex set of perturbations that you can go around where it's also clear that, a priori clear that you would have compactness. So let me explain here. So we will actually be able to make an infinite construction. And there's some kind of an inductive procedure to produce it. So it starts off with low energy things, which cannot split further off, which only contain buildings with height one. Then you go to the next one where you have buildings of maximal height two. But then the boundary phase is just by the algebraic structure, which you have have to be explainable in terms of the data you already dealt with. So you pull up these perturbations on the boundary, and then you extend it. Then you have to distinguish between sort of connected components alone together and others with added cylinders and so on. So there's a certain number of things which you have to do on this level. So then, of course, when you just see this procedure, I go on and on and on and on. So it's not surprising that my perturbation gets larger and larger and larger and larger. So now if I only can control things in an a priori given neighborhood of the original solution set, you can imagine that at some point you fall out of it. That actually happens in every other approach. So what do you do? Well, you use homological algebra. So you perturb up to some level, then you encode this data, then you take a smaller perturbation to begin with and you can run higher up, and so on. But that would require you already to come up with some packaging ideas to just, at the end, get a complete set of data. But it turns out that in this problem, actually, and that has to do something really, it's basically an abstract factor. You don't have to know much. You have to know a little bit. You have to exploit a little bit more about the Cauchy-Riemann operator than people usually do. So there's more compactness in the background. And actually, this infinite procedure can be done. So geometrically, you can run up. You never have to start smaller and go up. So that's a nontrivial effect. So you can make that infinite construction and get that infinite, smooth object. So that, of course, could be used in other approaches. So this finer compactness could be used in the other approaches as well, so there wouldn't be the need to do this. OK, so in order to deal with this aspect of the story, so you have to measure the size of perturbations and you have to come up with some idea how that controls compactness. OK, so we have heard what an auxiliary norm is in the case of an M polyfoil setup. So here, it's basically a functor. So it's actually defined on the things which have a little bit better quality in the fiber. But if they're not lying in this space, they'll just put it plus infinity. So a functor there restricted to this E is on each fiber norm. So if you go to orbit space, it should be continuous and should have the following properties. So if I take a sequence of objects so that n goes to 0, so that's the sequence of numbers, and if the isomorphism class of the base point below converges, then the isomorphism class of this vector converges to the isomorphism class of 0 over that point. So that's the requirement. It's some kind of local uniformity. So you take a sequence here. So if this is finite here, that means you're actually here. So then the size goes to 0. So this is just a sequence of objects, wherever. But now you look at the base of this object. So it lies over an alpha k takes the isomorphism class, assume this converges. Then the isomorphism class of this should converge to the isomorphism class of 0 over that object. So then there's something which I'm not going to explain. That's called reflexive auxiliary norm. That's an important refinement actually for this to utilize this more general compactness. So one would have to explain a little bit about that. But basically, this you can construct always if the fiber in your bundle category is actually reflexive, which is likely about space in our case. Then you can always get this. So one can presumably talk about an hour about the compactness issues. So then it should be compatible. So first of all, you define, if you have an auxiliary norm, you can define this on the fiber product. You just look at the individual factors, maximum. It's important that you take this and not the sum. That is what you can control here. If the maximum is controlled and you can control, and that means that you can control compactness here and here, you control compactness of the product on the boundary. Then the furtome property allows you to disperse that control a little bit to a neighborhood, because locally it looks like some finite dimensional times contraction. So if you're not too far away, you still have the contractive parts. So it still essentially looks finite dimensional and you can control this. So then you want compatibility with this structures which you have. Then you want to have something like this. It factors. So pi is this projection coming from this Whitney type decomposition, where you put the 0, 1 form on the non-trivial cylinder, on the trivial cylinder component either 0 or the other way around. Remember, that was one of our structural things. You want to have this. So it measures the size of a 0, 1 form on a non-trivial components or, in this case, over the cylindrical part. And you want to have that behaves like this. The reason is that you later perturb anyway where there's nothing on that side here, because over the cylinders, the perturbation will be 0. What is this E subscript 0, 1? Oh, this is E? This? No, cappy. Oh, yeah. So remember, for a strong bundle, K over O, has this bi-filtration, Mk where K is less than or equals M plus 1. And if you have the strong bundle structure on this and this E over C, it also gets a bi-filtration. So on the stable maps, you can talk about the regularity of the object. And then if you have something in the category of firing over it, you can talk about the regularity of the 0, 1 form over the underlying object. And that has an addition. So why do you allow the thing to actually go there infinitely on E? Yeah, because it's precisely finite if it's here, which is a sub-fiber. Can you say something about why is that? Yeah, so remember that, as Katrin pointed this out, that for a strong bundle, it's not just a bundle, but there's an additional bi-filtration. And if you lie on the one level, it's sort of compact perturbations. And the SC plus sections are sections of the one level. So that is where you know that things are compact. So in the fiber, you have 0 level, 1 level, and so on. But if you are on the one level, you go down by compactness on the 0 level and you perturb by sections which lie on the one level. And that is precisely there are sections of this, the multi-section functors. A priori are given by sections of this, but the images of this section actually are in that better part of the space. So that is how a compact perturbation is built into the whole thing. And then it should have this property. So if you forget about the stuff over trivial cylinders and take the norm, or you move, or you even throw the component away, which, of course, is more or less, you don't need the trivial cylinder component, then you see the same measurement. So then, of course, you want, if you have disconnected non-trivial configuration and move them against each other, then you would see this. The measurement of those guys, they are maximum. Doesn't that contradict the thing you have to bother about n equals n equals to pi plus n equals to the Iw1 time? That's a plus right there. What a plus, no, no, why? So if, do you mean this? And I mean that one. Oh, no, no, no, no, no, no, no. So these are non-trivial components. So then you take this, and then there would be some cylinders here, which you discard, and some cylinders here you could discard in order to compute this. And if you look what happens over the cylinder, you take that. So this bit only refers to the things which are sort of non-trivial. Yeah, these are non-trivial. Non-trivial components, yeah. So once here in this compatible reflexive auxiliary norm exists, actually construction is rather easy to take a local model. And then locally you have a retract in the Banach space. It just takes the norm and take a partition of unity times the norm. And then you restrict it to the retract. And if you add it up, it has the size of this property. So if you have to construct such thing for a Banach manifold, for a Banach-Bandor-Banach manifold, it's precisely what you would do there. Is that, might we as well just use the thing you just said all the time instead of the general auxiliary norm? Yeah, so yeah, that is what I mean. Whenever I would construct an auxiliary norm, I would construct it like this. And it would have the property required from a general auxiliary norm. Then if I do the same procedure and have reflexive fiber, this procedure would precisely produce the norms which has the property which I want to have. But it turns out that reflexive auxiliary norms have some additional features which I haven't described. So there's some more properties you can. Because when you have reflexive fiber, you can start talking about reconvergence in the fiber. And that is actually a useful thing. Because the opposite freedom operators have a closed graph. Even if you don't have full convergence, you can see that in some weak sense, certain things would satisfy the equation. And this means you get a better organization of compactness just by that. But as I said, one needs some time to explain it. So once we have such an auxiliary norm, then we can measure the size of a multi-section function as follows. So first of all, since these are functors, I can pass to the orbit spaces. And then you get a function on the orbit space of stable maps just by looking at the maximum of n absolute value where I look at all u for which is non-zero. So basically, I measure the maximum norm of the different branches which I have locally. Can you remind us the SC plus in this? So SC plus meant that this is a section functor, but the local picture is that it's SC plus section. So they go into the 0, 1 fiber. So OK, so here's a local. So let me see if there's any. Hold on. Stay back. There was a dictionary. You should just think of compact perturbation. Yes, but nevertheless, thank you, Katrin. But let me just say that here, so this is the polyfoil bond, the strong bond. So then here you had this filtration where k is less than or equals m plus 1. So for example, the Kuschelman operator goes from the m level to the mm level. But there is one better fiber, m plus 1. So you could look at sections which go here. And these are sort of the compact perturbations because they have the better regularity. And you can go down to the mm level, and then you get compactness. And the SC plus multi sections locally are given by sections of that kind. So then I have locally the sections. And then I precisely, this norm takes real values on this. So I look when I have this picture here. This is O. And this is the local section structure. I'll just look at the maximum norm I see. So this function of z is over. Since isomorphic objects have the same norm, it just is a function telling me what is the maximum norm I see over the isomorphism class. Michael, happy? Well, you look like you were unhappy. So we fix such a norm. And now we also have to exhibit a compatible neighborhood u of the cos-modelized space of geomorphic curves so that we can guarantee certain compactness properties for the perturb problem. So what I want to say, for example, there is a neighborhood of this modelized space. And if I have a perturbation which is supported in this neighborhood, and the norm measurement is small, then the solution space is automatically compact of the perturb thing. Of course, if the neighborhood is too stupid, it is not compatible with all the operations I have. Like if I construct something here and it is in the neighborhood, then I bring it to the boundary. And then this boundary part doesn't belong to the neighborhood of the next level, then I'm stuck. So the neighborhood, so if I have neighborhood of the solution space, say, on the level which never degenerates, then it gives a product on the boundary of the next level. But that should be, in some sense, the trace of the neighborhood which I have there. So that is it. You guys with me? Just a bell of definition. MJ is the one that is actually geomorphic, or the one that is a stator? Yeah, so these are just the origin through the holomorphic objects, yeah? So this is the one for which stator is the 0, 1, 1. Right. And now, so this is a subspace of that, close subspace of this, which on each connected component is compact. So you require as input this geomorphic object? Yes, yeah, OK. So now I want, so now compatibility. So the compatibility is best explained by the hierarchy I use for inductive constructions. So this is a set of connected components of z. Then this splits into a disjoint union of those which never further degenerates. So they only contain building of height 1. They contain building of maximal height 2. So they contain components where the maximal is 2, but there is a building which of height 2, or top floor number 1, and so on. So that is also how the inductions go. Here come objects which you actually can count. And then you start some schemes. So there are some intermediate steps, I think, which would be sort of interesting to see if something at that level works. So this is sort of the mother of all schemes. But I think for certain specific things and subcategories, you might be able to get more structure, but you have to adapt the perturbation thing to this. So this is not written in stone, but that always works. But if you want to have more fine structure, you should on certain subcategories modify this. So then we had this forgetful functor which forgets unstable, forgets trivial cylinder buildings. If you pass, and this actually maps connected components to connected components, actually preserving this decomposition. So I go from a component which I have and some trivial cylinders. I just forget the trivial cylinders. And then in the image, I get another component. And let me write, if u is a subset of that, that is the orbit space of s. Let me write it like this as an index. A is an element here. So a lot of the stuff, I mean, the connected components actually appear as indices in some way. So an open subset is compatible provided. So if after forgetting trivial cylinder components, I get this class, then the open set over a prime of the connected components should be just the preimage of that. So that means the neighborhood of a trivial cylinder is not restricted at all. But I don't have to, because I'm not going to perturb to in the de-bar where this is 0, then I will always find my holomorphic cylinder. So it can be generous on that side. If the objects in A have more than one non-trivial component, then ua is invariant under this local family structure when I move things around against each other. And if A, so for any A, it holds, for the phases of this A, it holds that if I look at the category with objects in U, fiber product is a category object in U, take its orbit space and pull this back by, of course, the image of this is in a subspace of that, and pull back by the covering functor, then I get that. Which basically means if I have an open set here, an open set here, takes a fiber product and view it as a boundary phase, then the neighborhood which I had in the full component restricted to this boundary phase is precisely what I already had, but I lifted up from below. Yeah, is that clear? So in this inductive thing, so you start on level 0, so you have two open neighborhoods of, let's say, of your modelized space. Then you go to the next level, there's a phase. Now the phase pulls back this product of the two neighborhoods, so on the boundary have a product of two neighborhoods. And then this should be the restriction of the neighborhood which I have on that component. OK, so given our reflexive, so this is important, but you can ignore that for most. So I would just say that the statements later would be wrong without that word. So I want to at least make sightable statements. It's a subtle thing you wouldn't realize. I mean, I could tell you, I mean, if I wouldn't have told you, I wouldn't have figured it out. So but I want to say that. So so on the level of lies which I heard in this course here, so I think that's I could have just forgot saying, not mentioned it, but no, I was referring to other speakers. Now, you know, the usual lies which is to say to get a point across, but it's on an approximation. So that is what I meant by that. So I tried, I didn't want to use an approximation here. So so then an open subset. So if you have an open subset of the orbit space, define, so the auxiliary norm is n, define u n to consist of all isomorphism classes of objects, which are belonging to you where the auxiliary norm on the Cauchy Riemann is less than or equal to 1. And the trivial cylinders occurring on any floor are j-halomorphic. So this j-halomorphic cylinders is black stuff. And this thing here has some data on it so that if I apply the Cauchy Riemann operator and evaluate its norm, it's less or equal to 1. So now, is that clear? So you have an open subset of isomorphism classes of objects and auxiliary norm. So now associated to this open set, you just take all objects in this open set for which the trivial cylinders occurring every floor are j-halomorphic. That means if you apply the debar, there's no contribution of a trivial cylinder. Then you apply n, and it should be less or equal to 1. So what is the condition of the trivial cylinders? Can you take a trivial cylinder? What is an open condition? Well, I mean, you'll say it's an open set. No, no, no. u is an open. I give you an open set. I don't say u n is open. u is open. u n is not open. It's definitely not open because of that condition here. And if I put less than not, then I would have to allow basically every cylinder here, or a neighborhood of those guys. Yes, yes, yes, so the thing is that the perturbations I know will precisely produce that, since I will never perturb over this. Yes, yes. So why do you ever then actually put trivial cylinders into the polyphonic if they're not j-trivial or morphic? Because if I have a building, suppose three floors, and then I have a cylinder here, and I do the gluing construction, I need the variation of those guys here, otherwise it's not gluing. So think about this picture here. And then maybe something non-trivial here. Then here is something non-trivial. But here is a trivial cylinder. And then something happens here. When I glue this part here, obviously, in the moment I glue, this thing will be close to this one, but it starts off deviating. So I need function space there. If you glue this in, I mean in the setup. And I need to vary the function here. No, it's glued. So I need this varying things in between here and the function space in the broken one in order that I actually get a frittongue problem. Otherwise you don't have too much enough input. The middle level, if you look at just the middle level, that is a trivial cylinder building on the far right. Oh, and you're saying the boundary stratum of your polyfoam to be a product of the polyfoam that you used before? Yes, yes, of course. Otherwise, OK. So I don't remember what and where can you set up? Well, just as implicit in the setup. No, not simple. They're just when I have, so I always distinguished between trivial cylinder components and j-holomorphic cylinders. The trivial cylinder component is just top and bottom. I have the same thing as homotopic to the j-holomorphic cylinder. I think there's a condition that the map from S to the breaking of it should be a finite problem. And that means you just. Sojective. Yeah, sojective and finite. OK, so now here. So you have the following strong compactness property given the reflexive compatible auxiliary norm. And there exists an open-neighbored U of the coarse-modelized space. So the unperturbed j-holomorphic curved-modelized space such that the closure of this UN, which I just described, has the following property. For every connected component A, the intersection is compact. So you can choose neighborhoods precisely with the properties which I described to you, which live under this inductive process. And you have a guarantee of compactness of the closure of this thing here, which satisfies this. So if you look at all those things, the closure of this associated UN is compact. And you can choose to be a neighborhood which is compatible so it satisfies these conditions. So what does it tell you? That tells you that in your inductive argument, the neighborhoods you are working in then appear as a product are the restrictions of neighborhoods you had already there. And then if you extend then in the perturbation by the construction before, the pullback of the perturbation by the covering transformation because of the definition of the norm maximum of the two things has the same norm on the boundary. So it's less than 1. So suppose the first we take a sequence of epsilon i's less than 1. So starting with epsilon 0, you perturb smaller than epsilon 0, then you get something on the boundary smaller than epsilon 0. Then extend smaller than epsilon 1. Then on the next level you get something smaller than epsilon 1. Extend smaller than epsilon 2, always controlled by this neighborhood so compactness never grows away. So let's see. OK, but I explained that in more detail. So then there is one issue. So let me first say if I have such a neighborhood u which has this property, I say n u as in the theorem that n u controls compactness. Yeah, so I think that's a good word. It controls compactness of the situation. Yeah? Can you say something about why the theorem is true? Yeah, because of reflexive auxiliary norms. It's almost true, but there's a certain problem in the transition when you lift up by the covering and then one extended to the boundary. So it's a little bit of a subtle argument. It's not true. So I wouldn't bet my life on it, neither your life, that such a property holds for arbitrary auxiliary norms. It's possible. Actually, nobody is alive in this room. OK, everybody happy now? OK, so. So now here is one issue which one should point out when one easily forgets about it. So if I have a perturbation below and it's supported in u here and here supported in some u prime, and then I take the product and go on the boundary, this is not supported in u cross u prime. If you take a product of two compactly supported perturbations, the product is generally not compactly supported. The solution set is still the same, because you could be here in the compact support and here outside. It's still non-trivial, the product. Can you draw a picture with your set? Yeah, so if you have a line and this is a support, cross a line where the support is here, then in the plane it looks rather like this. That's the support. But the solution set, of course, for the product still lies in here. So OK, so that's. I mean, if I take a function on one and a function on the other. Yeah, so if the support is here and here, then for the product the support lies in that. If f times f prime is non-trivial. No, no, no, it's a function now from. So you have a function from r into r, cross a function from r into r. So you get something from r2 into r2. Yeah, it's a Cartesian product. So that's what I have. But you're adding the function, right? No, no, no, no, no. I mean, you have a threat on problem here with the compactly supported perturbation, say infinite dimension in here. Then the product threat on problem would not be compactly supported. I mean, everybody likes this fact, so you find it in a lot of papers. But it thinks that compact support is compact support. So everybody is a generally accepted fact. But you have to be a little bit careful. What do you mean by saying that this solution set is convenient? No, the solution set is, of course, a product of the two solution sets. So if the perturbation was compactly supported and forces you to have a solution. So if you know that for each of the freedom problem, the solution set is compact, then the product set is compact. You want to know what that means? OK, suppose you have a map from the real line to the real line and the support is here. Suppose I have another map of this kind with the support last year. Takes the Cartesian product of the two things, you get a map. This means ff of x comma g of y. So it's a map from R2 to R2. So then the support is here, this. And that is precisely what we have here. So now, however, if you explain the notation that's in there, what is n sub lambda? What is t dot lambda? Yeah, I did, actually. There was a definition somewhere. t dot lambda was a rescaling by lambda when you were sort of multiplying by lambda. Oh, it was there. t dot lambda. So here, u n consists of all the morphism classes in u, such that this is true and this thing only has 12 cylinders. Little n sub lambda. Oh, a little n sub lambda. Here, it's just the maximal norm you see for your section. So you have a multi-section functor that goes down to orbit space. And over an object, you look at the maximal norm of one of the branches. And that's invariant under isomorphisms. So you get a function on the orbit space. So now, however, if you are in this situation and you know that this space, so let me give this in the simple case. So suppose your freedom problem is this. And then you take a perturbation. And you know that for this perturbation, the solution set where you allow any t between 0 and 1 is compact. So if you can switch it off, then so there's a little bit stronger assumption than just assuming that this is compact. But of course, it's true in these kind of sets of all ways. Then if you take the product which would not have of the perturbations, you take two such things. Suppose S of x has bounded support. Then of course, this was happened here. But it would be true with two parameters t and s. But because you can homotop away, you can put a bump function front of it, which actually is supported, which is just identically one here. And you cut everything off around it. And it would not influence the solution set. Why does two parameters lie in through that? No, no, that is even unimportant. So what is important is that it means if you put so, if you replace, so now replace, if you take the product, take just put then in front of this perturbation a cutoff function. And suppose the second problem was this, times S prime. So here's x times y. So because of that property here and the control compactness, you can, if you take the set of solutions of this problem for minus 1 to 1, then you can actually find a cutoff function which just on the solution set has to be 1, you can cut it off. So there's a different perturbation which actually would have bounded support. It would have the same solution set and the same linearizations as a solution. So there is this marginal difference. However, if I would require this, then this nice multiplicative property goes away. So if my perturbations are multi, I have a definition that on the fiber product and so on, then it's precisely the one which has this support. But it has the same function, then there's another perturbation inside which would have bounded support. So therefore, that is a little point one has to observe. So without loss of generality, you may assume that as far as compactness is concerned, the support is in you. And then we have compactness. So you need this property here, then this is well behaved. But that precisely we have because I have the statement about UN which precisely says, if this here is positive, which precisely says this. Let's see, where is this? Somewhere. Oh, here. This property. But this means if you perturb this by t times some section which has norm less 1 for t between 0 minus 1, you have that property here. So it's built in. So of course, I could have gotten the way without saying this because nobody really cares about that fact. But it's sort of good to know. But for the following argument, I will always say that the support is in you because I could modify it. So for the following, we assume we have compatible auxiliary norm, we have a compatible neighborhood of this, and you controls compactness. Now since the local models are built on SC Hilbert spaces, we have smooth partitions of unity and so that the supports are locally finite, of course, these things can be used to patch together and construct this SC plus multi-section functions that I explained to you last time, how to construct them. And you can construct them controlling derivatives and whatever you want. So precisely, so whatever you want to do in finite dimension. So the whole whatever you can dream up in finite dimension you can do with transversality you can do on this level. Because you have the same amount of freedom as far as perturbations are concerned. So on the previous slide, when you said N sub lambda, so this is the maximum norm? N sub lambda, yeah. So it means at every point, at every isomorphism class are less than 1. So again, over all of this? No, no, no. If this is less than 1, on what? Everywhere, everywhere, on the whole of S. But in addition, it has this property, which would be precisely, it would be an example here. If I just take the product, that would be a perturbation, which would be non-trivial on a rather large set, but it would have this property. So the end quality on evolving T in lambda, there's just an additional sign of T, isn't it? No, no, that's precisely this problem for each branch. So this is just the rescaling of the branches here. T in front of this. It's not T times lambda. T lambda is the rescaling of the local sections by multiplying them by T. So how does the point of lambda is supposed to be contained in U? Is that not true? No, no, no. This is still addressing the starting point of my discussion. So if you have this property here, then even if you arrive in this bad situation here, you can find a new perturbation, which has precisely the same solution set, with the same linearizations, which has support in U. So when I'm squaring back, so I now forget about this fact, and in my inductive procedure, I will always say the support is in U, which literally is not true, but would have to be true if it's compatible with the other structures. But since you can cut it off, and once you are a little bit inside this thing, then you can just go. So you have this big support here, then you go in, which is on the boundary, then you go inside, you just immediately decrease it to something that you line the set U. OK, so what I just said, transfers are like infinite dimensions in the case of sections. So now here's the perturbation algorithm. So the following result, given a compatible UN, so it's the things we just constructed, they exist for every epsilon between 0, 1, a compatibility SC plus multi-section function. So it's compatible with all the things we ever discussed, here controlled by N and U, satisfying the norm is less than epsilon, so that lambda and d bar are in general position. So that's the basic result. So here comes now the proof. So let's recall this. Then let me define pi 0 less or equals j, just the union of all things which have, so it consists of connected components, but the maximum occurrent flow number is j. So that is how we do the induction. Then of course I allow that, I don't know why I wrote this. Forget about this. So we say A has at least two non-trivial components if there exists an alpha, this d alpha equals 0, and alpha having two non-trivial components, so we know this already. We say two classes are related if the image in pi 0 is the same, that means this thing and this thing just differs by shedding a certain number of cylinders, triple cylinder buildings. Then E less or equals j is just E restricted to this, and this thing here just stands for the union of all. So it's S, the full subcategory associated to all objects which lie in some component, which have an isomorphism class lying in some component here. And lambda j is an SC plus multi-section function defined on this, and the inductive procedure starts, constructs a lambda 0. Then I go to the next step. It extends what I had before and goes to the next level, and so on and so on. OK, inductive statement. Given a strictly increasing sequence epsilon j in 0 epsilon, there exists for every j in SC plus multi-section function having the foreign properties. One level up, it's restricted to the previous level. It's the one which I constructed. The norm of the thing up to level j is less or equals epsilon j. The boundary of the thing on level j is the pullback of the data which I constructed up to level j minus 1, because the boundary faces are coming from stuff below. Then this restricted to Ej should be true. Then this has to be true, which precisely says that I don't perturb over the cylinder components, n is compatible with the local family structure. For some reason, that should be blue is black, I don't know. It's controlled by nu over the. So lambda j, the compactness is controlled over this set over the closed open subset of the orbit space associated to the connected components in j. And these things are in general position over this. So if you can go to the next step, then you have a perturbation which produces compactness and everything. A lot of statements here, but basically it says, OK, so let's start. So the induction first step is induction. So that's easy. So take first a component without trivial. Take first a connected component which does not represent object with trivial cylinder components. So these are the things which cannot decompose further. I have an open neighborhood around the solution set here, which lies in A. So of course I can have only one connected component because if I would have two, non-trivial could shift them against each other and I get boundary strata. I mean, it shifts them so that I get a building with two floors. So I would have boundary strata. So actually, if I exclude trivial cylinder components, I only have one connected component. Then you find a small perturbation lambda A on this thing with norm less than epsilon 0 so that this here over SA is in general position and the support of these sections occurring there is in you, the neighborhood which I constructed. So this is basically the statement. If you have a freedom operator with compact solution set, you take neighborhood of this. You take a perturbation with support in this neighborhood and you wiggle a little bit and it seems becomes transversal. So that's the classical freedom result just like that. So then now take a component which is related to A. So this differs from this by having trivial cylinder components. So this is how you define it. You just take the perturbation which you had on A and the extended so that over a trivial cylinder there's nothing. So now this is obviously transversal as well because the perturbation is transversal over the cylinders because you have automatic transversality and otherwise it's the previous space. So then the support of this lies in you because you intersect with A prime because there was no constraint whatsoever with respect to the cylinders by compatibility condition. So now the collection of these defines lambda 0, 1 and it satisfies ST0. That is straightforward. So now. I'm like, before you move on, can you briefly say what the theorem is saying? Because it was like. Oh, OK. So you are sure. So it says that in a consistent way I can perturb on each connected component. So there is a SC plus multi-section function globally which perturbs the freedom problem in general position and it has the additional compatibility conditions like the data before when it occurs as a fiber product of another component then the perturbation there is a pullback of the stuff which was constructed there. Then if I have a building which has two non-trivial components and I move it against each other and that was a solution, that was a solution, it's also a solution. Then if I have a solution, I add trivial cylinders to it which are geomorphic, it's also a solution. So this is precisely if you have this and now you compose, so if you have this thing and you compose this with the Cauchy-Riemann operator, you get a smooth weighted subcategory which locally can be represented by this file manifold, that is precisely what it says. And these are the things you can integrate over if you have orientations and so on. So that is precisely where you can sort of do what is in most papers when people normally are like we could do with transversality and then they end up at this point, then they say what they would do if things would have been generic. So there's a lot of documentation about what you have to do. So if you have this, you can basically actually take this paper about SFT and just start implementing the things rigorously. Okay, is that clear? Okay, so now. So assume this is constructed with this required property, so the norm is, for example, less or equal to epsilon n. So now we pick a component A, so one level higher for which the objects are connected. So which means there is, if you look at a building of height one, it looks like this, of course on the boundary it might have to compose this but it's still connected, yeah, in the boundary. So now consider the pullback via F theta. These are the covering functors of the previous perturbation. For all theta in the phase of A. So now for each, so the theta, that's, so your, so phase comes from the pullback, comes from a pullback of two problems which are in general position. And then, so for example, if you have two level one buildings, you have two of height one, and they are in general position on the boundary then, then you have still the normal direction to the boundary and you perturb it to make it in general position in the neighborhood with respect to the full space. Actually don't even, so near the boundary you wouldn't even need this because it's already onto there. And then you extend it. So, so they, so if you go, so there's one thing one should point out also that if you are, so there are certain things one should think about it, but they will not cause any difficulty. So, so I started in the induction with the components which don't compose further and obviously I can achieve general position there. Then I take a product of two of such guys. I'm on an honest boundary. So, so if I have a product of two things whose linearization is surjective, then if I just and pull it back, then on the boundary viewed as an operator with respect to the boundaries already surjective. And then you can extend it somewhat nearby would be surjective. And then inside you have to wiggle a little bit to make a surjective there. Then, so now what happens, so what happens if I'm at a corner? So when I have a corner I cannot argue like this. Then this part is restricted, so this is surjective and restricted to this is surjective, but if you linearize with respect to this direction, this direction, you know precisely what the linearization here is. You cannot say now my, I have an additional parameter to help you to make it surjective, so. But, but so you have to look what the linearization is and it's automatically surjective. So, so, so then the restriction to previous things are surjective, so it's in general position, so everything's fine near and you extend it, so. Oh, oh, I'm, so, what, oh, okay. So, I'm astonished how short the proof is. So, so, so then we take the phases and pull back perturbations. So, this first, first without traversing the components, but, okay, so I was here. Extend to related as before, related means I add trivial cylinders. So, now take a component also on the same level without trivial cylinder buildings, but disconnected components. Then you study the phases, it takes a pullback of the perturbations from EP, blah, blah, blah, and it turns out that this thing here, by just, because it can decompose it into disconnected components, which I do, and it just takes a product. That relation was already true on the boundary. It just extended. So then, so then you have to verify that that is smooth, yeah, but sort of, that was smooth, so you just write down what that means. And so then, that's defined. And then, then this buildings here, I extend by adding trivial cylinders to it. So, now, so now here, when I take products and so on, you, so there is this point with what I discussed, this difficulty which discussed here. But if you believe me that I could sort of modify it and bring it into something which has support in you, then this procedure never, will always have multi-sections which have support in you and are small and a priori as a compactness result says that my solution spaces are compact. So then, this is a step and this is an extension of the previous one and so on. So, now you have transversality. So now, remarks, so I started 20 minutes late because of you. Six. Okay, no, no. Okay, so I had actually two endings for my talk and one ending was to show you the homotopy argument which is actually between two perturbations which is actually more interesting than that one. So that's rather straightforward. So you cannot have general position when you homotop from one to the other and the reason is that you have two T dependent. So if you take, if you go from, so on the zero level you can homotop nicely, but you also want to homotop for example like this that if you look at the T projection that you only see more singularities for example. So if I look at the modular space with the most singularities for example. So then you do the product and go to the next level, but then of course the perturbation is the fiber product with respect to T. That causes some constraints in general, so things are not, and that prevents you from putting things into general position because of all the other structures which you have. And the problem are precisely situations like this where you, so also if you have two different components below then of course when the modular space and have T you can talk about that you only see most type singularities and on different components which are really different you can make sure that the critical levels are different by generosity. But if you have a component and you add trivial cylinders to it the most function is the same. So this thing's already produced the same critical point, a critical point, so like this. Suppose you have this thing here, say beta, this is a periodic orbit, gamma. So then you can glue that thing on it and you have a beta here, gamma and you put a cylinder on this. Now you see that this thing plus the cylinder here is related to that object. This differs from this object only by adding a trivial cylinder and gluing it on. So the perturbation you see here is the same perturbation you see here and you build a fiber product that causes immediately difficulties. But the deficiency in the cocoon is one and you have a normal direction if you glue these two things you can kill this but you have to find out what a good perturbation is because when you iterate this process then precisely this kicks in, this problem. So the result of this is that you cannot achieve general position but you can achieve identifiable garbage plus good position and you only have to integrate about good stuff. The other is just, so identifiable garbage is something like this. You might have a solution here but where would the solution space be? It would be something like this, it lies outside. The stuff you see as too small dimension. Whereas other stuff looks like this. Yeah, you can, yeah, you throw the garbage out and yes, that's what we do. I mean, if you're just interested in zero-dimensional components it's what happens is precisely you might have this picture where technically it looks like there would be something lying outside of the space so that doesn't play a role but then all you have something like this which is good. Which of course in general position would have been something like that. So you come out of a corner. Okay, so there you have to actually explain more in the transversality during the homotopy which was one ending but after my disaster's first lecture there only came through half of it. I had to cancel that plan and presumably now I have to cancel that plan here as well because our chairman is really tough. So remarks on, I have to stop? Okay, so here, okay, so let me just make some one minute. We do have a discussion. Right, but I have to prepare everybody for the discussion. So let me just say, so at that point you want to introduce invariance so what you have to do is work out orientation things and you run, you see that there are some issues and the best thing is to go to a covering which is precisely in the SFT paper where you number the punctures, you put asymptotic markers, you put random mark points on the periodic orbits and have some requirement, then you see that you're basically oriented except with the occurrence of bad orbits and then you can write down integrals and then you get the relations by this modular space is how data is related and then you can represent the data in whatever we said there in that paper. That's a remark. Great questions. Yes, so critical cylinder buildings, right? So you're saying you want an ampere 40-fold in which the constant cylinders sit. So in my world, what I'm hearing when I hear that is you want local slices to the R action on maps from R times S1 to a manifold M near the ones that happen. I mean, you definitely have to put, okay. So when I look at stable maps, then I would have maps from... Oh, the question is what is your ambient polyfold for just for the trivial cylinder? It doesn't. It never occurs by itself. It only occurs in combination with something else. But there is one, right? And it's sort of... No, but the construction you have is not a product construction. So you cannot have a trivial cylinder by itself because it's an unstable object. So moment, but a trivial cylinder occurs always together with a non-trivial component. And then when I construct a global slice, it's not a product of a construction on a cylinder and the other one. It takes data from both and mixes them up. It's not a product. So therefore, trivial cylinders never appear alone because they are unstable. They can only appear together with a non-trivial object. And then when I make a slice with respect, because when you make a slice with respect to the R action then you fix say the non-trivial part but you can slide the cylinder against it. Yes, yes, and that has to be built in but it's not a product situation. Because you also have to divide, you have to take a transversal constraint, you have to divide out the automorphism group for the cylinder as well. So there are several things which you have to do at the same time, which go into the construction of the uniformized. So let me get this right. You're saying there's not an ambient M-polypone in which the trivial cylinder sits. No. Okay, that's just what we need to break out of. I'm just going to push this over here.