 Well, it's a participatory democracy kind of talk, so everyone is welcome to choose his own title. The only thing is that this is joint work with Alexandre Roantcher, and I would like to thank the organizer for the invitation. Ah, merci beaucoup. So this started with two sources. The first one is a paper by Elias Berg and Hofer, which is called Towards the Definition of Simplactic Boundary, and which asks the following question. Well, if you're given, I will be more specific on the assumption, but a symplectic manifold. So just for a second, the symplectic manifold will be open, but such that it has a boundary, and the question is, does w omega determine sigma? So that's one, and actually the second one is a paper by, or, well, a result, let's say, by Fleur, Dusa and Elias Berg, about the following says that if you have m omega, which is symplectically as spherical, yeah. Do you mean the interior determines the boundary? Yeah, does the interior determine the boundary? That's what I mean. So it's, if m omega... And determine the boundary, which means it's an abstract manifold? No, as, well, if it's a contact boundary with the contact structure, for example, what can you say about that? With omega over pi 2 is zero, so simplatically as spherical. And if m omega is symplectomorphic to r2n with the standard symplectic form, then m is diffeomorphic to r2n. You haven't said that correctly. You mean it's symplectomorphic to that, at infinity? At infinity, yes. I said that. I didn't write it down, but I said it. Well, maybe. But as I said, it's participatory democracy talk, so you're welcome to add your favorite assumptions to, or maybe the necessary assumptions to the statements. And also to modify the conclusion, if you like. Then m is diffeomorphic to r2n, actually, for n equal 4. For n equal 2, sorry, there's a stronger and older result by Gromov, which implies that m is actually symplectomorphic to r4 with the standard symplectic form. But this is very specific to four-dimensional situations, and I will not talk about that, even though what I'm going to explain could be maybe adapted in dimension 4 to get stronger results in this kind of spirit. So just one word about this. Well, first it uses modular spaces of curves, which is the excuse for discussing this kind of problems here. And the second is that the proof is actually in two parts. First, you prove that the homology of m is zero, and then, except in dimension zero, and then you prove that the pi1 is actually zero, which is, if I'm correct, which is what Yasha proved. So I will, again here, not be interested in pi1 questions, although it could be interesting to check that. So what you should remember about this theorem is that if m omega is like r2n with the standard structure at infinity, then the homology of m is like the homology of r2n. And there's a slightly different way to rephrase that, which is the following is that if w omega, and I will again explain a little bit more about that, as contact boundary, which is the sphere with the standard contact structure, then is the thermomorphic to r2n or to the, sorry. First, it's not m, it's w. And here, let's say it's the same, of course, but to the unit ball. Makes it more. You do need omega with, still, again, with omega over pi2. Well, just because if you have such a manifold, you add the simpletization of the sphere, and then you get something which is, so you have your something which bounds the sphere here inside, and then you just have the simpletization, and you get some manifold which is simplectomorphic to r2n with the standard structure at infinity. So this is the kind of question I will be interested in. And in fact, let me just, unfortunately, we have to write down some definition, which are pretty much standard. If you have sigma psi or contact manifold, and if you have an embedding in a simpletic manifold, then this is a contact embedding. If there exists some form alpha near sigma, such that psi is determined by alpha, so is the kernel of alpha, and the alpha is omega, well, of course, near sigma, since that's where it's defined. And the embedding is exact, or sometimes it's a restricted contact type, if alpha extends to m, and, of course, as a primitive of omega. So that's definition one. Definition two, which is basically the same, is that the simpletic filling of sigma psi is just a simpletic manifold m omega, and I will add with no closed component such that the boundary of w is sigma. When by slight abuse of language, I will say that sigma psi is contact embedded in w omega. Well, there's slight abuse of language because w only exists on one side of sigma, not on the other side. So it's not a usual embedding. So the existence of alpha is only on one side, but I think it's pretty clear what it means. Can you, in fact, always extend by some such activation? Yeah, you can always extend by some small simpletization, so there's no real, there's no issue there. And so it's exact, the filling is exact, if, again, as before, let me say, if it's exact as above. And the last thing I need is, can everyone see when I write here? You can't, but it's okay. Well, it only says definition three. And the filling of sigma psi is stein. Well, if it's a stein manifold, so if I have w omega, the simpletic manifold, psi is a function. So psi is j plurisobarmonic, and j times omega, sigma is psi minus one of zero, and psi is negative on w. So w is a sub-level of plurisobarmonic. So as a consequence of being plurisobarmonic, sorry, I missed something here, and psi is j star of deep psi. Okay, so, well, sorry, sorry, the kernel, I should say. So the contact form is given by the kernel of j star of deep psi. So if you have a plurisobarmonic function, all critical points have index less or equal to n, and it's time sub-critical if and only if all critical points of psi have index strictly less than n. Okay, this is... Are you asking for j to be integrable? No, but... So I will start on one of the blackboards. Let's start on this one. By something which is an element we remarked based on the Florian-McDuff-Eliashberg theorem, so some of you may say very elementary remark, maybe. So the theorem is the following. So if sigma psi has a contact embedding in R2N with the standard symplectic form and interior component z. So if you have an hypersurface in R2N, it has an inside and an outside, and the inside is called z. Now let w omega be a symplectic filling of sigma psi, and let's say such that the second relative homology of w with respect to sigma vanishes. Then the map from HP to W, well, let's say 1 first, to HP of sigma induced by inclusion is injective. And the second remark is that any two such fillings have the same betty numbers. And the third remark is that the p-betty number of sigma is the p-betty number of w plus the 2N minus p minus 1 betty number of w. So I must say we are a little bit surprised to discover that. So it's not just the sphere, and you will see that it's totally, I mean, once I tell you it's totally elementary, you can actually prove it in two lines by yourself. But it's not just that the sphere that has a unique contact filling in the sense, unique in the sense that the topology is completely determined by sigma, but you have a whole class here. And in fact, if you have a contact embedding of something into R2N, it's very easy to construct many more. So let's say it's very hard if you have, if you give yourself a sigma, a contact manifold, it's very hard to embed it in R2N a priori. I mean, unless it's already given of something embedded in R2N. But once you have something embedded in R2N, it's very easy to construct new ones by just doing surgery along the handles of index strictly less than n. You have something like this. You can start with a sphere, and then you can add the handle like this just as much as you like. Provided the index is less than n. So you're adding handles to what? Well, you're doing surgery. I mean, so you're adding a Lagrangian or actually isotropic because I say it's less than n. So isotropic. So you're modifying sigma. So you're modifying sigma. So once you have a sigma, for example, in the unit sphere, then you can construct lots of things with quite different topologies. So the only thing that is delicate to touch is the n-dimensional homology. Can I ask about the definition of fulfilling? You don't distinguish between sigma being convex or concave boundary? Sorry, it's always a convex boundary. I should have said that from the beginning that all my contact structures are oriented and everything here is considered compatible with orientation. So indeed, they're all convex boundaries. We're talking about the surgery. So you have sigma as a contact paper surface. And then you're modifying... Well, you attach an isotropic handle and make surgery on this handle. And because it's isotropic, you can just put it in any way you want without by still being embedded. This is something you can find in a paper by... Well, an old paper by Lodenbach in the ATE. But why is it still embedded? Well, because it's embedded... I mean, you start with something embedded, you add an isotropic handle here and basically isotropic, but not Lagrangian. So isotropic in dimension i-1 at most satisfies each principle. So you can just realize that as something embedded and not crossing again the sphere and then you do the surgery around this. So you're finding an isotropic sub-nath on the R2M which is going to be the core of the handle. Yeah, and you can always do that because you're in the dimension where you have each principle. Okay? Did Z enter into this anywhere? Sorry? Well, you could say that because they all have the same Betty number here from 2. You could say that the Betty numbers are the Betty number of Z, if you want. So do you prove that the filling, you've got sigma fixed and you have HP back it's injected into HP sigma. Is the image of that determined? Is that determined? Almost. You're not. You claim that quite. Almost except in dimension... Sorry, no. Not in general. Let me maybe state... Yeah, before I prove that which is... So here how many blackboards? Ah, two already. Okay. Let me state a corollary. Here you probably don't see. Ah, it's great. Is there a blackboard on which you can't see? Ah, it's the one behind. Well, I put myself in between the board and you. So the corollary is the following, is that if sigma psi has a contact embedding in the standard R2M and W omega is a time filling, then I need... Actually, I forgot Y, but I need N to be greater than 3. BP of sigma, so the betty number of sigma is the same as the betty number of W for P between 0 and N minus 2 and BN minus 1. So N minus 1 betty number of sigma, which is the same as the N betty number of sigma by Poincare duality, is the sum of BN of W and BN minus 1 of W. So in particular, if W is time subcritical, then BN minus 1 of sigma is BN minus 1 of W and BN of sigma is BN. No, I don't believe that. BN minus 1 of sigma and BN of sigma is... Here? Here? Equal? On the left? Here? Equal? BN minus 1 and BN are the same because it's 2N minus 1 dimensional manifold and you have Poincare duality. That's the only... So plus sign? Here it's a plus sign. The sum of the two, and here is an equality sign. I think I missed the explanation as to why this elementary mark was elementary. I haven't proved it is elementary yet. You're not going to do some sort of surgery on the... It's almost more trivial than that. So this one, you can't write in the corner like that. Here I am not supposed to write here. Well, it doesn't matter it's elementary anyway. Maybe I'll do my morning exercise then. Maybe I want to send this one. So the proof of the theorem is the following. Just look at W union R to N minus Z. So you take R to N, you have... You remove sigma, you get something with bound... Remove Z, you get something with boundary sigma and you plug in W. At the end, you write... Myerviator is except that instead of doing that with... R to N, you... Well, it's the same R to N or the bowl. So you get this sequence here. HP of R to N minus Z. And goes to HP of sigma. And then, well, there's no need to go back because this is going to be HP plus one of R to N. And so this is zero. This is also zero, at least for most values of P. And so here you get that this is an isomorphism. And so first you get that this map here is injective. And then you get... So this has to be injective. And then you get that BP of sigma is BP of W plus BP of R to N minus Z. And this implies by Alexander duality that BP of sigma is BP of W plus B to N minus P minus one... Sorry, of Z. So I don't need Z in the theorem, but I need Z. And now what do we have? Well, you can apply the same thing for W equals Z because it gives you a feeling. And so what you get is that BP of sigma is BP of Z plus B to N minus P minus one of Z. And as a result, BP of Z is BP of W. Well, this is actually true for P between zero and two N minus one. You have to deal with the non-zero homology of R to N, but you do that as an exercise. So I must... Yes, of course. The fact that the last term... Sorry, the last term is R to N. So it's the fact that the union... Yes, sorry. So it's not only my observatories. I would argue that that means it's not elementary. It's elementary from... It's an elementary step starting from Flora Magda Feliaspar. That's what I meant. Because it's Yasha. Because what? Because it's Yasha. No, I think because Flora Magdaf proved that the homology is zero and then after that Yasha will prove... Am I correct? Proved that the pi one is zero. Is that correct? I think that's right. And the paper is actually in your paper. We were talking at MSRI and I was talking to Andreas and he had this and then Yasha came. Right. So we could say it's elementary, mod... Yasha's Yaha. That's what it uses, sort of jail... Yeah, jail homophic curve and modular spaces. Okay. Yeah, that's the... And the fact that the only breaking can come from... Exactly. In addition, on relative h2, is what gives you some kind of a spherical... No, what you actually need is that the union... that I picture here, so W union R to N minus Z should be still simulactically as spherical. So this assumption on the h2 implies that. Actually in the paper with Alex, we have other conditions that imply that. But you need some condition that will guarantee that this... this manifold is as spherical and not enough that W and the W itself is as spherical. So let me make a number of remarks. One remark, I don't know how much it fits here, but so according to some result by Meilin Yao, if sigma psi is contact and satisfies the third churn class of psi is zero and W is time subcritical, then we have that the cylindrical contact homology of sigma and psi is isomorphic to the homology of W relative to sigma, tensor well basically with the homology of Cp infinity, I think shifted by two, but that's... And it's interesting to compare the kind of result that you get. So here what you get is that if you know the cylindrical contact homology of sigma and psi, then you know something. I mean from this you can extract the homology of W basically. So you basically know the homology of W and twice versa, provided you have this time subcritical condition. This kind of statement is different. You ask for an information about the contact structure by saying it can be embedded somewhere, in this case it's in R2N, and then from that just the differential or the homology of sigma gives you information about the homology of W. But of course you can also somehow combine the two and say well, if for example sigma has the contact embedding in R2N, then you're going to know the relative homology of W sigma, and therefore you're going to know something. I mean if you're in the situation of the corollary where you have a subcritical stein feeling, then you know also the cylindrical contact homology. But if you look at the two statements somehow separately, they're telling you different things. One starts really from the contact structure that you're supposed to know quite well to compute this and gives you information about this, and the other one you have a sort of much coarser assumption by just saying well there's a contact embedding somewhere and then I know the homology of sigma and from this I know the homology of W basically. So this, yeah. This result is for any dimension, right? Maybe I don't know, and it's greater than something at least. I don't. So let me just continue on similar questions and then, so another obvious question somehow, well I mean the question is obvious, I mean the answer, is the following take L to be a compact manifold? Look at the sphere bundle of the cotangent, the sphere cotangent bundle of L with the standard contact form. And then the question is what are the possible feelings of this, of S t star lambda, S t star L with lambda. Of course the question is well at least homologically should they have the same homology as D, so is it D t star L? And well we have the proposition again I think N has to be greater than trees that if L, which obviously follows from what? If L has a Lagrange embedding in R to N, unfortunately I didn't say exact Lagrange. So you have non-empty set of examples. W, a feeling of S t star L lambda such that the relative homology of W with respect to the boundary is 0, then from what follows W as the homology? And you can construct of course many of these if you have L which has an immersion in R to N then L times S1 for example has an embedding in R to N plus 2 and the assumption of having an immersion in R to N is just an assumption of the tangent bundle. So if the tangent bundle, the complexified tangent bundle is trivial then you have such an immersion for example. So what did I want to say? Okay so of course it's a bit frustrating if you say that all these comes from one a well-known theorem and two Maya vietoris. And then you think well maybe we can work a little bit more and get some... Can you use this S1 trick to upgrade the ring of... No, no because if you try to multiply by D2 so the problem is that the boundary of D t star this is this term which is okay but it concerns also this term which is not okay because it really comes from the feeling that you had. So yeah maybe I can... So the idea is well let's try to generalize this theorem the Fleur-McDuff-Eliashenberg theorem and then apply again Maya vietoris and maybe we get to a wider class of examples. So the theorem is the following. So assume sigma psi admits a contact embedding in a subcritical stein manifold. Now let W be an asphherical feeling such that the relative H2 of W on sigma vanishes then the map I hope this is correct from Hg of W to Hg of sigma is injective for all j. And well if sigma psi is just S-1 S-1 and S-1 with the standard sphere you essentially get well up to this pi one thing which I must admit we were too lazy to investigate we get the Fleur-McDuff-Eliashenberg theorem. Of course S2 and S-1 with the standard structure has a contact embedding in R2M so in a subcritical stein and what this says that if you have an asphherical feeling then the map from Hg of W to Hg of sigma is injective but this is zero for most values so it means that this is zero for basically all values. So just one question. Subcritical stein manifold does not mean it from now is it subset of R2M already? No. It means that well implies that it's a product of C by your theorem by Chilibach is the product of C and I think a stein manifold. And so in particular subcritical stein implies that the symplectic homology so let's say this one is N omega subcritical stein the symplectic homology of N well the plus part let's say and constant part vanishes. So let me state the corollary of this if sigma is a rational or just a homology sphere and if sigma psi embeds in subcritical stein then any symplectically asphherical feeling will be a rational homology ball. Well I had plans to describe a little bit the proof but I think that was overly optimistic. Oh the clock is fast we started five minutes late so you have till 40. I have till 40. On this clock. Oh yes my says 20. Past. Okay then so I was about to skip many things so we'll just skip half of what I plan to skip. Still I don't think I want to embark on the idea of the proof. Can you tell us what the ingredients are? What the ingredients are? They're basically the same as well let's say no for this no. So what we use is that so the first thing is that is a result by Chile back which says that subcritical stein is the product of something which I will denote by M and C. So that's how I start. The second point is result by Liska and Matich which tells me that one can close M to a manifold P and then the idea is to work in P times S2 and now the idea is essentially the same as the one but requires more work in the theorem by Flor Macduff and Eliashberg looking at rational curves which are in the homology class of S2. So what actually we do is to replace and then consider V as well as Z was embedded in M times C here so becomes embedded now in P times S2 so you take P times S2 removes Z and then you glue over sigma your manifold W and then you somehow count homomorphic curves which are so somehow at infinity you have the curves in P times S2 you use some analog of hyperplane section to sort of normalize the homomorphic the homomorphic serves by fixing three points so that you don't have to deal with reparameterization and then you have also some homological argument but basically by kind of continuation you can fill the space V by this homomorphic curves. That's exactly the same idea. Z is the interior of no for once it was not on the board everybody asked what in all previous statements there was a Z and since you complained I dropped it but now you see it's useful so it embeds in a subset in the in a subcritical stein and the interior is Z so here you would say with interior component equal to Z in the other case you have like a choice of which thing to make your S2 no here you have no choice on which thing to make to S2 but I mean that's not so different and then the sort of homological argument in the end is slightly different but basically the ideas are essentially the same so is it an assumption that there is an interior and just being completely different why is there an interior? Is it an assumption? It says no because if it's a critical stein H2N-1 is zero and then by Alexander duality the hypersurface will have an inside and an outside so it follows from the hypothesis here so there are two more things I wanted to discuss about this kind of problem the first one are symplectic, let's say symplectic homology obstructions because so far except for, there's a sort of starting point which uses rational curves in the symplectic manifold but apart from that it's all ordinary homology somehow there's no flow of homology here there's a hidden gram of width and invariant but there's not much so what can we say if we use more about symplectic homology? Well there was the statement I quoted before by Meilin Yao but let me say a few more things so I will say, actually I don't know what this W omega is I would like to say flow as spherical but if it satisfies that the symplectic homology of W vanishes I thought of symplectic, no not as spherical sorry I meant acyclic I don't know if there's, yes P is the manifold you get from M which was probably so you start from the subcritical stein, you write it as M times C so in the original situation of Fleur Macduff and Delyashberg the subcritical stein is just R to N so it's easy to write it as a product of something and C so it's going to be CN minus 1 times C and then the CN minus 1 you close it as a compact symplectic manifold and this is P and so in the original paper it's closed as a torus sorry as a torus as a product of spheres but it doesn't matter what you close it has and then you have that sigma is embedded, of course it was embedded in M times C so it's embedded in P times S2 and then you make surgery over P times S2 by removing this Z which is the interior and gluing W instead and see what you get from there by looking at homomorphic curves such a manifold without a C factor what, subcritical stein? no I mean a fleuric cyclic manifold which is not something that's C ah share our, yeah it's a, every flexible range of domain is where a cyclic ah, but they're not all subcritical so subcritical is a good project for this product so pretty good to do product by, by Chili-Buck theorem and ok thanks so let me state the proposition here so let sigma psi bound a subcritical stein W omega let M omega be a fleuric cyclic and sigma be embedded in such a way that you have a separating embedding and sigma be separating in M with interior Z and then the homology are the same the homology of Z is the same as the homology of W and well, if you want to know the ingredients well there's a result by McLean which says that when you're embedded where you have an exact embedding in fleuric cyclic you're again fleuric cyclic ah, there's the, you use the long exact sequence in symplectic homology and then you use a result by Bourgeois, sorry this is the second Bar Bourgeois and the one chair which says that sigma psi determines the positive part of W and there are some assumptions you need that there are no conlesender orbits on the boundary of index I think less than 3 minus N and you can prove that it's satisfied here if sigma bounds a subcritical stein and so the last thing I wanted to say are applications which may be in a way is most fun and also maybe asks so somewhere here what's hidden is the idea that you, if you have a a co-bordism so if you have a contact manifold which is concave on one side and another which is convex on the other side one of them is more complicated than the other there should be sort of increasing complexity in this way and so there are sort of obvious questions that come up about complex singularities so for example let's f be a complex polynomial zero be an isolated singularity then you can look at this singular hypersurface here you can look at the intersection with the small sphere and this is actually well here it's going to be let's say plus so a complex polynomial on CN plus one so that this now is in S to N minus one this has a contact structure by taking the maximal complex sub spaces here and there's something which is called a symplectic manifold which is the Milner fiber here which essentially you get by instead of looking at f minus one of zero you just move a little bit so you have your singularity here you have a sphere here where you would like to look at this feeling here but you don't you actually look at something which is just below here and so w will just be this this thing here is w so w is a symplectic manifold that bounds sigma and w is a wedge of sieve spheres and it's a wedge of mu spheres and mu is the Milner number and so here the idea is that the biggest mu the more complicated singularity you have I mean if you have no singularity actually mu is equal to two and so the proposition I would like to state is the following and then I prove so again for N greater than three sorry so sigma psi is called the link a singularity such that the intersection form on w is nonzero then sigma psi has no embedding so no contact embedding of course in a subcritical style and the second theorem is that if you're looking at briskorn manifolds and mu is at least equal to two there are no subcritical for example stein embeddings and the idea is that I mean this is somehow the result for mu equal to one it says that if you have a trivial singularity then w is essentially Cn and so you cannot have a larger singularity so something which with the larger mu with some assumption here that will embed of course in Cn what you would like to say is that if you look at two singularity with mu strictly less than mu prime and you take the manifolds w corresponding to mu and to mu prime you cannot embed the more complicated one into the simplest one that's what you would like to know but I have absolutely no idea on how to can you describe it by the statement before that to now? well if you have sigma one w one with mu one sigma two w two with mu two so this one comes with xi one and this comes with omega one this comes with xi two and this one with omega two and then you would say that something like there's no contact embedding or exact contact embedding I don't know sigma two xi two one omega one if mu one is less than mu two so in this sense sigma two is more complicated than sigma one there's one of them degenerates into the other one does any of us avoid it as a no no the only thing I know is is this so with this extra assumption about the intersection and it's essentially the trivial case I mean it says you cannot embed in a subcritical style okay my time is more than up now thank you very much for your attention