4/4 Polyfolds and the construction of Symplectic Field Theory
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Hofer, Helmut
Topics: 1 Perturbation algorithm in the homogeneous case. 2 Perturbation algorithm for relating two perturbations. 3 Remarks on orientations. 4 Representation theory and SFT.
 The last lecture. Okay, so in lecture one to three, we have described how the algebraic, so we've introduced a category of stable maps, a bundle category over it, Kuschelrieman section functor. We have described the algebraic relationships and we have put smooth structures on this and sort of the packaging is essentially this of the SF, of when you want to resolve the transversality questions for SFT. So you have this diagrams of covering of actually is it now smooth covering, SC smooth covering functors, some other relationships formalized by having some functors which behave in some compatible way. This is the Kuschelrieman operator. So now, so the goal then is so in so now we have a complete setup to perturb the, the cat is a subcategory of sort of J holomorphic objects into some other sub weighted subcategory, which actually will be smooth as C smooth. So that will be enough to extract 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 concerned. Now, if you think of this here locally as a finite number of sections, you know, so you get a finite number of equations and you want that each of them is in general position, which is, can be achieved for this. It cannot be achieved when you homotope one 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, that's still possible. So now, 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. Yeah? And we perturb 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, so now, in order to actually make sure that that is the case, you ideally distinguish something like a convex set of perturbations that you can go around where it's also clear that, up here, clear that you would have compactness. So let me explain here. So, so we will actually be able to make an infinite construction and there's some kind of an inductive procedure to, 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, the, 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, then you have to distinguish between so, connected components alone together with, and others with added cylinders and so on. So there's a certain number of things which you have to do on this, 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, so now if I only can control things in the neighborhood of the, 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, you perturb up to some level, you know, 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. So, but it turns out that in this, in this problem actually, and that has, 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 Kurscher 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 non-trivial effect. So you can make that infinite construction, get that infinite smooth object. Okay. So that of course could be used the other approaches. So this finer compactness it could be used in the other approaches as well, so there wouldn't, there wouldn't be the need to do it. Okay. So, so in order to deal with this aspect of the story, so you have to measure perturbation, you have to measure the size of perturbations and you have to come up with some idea how that com, controls compactness. Okay. So, so we have heard what an auxiliary norm is in the case of an M-polyphol set up. So here it's basically a functor. So it's actually defined on, on the things which have a little bit better quality on the, in the fiber. But, but if they're not lying in this space, we just put it plus infinity. So, 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 pro, property. So if I take a sequence of objects so that n goes to zero, so that's a num, the sequence of numbers. And if the, if the isomorphism class of the base point below converges, then the isomorphism class of this vector converges to the isomorphism class of zero over that point. So that's a requirement. It's some kind of local uniformity. If, so you have, so you take a sequence here. So if this is finite here, that means you like actually here. So then the size goes to zero. So this is just sequence of objects, but now you look at the base of this object. So 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 zero 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, one has, 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. So, so, so then you can always get this. So, so there's a, one can talk about, we can presumably talk about an hour about the compactness issues. So then it should be compatible. So, so first of all you define, if you have an auxiliary norm you can define this on a 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. It's 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 photon property has, has allows you to disperse that control a little bit to a neighborhood because locally looks like some finite dimensional times contraction. So if you're not too far away you, you still have the contractive parts. So, so it's still essentially looks finite dimension 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 zero one form on the non-trivial cylinder, on the trivacinitinal composition either zero or the other way around. Remember that was one of our structural things. You want to have this. So it measures the size on non-trivial of a zero one form on a non-trivial, on 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 zero. What is this E subscript zero? Oh, this is E. Oh, yeah. So remember for strong, a strong bundle K over O has this by filtration Mk where K is less or equals M plus one. And if you have the strong bundle structure on this and this E over C also gets a by filtration. So on the stable maps you can talk about the regularity of the object and then in the file, then you can, and if you have something in the, in the, the category of firing of it, you can talk about the regularity of the zero one form over the underlying object and that has an addition. So why do you allow the thing to actually go there? Infinity or E? Yeah, because this is only, it's precisely finite if it's here, which is a sub, a subfiber. Can you say something about why is that? Yeah, so, so remember that as, as I think Katrin pointed this out that for strong bundle there is an, it's not just a bundle but there's an additional, this by filtration and if you lie on the one level it's sort of compact perturbations and the SC plus sections are, are sections of the one level. So that is where, where you know that things are compact. So you, so in the fiber you have zero level, one level and so on. But if you are on the one level you go down by compactness on the zero level and you perturb by sections which lie on the one level. And that is precisely there are sections of this. The multi-section funtors are priori are given by sections of this but the images of this section actually lie in that better part of the space. So that is, that is how compact perturbations built into the whole thing. And then, and it should have this property. So if you forget about the stuff over trival cylinders and fix the norm or you move, or you even throw the component away, which of course is more or less, you don't need the trival cylinder component then, then you see the same measurement. So then of course you want, if you have, if you have disconnected conf, non-trival configuration and move them against each other then you would see this. The measurement of those guys, we are maximum. Doesn't that contradict the thing you have a problem about n equals n equals to pi plus n equals to w times? That's a plus right there. Well, a plus, no. So, no, why? So, so if you mean this or that. Oh, no, no, no, no, no, no, no. So this are non-trival components. So then, then you take this and then there would be some cylinders here which it discard and some cylinders which you, 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 is sort of non-trivial. Yeah, these are non-trivial. Yeah, so non-trivial components. So once here in this compatible reflexive auxiliary norm exists, actually construction is rather easy to take a local model and then you, you, you use 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 precisely this property. So it's, if you have to construct such thing for a Banach manifold, for a Banach-Bandau-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 I have reflexive fiber, this procedure would precisely produce the norms which has the property which I want to have. So, but, 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 the convergence in the fiber. And that is actually useful thing. Because the, the opposite, freedom operators have sort of a closed graph. And then you can, 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 better organization of compactness just by, by that. But as I said, one needs some time to explain it. So once you 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 functions I can pass to the orbit spaces. And then you get a, 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 norm of the, the maximum norm of the different branches which I have locally. Can you remind me what SC plus means? Okay. So SC plus meant that this is a section function, but the local picture is that it's SC plus section. So they go into the zero one fiber. So, so, okay, so, okay, so, okay, so here's the local, so here's, so let me see. Hold on, there was a dictionary, you should just think of compact perturbation. Yes, so, but nevertheless thank you, Katrin, but let me just say that, that here, so this is the polyphonic bond, the strong bond. So then here you had this filtration where k is less than equals m plus one. So, so, so for example, the Kuschelman operator goes from the m level to the m level, to the m, m level, but there is one better fiber, m plus one. So you could, you could look at sections which go here and this is sort of the compact perturbations because they have the better regularity and you can go down to the m, m level and then you get compactness. And the SC plus multi sections locally are given by sections of that kind. So they're really, so, so then I have locality sections and then I precisely, this norm takes real values on this. So I look, look when I, when I have this picture here, this is, and this are the local section structure, I'll just look at the maximum norm I see. So, so this function of z is over, since isomorphic objects have the same norm, it just is a function of telling me what, what is the norm, the maximum norm I see over the isomorphism class. Michael, happy? Well, you look like you were unhappy. So, so given it, so, so we fix such a norm and now we also have to exhibit a compatible open neighborhood u of the course modelized space of geomorphic curves so that we can guarantee certain compactness properties for the particular problem. So, so I want to say, 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, the norm measurement is small, then the solution space is automatically compact of the perturbs. 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, so if I have a 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. Yeah? So, guys with me? Just a level of definition. MJ is the one that is actually J-holomorphic or the one that is a stator? Yeah, so these are just the origin pseudo-holomorphic objects, yeah? So this is the one for which stator is the zero one? Right. And now, so this is a subspace of that, a closed subspace of this which on each connected component is compact. So you require as input this ground-mortar? Yes. Yeah? Okay. So now, I want, so now compatibility. So I have to, 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 degenerate, so they only contain building of height one. They only contain, they contain building of maximal height two. But so they contain components where the maximal is two, but there is a building which of height two or top floor number one and so on, yeah? 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 is sort of interesting to see if something at this, at that level works. So, so just, 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 out, but you have to adapt the perturbation thing to this. So, so this is not written in stone, but that is, that always works, but if you want to have more fine structure, you should on certain subcategories modify this. So then, then we know that we have, then we have 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, well, I go from a component which I have, so, 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, of that, that is the orbit space of s, let me write it like this as an index. A is an element here. So, so, so a lot of this stuff, I mean, the connected components actually appears indices in some way. So, so an open subset is compatible provided. So if, 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, not restricted at all. But I don't have to because I'm not going to perturb to, in the debaubo this is zero, then I will always find my holomorphic cylinder. So, 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, if I take, 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, and u would take, take its orbit space and pull this back by, of course the image of this is in a, in a subspace of that and pull back by the covering functor, then I get that. Which basically means if, if I have an open set here, an open set here, takes a fiber product and view it as, 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. Construct, but I lift it up from below. Yeah? Is that clear? So in this inductive thing, so you start on level zero. 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, on the boundary, you have a product of two neighborhoods. And then this should be the restriction of the neighborhood which I have on that component. Okay. Okay, so given, given our reflexive, so this is important, but you can ignore that for most. So I would just say that the statement 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, you wouldn't have figured it out. But, but I want to say that. 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 mention it, but. You're denying to it? Huh? You're denying to it? No. I was referring to other speakers. Right. Now, you know, the usual lies which is to say to get a point across, but, you know, it's on an approximation. So that is what I meant by that. So I, I, 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 Kusch-Riemann is less or equal to one. And the trivial cylinders occurring on any floor are j-halomorphic. So this are j-halomorphic cylinders, this black stuff here. And this thing here, the, has some data on it so that if I apply the Kusch-Riemann operator and, and evaluate its norm, it's less or equal to one. Yeah? So now, the bay, is that, 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 one. So how much, so like, what is that, the condition of the trivial cylinders? Can you take a trivial cylinder? What is an open condition? Well, I mean, you're saying it's only set. I don't say, no, no, no, you is an open, I give you an open set, I don't say UN is open. Ah. You is open, UN is not open, it's definitely not open. Okay. Oh, well, because it's less than one. Because of that condition here. Yeah, but even, even if you, less than one. And if I put, if I put less than not, then I would have to allow basically every cylinder here. Yeah, that. Or a neighborhood of those guys. I'm not saying, can I not allow them to do that. Yes, yes, so, so the thing is that the perturbations I know will precisely produce that, since I will never perturb. You're not to turn. Over, over, yes. So you might, so. Yes. So. Okay. So why don't you, why do you ever then actually put trivial cylinders into the polyfoil, if they're not J-tool or more like? Because if, 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, there's no gluing. Yes. Yeah, so, if, so think about this picture here. So 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 has to, 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, in, in this, in the setup. And they, I need to vary the function here. Huh? No, it's glued. No, this is now. So I need this varying things in between here and the function space in the broken one in order that I actually get a Fritton problem. Otherwise, you don't have too much, enough input. Yeah. 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 you want your poly, the boundary stratum of your polyfoam to be a product of the polyfoam that you used before. Yes. Yes, of course. How does it, okay. I don't remember. What, and where, where can you set up? Well, just as implicit in the setup. Not, not, not simple. They're just, when I, when I have, so I always distinguish between trivial cylinder components and J-holomorphic cylinders. The trivial cylinder component is just top and bottom, have the same thing as homotopic to the, to a 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 in this. Ah, subjective. Yeah, subjective. Okay. Okay, so now here, so, so you have the foreign stock, strong compactness property, given the reflexive compatible auxiliary norm, and there exists an open-neighborhood U of the coarse-modelized space, so the unperturbed J-holomorphic curve-modelized space, such that the closure of this UN, which I just described, has a following property. For every connected component A, the intersection is compact. Yeah. So, so you can choose neighborhoods precisely with the properties which I described to you, which live under this inductive process, and you have compact, guarantee of compactness of the closure of this thing here, which satisfies this. If you look at all those things, the closure of this associated UN is compact, so, and you can choose to be a neighborhood which is compatible, so, which is compatible, so it satisfies these conditions. So, so that, 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, and the perturbation by the, by the construction before, the, 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 one. So suppose the first, we take a sequence of epsilon i's less than one on the, so starting with epsilon zero, you perturb smaller than epsilon zero, then you get something on the boundary smaller than epsilon zero, then extend smaller than epsilon one, then on the next level you get something smaller than epsilon one, extend smaller than epsilon two, always controlled by this neighborhood, so compactness never grows away. So let's see. But I explained that in more detail. So, so then there is one issue. So let me first say, if I have such a neighborhood, you, which has this property, I say, and you, as in the theorem that and you controls compactness. Yeah, so I think that's a good word. It controls compactness of, 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 as a covering and then one extended to the boundary. So you need some, it's a little bit of a subtle argument. It's not true. So I, I wouldn't bet my life on it, neither your life, that such a property holds for arbitrary auxiliary norms could be possible. Actually nobody's life in this room. Okay. Everybody happy now? Okay. So, so now here, here's one issue, which one should point out when it went easy forgets about it. So if I have a perturbation below and it's, it's supported in you here and here supported in some U prime and then I take the product and go on the boundary. This is not supported in you 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 the compact support and here outside. It's a non, it's still non-trivial, the product. Can you draw a picture with me? Yeah. So, so if you have a line and this is a support, cross a line, there's a support is here, then the plane, it looks rather like this, that's a support. But the solutions, the solution set of course for the product still lies in here. So okay. So that's, I mean, if I take a function on one and a function on the other side. Yeah. So, and if the support is here and so if the support is here and here, then for the product the support lies and that's this cross. If f times f prime is not, is long enough? No, no, no, it's a function now from, so, so you have a function from r into r, cross a function from r into r. So you get, you get something from r2 into r2. It's a Cartesian product. Yeah, it's a Cartesian product. Yeah, so that's what I have. But it's like you're adding the function, right? No, no, no, no, no. I mean, you have a, you have a, you have a threat, you have a threat on problem here with a compactly supported perturbation, say infinite dimension in here. And the product, 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, but it's like, like sort of thinks that compact support is compact support. So, so everybody, it's a generally accepted fact, yeah. But you have to be a little bit careful. Where do you imagine that this solution set is contained? 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, yeah, so if you know that for each of the threat on problem, the solution set is compact, then the product set is compact. Again, what that picture means? You want to know what that means? You put all those pictures in. Okay. 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 where the support lies here. Takes a Cartesian product of the two things. You get a map. This means f of x, g of y. So it's a map from R2 to R2. So then, then the support is here. And that is precisely what we have here. So, so now, however, if- Sorry, can you explain the notation that's in there? What is n sub lambda? What is t dot lambda? Yeah, this, yeah, I did, actually. There was a definition somewhere. T dot lambda was a rescaling by lambda when you, when you were sort of last time by lambda. So it makes a score. Oh, it was, it was there. Oh. It's t dot lambda. So it's, ah, here, u n. Consists of all the morphism classes in u. Such a delus true and this thing only has 12 cylinders. A little n sub lambda. Oh. A little n sub lambda. Here it's just the, the maximal norm you see for your section. So you have a multi-section functor that, that goes down to orbit space and you just look at, over an object you look at the maximal norm of one of the branches and that's invariant under isomorphisms. So you, so you get a function on the orbit space. So, ah, so now, however, if, if you are in this situation and you know that you, that, that this space, so if, so let me give this in the, in the simple case. So if you, so, 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, any t between 0 and 1 is compact. So if you can switch it off, then if you, then, so there's a little bit stronger assumption than just assuming that this is compact, but of course it's true in, in this kind of sets of, always. Then if you take the product which, which would not have of the perturbations, it takes two such things suppose S of t has bounded support, S of, 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 then, but you can put a bump function in 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 this two parameters are in through that? No, no, just that is even unimportant. So what is important is that it means if you put, so, if you replace, so, so now replace if you take the product, yeah, take, just put then in front of this perturbation a cutoff function and of the, 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, the set of solutions of this problem for minus one to one, then you can actually find a cutoff function which just on the solution set has to be one, you can cut it off. So, so there is a, there's a, there's a different perturbation which actually would have bounded support. It would have the same solution set and the same linearizations as the solution. So, so there is this marginal difference. However, if I would require this, then this nice multiplicative property goes away. So I, if my perturbations are multi, I have a definition that on the, on the fiber product is the product and so on. And it's precisely the one which has this support. But, but it has the same, it has the same function then there's another perturbation inside if, which would have bounded support. So, so therefore that is a little point one has to, 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, okay. So, so you need this property here then this is well behaved. Okay, so, so, but that precisely we have because I have estimates for my, 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 one for t between zero minus one, you have that property here. Yeah? So, so it's built in. Okay, so, so of course I could have gotten the way without saying this because nobody really cares about that fact, but, but it's sort of good, good, good to know. So, 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, this things can be used to patch together and construct this SC plus multi-section function 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. So, because you have the same amount of freedom as far as perturbations are concerned. So, 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 one. So again, over all of this. No, no, no, no, if this is less than one. On what? Everywhere. Everywhere, everywhere, everywhere, on the whole of S. And, but, and in addition it has this property. Which would be precisely, I would be an example here if I just takes a product that would be a perturbation which would be non-trivial on a, on a rather large set but it would have this property. So, the, the, the, the inequality on evolving t in lambda, there's just another sign of t, isn't there? No, no, no, 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, just t lambda is, is a rescaling of, 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, 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, which has support in u. Yeah. So, so, so, so now I'm squaring back, so, 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. Then it would, but, but you can, but since you sort of can cut it off and once you are a little bit inside this thing, then you can just go, yeah, so, 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 your line just said u. Okay. So, now, so what I just said, like infinite dimensions in the case of sections. So, now here's the perturbation algorithm. So, the following result, given a compatible u n, so, these are things we just constructed, they exist for every epsilon between 0 and 1, a compatibility s c plus multi-section functions, this compatibility is 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, 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 maximal current 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 they exist in 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, trivial cylinder buildings. Then e less or equal to j is just e restricted to this and this thing here is just stands for the union of all. So s, 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. Okay, inductive statement, given a strictly increasing sequence epsilon j in 0 epsilon, they exist for every j and 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 one because the boundary phases 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 and is compatible with the local family structure for some reason that should be blue is black, I don't know, is controlled by n u 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 have an induct, if you can go to the next step, then you have a perturbation which produces compactness and everything. Okay, a lot of statements here but basically it says, okay, so let's start. So the induction first step is induction, okay, so that's easy. So take first a component without triv- take first a connected component which does not represent object with trival 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-trivia could shift them against each other and I get boundary strata. I mean, shifts them so that I get a level, I get a building with two floors, so I would have boundary strata. So what I actually, if I exclude trival 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 S A is in general position and the support of the sections occurring there is in U, the neighborhood which I constructed. So this is a basic of the statement. If you have a freedom operator with compact solutions set, you take neighborhood of this, you take perturbation with support in this neighborhood and you wiggle a little bit and this thing 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 trival cylinder components. So this is how you define it. You just take the perturbation which you had on A and on the extended so that over a trival cylinder there's nothing. Okay, 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 U because U intersects 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 01 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... Okay, so you are sure. Oh my God. Yeah. So it says that in a consistent way I can perturb on each connected component. So there is an 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 finite 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 SF-t 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 equals 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 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 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 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 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 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 it would be surjective and then inside you have to wiggle a little bit to make it surjective there. Then so now what happens if I'm at a corner. So when I have a corner I cannot argue like this then this part is restricted to this 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 I have an additional parameter to help you to make it surjective. But so you have to look what the linearization is and it's automatically surjective. So then the restriction to previous things are surjective so it's in general position so everything's fine near and you extend it. Oh, oh, I'm sorry. Okay so I'm astonished how short the proof is. So then we take the phases and pull back perturbations. So this first is our trivial cylinder 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 as our 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 you can decompose it into disconnected components which I do and it just takes a product that relation was already true on the boundary. So just extend it. 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 this buildings here I extend by adding trivial cylinders to it. So now here when I take products and so on so there is this point with what I discussed here, this difficulty which discussed here but if you believe me that I could sort of modify it and bring it into something which has supported you then this procedure never will always have multi sections which have supported 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. 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, 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 modellized space which is where you want the more singularities for example. So then you do the product and go to the next level but then of course the perturbation is a fireball 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, then of course this, when the modellized space and have T you can talk about that you only see more 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 already produces 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 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 fireball 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 is too small a mention. This other stuff looks like this. So you just throw the garbage out? You can, you throw the garbage out and yes, that's what you do. 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. So I have to cancel that plan and presumably I have to cancel that plan here as well because our chairman is really tough. So the marks on. After stop. Yeah okay. So here, so let me just make some one minute. We 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 an oriented except with the occurrence of bad orbits. And then you can write down integrals and then you get the relations by this, the model of 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. Quick questions. Yes. So, you're saying you want an amniotic form in which the, the, the, the, the, the, the constant cylinders. 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 have. I mean you, you, you definitely have to put, okay. So, so, so when I, when I look at stable maps then I would have maps from. The question is what is the ambient polyfold for just for the trivial cylinder? It doesn't. It only, 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, but, but, but the construction you have is not a product construction. So you cannot, you cannot have a trivial cylinder by itself because it's an unstable object. So, so, so, so, moment, but a trivial cylinder occurs always together with a non-trivial component. And then when I, when I construct a global slice, it's not a product of a construction on a cylinder and the other one. It's a more, it takes both, it takes data from both and mixes them up. It's not a product. So, 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 the slice with respect, because when you make a slice with respect to the R action, then you, you fix say the non-trivial part, but you can slide the cylinder against it. Exactly. 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 make, you have to take a transversal constraint. You have to divide out the automorphism group for the cylinder as well. So, so there are several things which I have to do at the same time, which go into the construction of the uniformizing. So, so let me get this back to you saying there's not an ambient and polyphonic in which just the trivial cylinder sits. No. All right. So, so, all right. Okay, I suggest that we break out of our precious rubric. Thank you.
 The last lecture.  Okay, so in lecture one to three, we have described how the algebraic, so we've introduced  a category of stable maps, a bundle category over it, Kuschelrieman section functor. We  have described the algebraic relationships and we have put smooth structures on this  and sort of the packaging is essentially this of the SF, of when you want to resolve the  transversality questions for SFT. So you have this diagrams of covering of actually is it  now smooth covering, SC smooth covering functors, some other relationships formalized by having  some functors which behave in some compatible way. This is the Kuschelrieman operator. So  now, so the goal then is so in so now we have a complete setup to perturb the, the cat is  a subcategory of sort of J holomorphic objects into some other sub weighted subcategory, which  actually will be smooth as C smooth. So that will be enough to extract 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 concerned. Now, if you think of this here locally as  a finite number of sections, you know, so you get a finite number of equations and you  want that each of them is in general position, which is, can be achieved for this. It cannot  be achieved when you homotope one 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, that's still possible. So now,  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. Yeah? And we perturb 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, so now, in order to actually make sure  that that is the case, you ideally distinguish something like a convex set of perturbations  that you can go around where it's also clear that, up here, clear that you would have compactness.  So let me explain here. So, so we will actually be able to make an infinite construction and  there's some kind of an inductive procedure to, 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, the, 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, then you have to distinguish between  so, connected components alone together with, and others with added cylinders and so on.  So there's a certain number of things which you have to do on this, 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, so now  if I only can control things in the neighborhood of the, 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, you perturb up to some level, you know, 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. So, but it turns out that in this, in this problem actually, and  that has, 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 Kurscher 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 non-trivial effect.  So you can make that infinite construction, get that infinite smooth object. Okay. So  that of course could be used the other approaches. So this finer compactness it could be used  in the other approaches as well, so there wouldn't, there wouldn't be the need to do  it. Okay. So, so in order to deal with this aspect of the story, so you have to measure  perturbation, you have to measure the size of perturbations and you have to come up with  some idea how that com, controls compactness. Okay. So, so we have heard what an auxiliary  norm is in the case of an M-polyphol set up. So here it's basically a functor. So it's  actually defined on, on the things which have a little bit better quality on the, in the  fiber. But, but if they're not lying in this space, we just put it plus infinity. So, 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 pro, property. So if I take a sequence  of objects so that n goes to zero, so that's a num, the sequence of numbers. And if the,  if the isomorphism class of the base point below converges, then the isomorphism class  of this vector converges to the isomorphism class of zero over that point. So that's a  requirement. It's some kind of local uniformity. If, so you have, so you take a sequence here.  So if this is finite here, that means you like actually here. So then the size goes  to zero. So this is just sequence of objects, but now you look at the base of this object.  So 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 zero 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, one has, 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. So, so, so then you can always get this. So, so there's  a, one can talk about, we can presumably talk about an hour about the compactness issues.  So then it should be compatible. So, so first of all you define, if you have an auxiliary  norm you can define this on a 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.  It's 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 photon  property has, has allows you to disperse that control a little bit to a neighborhood because  locally looks like some finite dimensional times contraction. So if you're not too far  away you, you still have the contractive parts. So, so it's still essentially looks  finite dimension 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 zero one form on the non-trivial  cylinder, on the trivacinitinal composition either zero or the other way around. Remember  that was one of our structural things. You want to have this. So it measures the size  on non-trivial of a zero one form on a non-trivial, on 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 zero. What is this E subscript zero? Oh, this is E.  Oh, yeah. So remember for strong, a strong bundle K over O has this by filtration Mk  where K is less or equals M plus one. And if you have the strong bundle structure on  this and this E over C also gets a by filtration. So on the stable maps you can talk about the  regularity of the object and then in the file, then you can, and if you have something in  the, in the, the category of firing of it, you can talk about the regularity of the zero  one form over the underlying object and that has an addition.  So why do you allow the thing to actually go there? Infinity or E?  Yeah, because this is only, it's precisely finite if it's here, which is a sub, a subfiber.  Can you say something about why is that? Yeah, so, so remember that as, as I think  Katrin pointed this out that for strong bundle there is an, it's not just a bundle but there's  an additional, this by filtration and if you lie on the one level it's sort of compact  perturbations and the SC plus sections are, are sections of the one level. So that is  where, where you know that things are compact. So you, so in the fiber you have zero level,  one level and so on. But if you are on the one level you go down by compactness on the  zero level and you perturb by sections which lie on the one level.  And that is precisely there are sections of this. The multi-section funtors are priori  are given by sections of this but the images of this section actually lie in that better  part of the space. So that is, that is how compact perturbations built into the whole  thing. And then, and it should have this property. So if you forget about the stuff over trival  cylinders and fix the norm or you move, or you even throw the component away, which of  course is more or less, you don't need the trival cylinder component then, then you see  the same measurement. So then of course you want, if you have, if you have disconnected  conf, non-trival configuration and move them against each other then you would see this.  The measurement of those guys, we are maximum.  Doesn't that contradict the thing you have a problem about n equals n equals to pi plus  n equals to w times? That's a plus right there.  Well, a plus, no. So, no, why? So, so if you mean this or that.  Oh, no, no, no, no, no, no, no. So this are non-trival components.  So then, then you take this and then there would be some cylinders here which it discard  and some cylinders which you, 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 is sort of non-trivial.  Yeah, these are non-trivial. Yeah, so non-trivial components.  So once here in this compatible reflexive auxiliary norm exists, actually construction  is rather easy to take a local model and then you, you, you use 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 precisely this property.  So it's, if you have to construct such thing for a Banach manifold, for a Banach-Bandau-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 I have reflexive fiber, this procedure would precisely  produce the norms which has the property which I want to have.  So, but, 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 the convergence in the fiber.  And that is actually useful thing.  Because the, the opposite, freedom operators have sort of a closed graph.  And then you can, 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 better organization  of compactness just by, by that.  But as I said, one needs some time to explain it.  So once you 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 functions I can pass to the orbit spaces.  And then you get a, 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 norm of the, the maximum norm of the different branches which  I have locally.  Can you remind me what SC plus means?  Okay.  So SC plus meant that this is a section function, but the local picture is that it's SC plus  section.  So they go into the zero one fiber.  So, so, okay, so, okay, so, okay, so here's the local, so here's, so let me see.  Hold on, there was a dictionary, you should just think of compact perturbation.  Yes, so, but nevertheless thank you, Katrin, but let me just say that, that here, so this  is the polyphonic bond, the strong bond.  So then here you had this filtration where k is less than equals m plus one.  So, so, so for example, the Kuschelman operator goes from the m level to the m level, to the  m, m level, but there is one better fiber, m plus one.  So you could, you could look at sections which go here and this is sort of the compact perturbations  because they have the better regularity and you can go down to the m, m level and then  you get compactness.  And the SC plus multi sections locally are given by sections of that kind.  So they're really, so, so then I have locality sections and then I precisely, this norm takes  real values on this.  So I look, look when I, when I have this picture here, this is, and this are the local section  structure, I'll just look at the maximum norm I see.  So, so this function of z is over, since isomorphic objects have the same norm, it just is a function  of telling me what, what is the norm, the maximum norm I see over the isomorphism class.  Michael, happy?  Well, you look like you were unhappy.  So, so given it, so, so we fix such a norm and now we also have to exhibit a compatible  open neighborhood u of the course modelized space of geomorphic curves so that we can  guarantee certain compactness properties for the particular problem.  So, so I want to say, 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, the norm measurement is small, then the solution space is automatically compact  of the perturbs.  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, so if I have a 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.  Yeah?  So, guys with me?  Just a level of definition.  MJ is the one that is actually J-holomorphic or the one that is a stator?  Yeah, so these are just the origin pseudo-holomorphic objects, yeah?  So this is the one for which stator is the zero one?  Right.  And now, so this is a subspace of that, a closed subspace of this which on each connected  component is compact.  So you require as input this ground-mortar?  Yes.  Yeah?  Okay.  So now, I want, so now compatibility.  So I have to, 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 degenerate, so they only contain building of height one.  They only contain, they contain building of maximal height two.  But so they contain components where the maximal is two, but there is a building which of height  two or top floor number one and so on, yeah?  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 is sort of interesting to see if something  at this, at that level works.  So, so just, 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 out, but you have to  adapt the perturbation thing to this.  So, so this is not written in stone, but that is, that always works, but if you want to  have more fine structure, you should on certain subcategories modify this.  So then, then we know that we have, then we have 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, well, I go from a component which I have, so, 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, of that, that is the orbit space of s, let me write  it like this as an index.  A is an element here.  So, so, so a lot of this stuff, I mean, the connected components actually appears indices  in some way.  So, so an open subset is compatible provided.  So if, 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, not restricted at all.  But I don't have to because I'm not going to perturb to, in the debaubo this is zero,  then I will always find my holomorphic cylinder.  So, 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, if I take, 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, and u  would take, take its orbit space and pull this back by, of course the image of this  is in a, in a subspace of that and pull back by the covering functor, then I get that.  Which basically means if, if I have an open set here, an open set here, takes a fiber  product and view it as, 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.  Construct, but I lift it up from below.  Yeah?  Is that clear?  So in this inductive thing, so you start on level zero.  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, on the boundary, you have a product of two neighborhoods.  And then this should be the restriction of the neighborhood which I have on that component.  Okay.  Okay, so given, given our reflexive, so this is important, but you can ignore that for  most.  So I would just say that the statement 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, you wouldn't have figured it  out.  But, but I want to say that.  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 mention it, but.  You're denying to it?  Huh?  You're denying to it?  No.  I was referring to other speakers.  Right.  Now, you know, the usual lies which is to say to get a point across, but, you know, it's  on an approximation.  So that is what I meant by that.  So I, I, 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 Kusch-Riemann is less or equal to one.  And the trivial cylinders occurring on any floor are j-halomorphic.  So this are j-halomorphic cylinders, this black stuff here.  And this thing here, the, has some data on it so that if I apply the Kusch-Riemann operator  and, and evaluate its norm, it's less or equal to one.  Yeah?  So now, the bay, is that, 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 one.  So how much, so like, what is that, the condition of the trivial cylinders?  Can you take a trivial cylinder?  What is an open condition?  Well, I mean, you're saying it's only set.  I don't say, no, no, no, you is an open, I give you an open set, I don't say UN is  open.  Ah.  You is open, UN is not open, it's definitely not open.  Okay.  Oh, well, because it's less than one.  Because of that condition here.  Yeah, but even, even if you, less than one.  And if I put, if I put less than not, then I would have to allow basically every cylinder  here.  Yeah, that.  Or a neighborhood of those guys.  I'm not saying, can I not allow them to do that.  Yes, yes, so, so the thing is that the perturbations I know will precisely produce that, since  I will never perturb.  You're not to turn.  Over, over, yes.  So you might, so.  Yes.  So.  Okay.  So why don't you, why do you ever then actually put trivial cylinders into the polyfoil,  if they're not J-tool or more like?  Because if, 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, there's no gluing.  Yes.  Yeah, so, if, so think about this picture here.  So 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 has to, 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, in, in this, in the setup.  And they, I need to vary the function here.  Huh?  No, it's glued.  No, this is now.  So I need this varying things in between here and the function space in the broken one in  order that I actually get a Fritton problem.  Otherwise, you don't have too much, enough input.  Yeah.  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 you want your poly, the boundary stratum of your polyfoam to be  a product of the polyfoam that you used before.  Yes.  Yes, of course.  How does it, okay.  I don't remember.  What, and where, where can you set up?  Well, just as implicit in the setup.  Not, not, not simple.  They're just, when I, when I have, so I always distinguish between trivial cylinder components  and J-holomorphic cylinders.  The trivial cylinder component is just top and bottom, have the same thing as homotopic  to the, to a 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 in this.  Ah, subjective.  Yeah, subjective.  Okay.  Okay, so now here, so, so you have the foreign stock, strong compactness property, given  the reflexive compatible auxiliary norm, and there exists an open-neighborhood U of the  coarse-modelized space, so the unperturbed J-holomorphic curve-modelized space, such  that the closure of this UN, which I just described, has a following property.  For every connected component A, the intersection is compact.  Yeah.  So, so you can choose neighborhoods precisely with the properties which I described to you,  which live under this inductive process, and you have compact, guarantee of compactness  of the closure of this thing here, which satisfies this.  If you look at all those things, the closure of this associated UN is compact, so, and  you can choose to be a neighborhood which is compatible, so, which is compatible, so  it satisfies these conditions.  So, so that, 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, and the perturbation by the, by the construction before, the,  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 one.  So suppose the first, we take a sequence of epsilon i's less than one on the, so starting  with epsilon zero, you perturb smaller than epsilon zero, then you get something on the  boundary smaller than epsilon zero, then extend smaller than epsilon one, then on the next  level you get something smaller than epsilon one, extend smaller than epsilon two, always  controlled by this neighborhood, so compactness never grows away.  So let's see.  But I explained that in more detail.  So, so then there is one issue.  So let me first say, if I have such a neighborhood, you, which has this property, I say, and you,  as in the theorem that and you controls compactness.  Yeah, so I think that's a good word.  It controls compactness of, 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 as a covering  and then one extended to the boundary.  So you need some, it's a little bit of a subtle argument.  It's not true.  So I, I wouldn't bet my life on it, neither your life, that such a property holds for  arbitrary auxiliary norms could be possible.  Actually nobody's life in this room.  Okay.  Everybody happy now?  Okay.  So, so now here, here's one issue, which one should point out when it went easy forgets  about it.  So if I have a perturbation below and it's, it's supported in you here and here supported  in some U prime and then I take the product and go on the boundary.  This is not supported in you 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 the compact support and  here outside.  It's a non, it's still non-trivial, the product.  Can you draw a picture with me?  Yeah.  So, so if you have a line and this is a support, cross a line, there's a support is here, then  the plane, it looks rather like this, that's a support.  But the solutions, the solution set of course for the product still lies in here.  So okay.  So that's, I mean, if I take a function on one and a function on the other side.  Yeah.  So, and if the support is here and so if the support is here and here, then for the product  the support lies and that's this cross.  If f times f prime is not, is long enough?  No, no, no, it's a function now from, so, so you have a function from r into r, cross  a function from r into r.  So you get, you get something from r2 into r2.  It's a Cartesian product.  Yeah, it's a Cartesian product.  Yeah, so that's what I have.  But it's like you're adding the function, right?  No, no, no, no, no.  I mean, you have a, you have a, you have a threat, you have a threat on problem here  with a compactly supported perturbation, say infinite dimension in here.  And the product, 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, but it's  like, like sort of thinks that compact support is compact support.  So, so everybody, it's a generally accepted fact, yeah.  But you have to be a little bit careful.  Where do you imagine that this solution set is contained?  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, yeah, so if you  know that for each of the threat on problem, the solution set is compact, then the product  set is compact.  Again, what that picture means?  You want to know what that means?  You put all those pictures in.  Okay.  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 where the support lies here.  Takes a Cartesian product of the two things.  You get a map.  This means f of x, g of y.  So it's a map from R2 to R2.  So then, then the support is here.  And that is precisely what we have here.  So, so now, however, if-  Sorry, can you explain the notation that's in there?  What is n sub lambda?  What is t dot lambda?  Yeah, this, yeah, I did, actually.  There was a definition somewhere.  T dot lambda was a rescaling by lambda when you, when you were sort of last time by lambda.  So it makes a score.  Oh, it was, it was there.  Oh.  It's t dot lambda.  So it's, ah, here, u n.  Consists of all the morphism classes in u.  Such a delus true and this thing only has 12 cylinders.  A little n sub lambda.  Oh.  A little n sub lambda.  Here it's just the, the maximal norm you see for your section.  So you have a multi-section functor that, that goes down to orbit space and you just  look at, over an object you look at the maximal norm of one of the branches and that's invariant  under isomorphisms.  So you, so you get a function on the orbit space.  So, ah, so now, however, if, if you are in this situation and you know that you, that,  that this space, so if, so let me give this in the, in the simple case.  So if you, so, 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, any t between 0 and  1 is compact.  So if you can switch it off, then if you, then, so there's a little bit stronger assumption  than just assuming that this is compact, but of course it's true in, in this kind of sets  of, always.  Then if you take the product which, which would not have of the perturbations, it takes  two such things suppose S of t has bounded support, S of, 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 then, but you can put a bump function  in 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 this two parameters are in through that?  No, no, just that is even unimportant.  So what is important is that it means if you put, so, if you replace, so, so now replace  if you take the product, yeah, take, just put then in front of this perturbation a  cutoff function and of the, 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, the  set of solutions of this problem for minus one to one, then you can actually find a cutoff  function which just on the solution set has to be one, you can cut it off.  So, so there is a, there's a, there's a different perturbation which actually would have bounded  support.  It would have the same solution set and the same linearizations as the solution.  So, so there is this marginal difference.  However, if I would require this, then this nice multiplicative property goes away.  So I, if my perturbations are multi, I have a definition that on the, on the fiber product  is the product and so on.  And it's precisely the one which has this support.  But, but it has the same, it has the same function then there's another perturbation  inside if, which would have bounded support.  So, so therefore that is a little point one has to, 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, okay.  So, so you need this property here then this is well behaved.  Okay, so, so, but that precisely we have because I have estimates for my, 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 one for t  between zero minus one, you have that property here.  Yeah?  So, so it's built in.  Okay, so, so of course I could have gotten the way without saying this because nobody  really cares about that fact, but, but it's sort of good, good, good to know.  So, 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, this things can be used to patch together and construct this SC plus multi-section  function 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.  So, because you have the same amount of freedom as far as perturbations are concerned.  So, 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 one.  So again, over all of this.  No, no, no, no, if this is less than one.  On what?  Everywhere.  Everywhere, everywhere, everywhere, on the whole of S.  And, but, and in addition it has this property.  Which would be precisely, I would be an example here if I just takes a product that would  be a perturbation which would be non-trivial on a, on a rather large set but it would have  this property.  So, the, the, the, the inequality on evolving t in lambda, there's just another sign of  t, isn't there?  No, no, no, 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, just t lambda is, is a rescaling of, 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, 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, which has support in u.  Yeah.  So, so, so, so now I'm squaring back, so, 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.  Then it would, but, but you can, but since you sort of can cut it off and once you are  a little bit inside this thing, then you can just go, yeah, so, 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 your line just said u.  Okay.  So, now, so what I just said, like infinite dimensions in the case of sections.  So, now here's the perturbation algorithm.  So, the following result, given a compatible u n, so, these are things we just constructed,  they exist for every epsilon between 0 and 1, a compatibility s c plus multi-section  functions, this compatibility is 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, 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 maximal current 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 they exist in 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, trivial cylinder  buildings.  Then e less or equal to j is just e restricted to this and this thing here is just stands  for the union of all.  So s, 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.  Okay, inductive statement, given a strictly increasing sequence epsilon j in 0 epsilon,  they exist for every j and 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 one because the boundary phases 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 and is compatible with the local family structure  for some reason that should be blue is black, I don't know, is controlled by n u 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 have an induct, if you can go to the next step, then you have a perturbation  which produces compactness and everything.  Okay, a lot of statements here but basically it says, okay, so let's start.  So the induction first step is induction, okay, so that's easy.  So take first a component without triv- take first a connected component which does not  represent object with trival 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-trivia could shift  them against each other and I get boundary strata.  I mean, shifts them so that I get a level, I get a building with two floors, so I would  have boundary strata.  So what I actually, if I exclude trival 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 S A is in general position and the support of the sections occurring  there is in U, the neighborhood which I constructed.  So this is a basic of the statement.  If you have a freedom operator with compact solutions set, you take neighborhood of this,  you take perturbation with support in this neighborhood and you wiggle a little bit and  this thing 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  trival cylinder components.  So this is how you define it.  You just take the perturbation which you had on A and on the extended so that over a trival  cylinder there's nothing.  Okay, 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 U because U intersects 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 01 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...  Okay, so you are sure.  Oh my God.  Yeah.  So it says that in a consistent way I can perturb on each connected component.  So there is an 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 finite 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 SF-t 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 equals 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 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 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 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 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 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 it would be surjective and then inside you  have to wiggle a little bit to make it surjective there.  Then so now what happens if I'm at a corner.  So when I have a corner I cannot argue like this then this part is restricted to this  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 I have an additional parameter to help you to make it surjective.  But so you have to look what the linearization is and it's automatically surjective.  So then the restriction to previous things are surjective so it's in general position  so everything's fine near and you extend it.  Oh, oh, I'm sorry.  Okay so I'm astonished how short the proof is.  So then we take the phases and pull back perturbations.  So this first is our trivial cylinder 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 as our 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 you can decompose it  into disconnected components which I do and it just takes a product that relation was  already true on the boundary.  So just extend it.  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 this buildings here I extend by adding trivial cylinders  to it.  So now here when I take products and so on so there is this point with what I discussed  here, this difficulty which discussed here but if you believe me that I could sort of  modify it and bring it into something which has supported you then this procedure never  will always have multi sections which have supported 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.  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, 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 modellized space which is where you want the more singularities for  example.  So then you do the product and go to the next level but then of course the perturbation  is a fireball 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, then of course this, when the modellized space and have T you can  talk about that you only see more 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 already produces 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 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  fireball 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 is too small a mention.  This other stuff looks like this.  So you just throw the garbage out?  You can, you throw the garbage out and yes, that's what you do.  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.  So I have to cancel that plan and presumably I have to cancel that plan here as well because  our chairman is really tough.  So the marks on.  After stop.  Yeah okay.  So here, so let me just make some one minute.  We 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 an oriented except with the occurrence of bad orbits.  And then you can write down integrals and then you get the relations by this, the model  of 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.  Quick questions.  Yes.  So, you're saying you want an amniotic form in which the, the, the, the, the, the, the  constant cylinders.  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 have.  I mean you, you, you definitely have to put, okay.  So, so, so when I, when I look at stable maps then I would have maps from.  The question is what is the ambient polyfold for just for the trivial cylinder?  It doesn't.  It only, 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, but, but, but the construction you have is not a product construction.  So you cannot, you cannot have a trivial cylinder by itself because it's an unstable object.  So, so, so, so, moment, but a trivial cylinder occurs always together with a non-trivial  component.  And then when I, when I construct a global slice, it's not a product of a construction  on a cylinder and the other one.  It's a more, it takes both, it takes data from both and mixes them up.  It's not a product.  So, 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 the slice with respect, because when you make a slice with respect  to the R action, then you, you fix say the non-trivial part, but you can slide the cylinder  against it.  Exactly.  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 make, you have to take a transversal constraint.  You have to divide out the automorphism group for the cylinder as well.  So, so there are several things which I have to do at the same time, which go into the  construction of the uniformizing.  So, so let me get this back to you saying there's not an ambient and polyphonic in which  just the trivial cylinder sits.  No.  All right.  So, so, all right.  Okay, I suggest that we break out of our precious rubric.  Thank you.
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