An Integral lift of contact homology
Nelson, Joanna
Cylindrical contact homology is arguably one of the more notorious Floer-theoretic constructions. The past decade has been less than kind to this theory, as the growing knowledge of gaps in its foundations has tarnished its claim to being a well-defined contact invariant. However, jointly with Hutchings we have managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. This talk will highlight our implementation of non-equivariant constructions, domain dependent almost complex structures, automatic transversality, and obstruction bundle gluing, yielding a homological contact invariant which is expected to be isomorphic to SH^+ under suitable assumptions, though it does not require a filling of the contact manifold. By making use of family Floer theory we obtain an S^1-equivariant theory defined over Z coefficients, which when tensored with Q yields cylindrical contact homology, now with the guarantee of well-definedness and invariance.
 Thanks for the introduction. I'd like to thank the organizers for allowing me to talk. Okay, so a lot of this will be joined with Michael and it was helpful he came with his obstruction, bumble gluing and intersection theory from ECH. So he was kind enough to lend them to the contact tomology. Alright, so we're going to start off with a contact manifold. For most of the talk it will be three-dimensional, but in some parts I might get to tell you about what we can do in higher dimensions. This is going to be non-degenerate contact and it's going to be co-oriented meaning that there's a global one form that defines the contact structure and for grading reasons let's pretend or assume that the first turn class of the contact distribution vanishes and then R is going to be my rave vector field. Okay, and then I'm going to have J is an almost complex structure on r cross m and I'll take d e to the tau lambda. So here tau is going to be my coordinate on r. It's going to be the translation invariant so I have d tau r is going to be equal to r and I want J restricted to, I'll just say J is d e to the tau lambda compatible. Okay, then we want to count cylindrical curves so crystallized asymptotically cylindrical curves are curves the domain is just going to be a cylinder so it's not going to be super exciting but they're still fun to look at. So I'll denote the space of, so let's gamma plus gamma minus be periodic rave orbits. Mj gamma plus gamma minus is going to be the set of J holomorphic curves from a cylinder r cross s1 j0 into the simpletization with this d lambda r invariant almost complex structure and then we want du of J to be J of du so we want them to be pseudo holomorphic and then we want the limit so if we take r to have the t coordinate s1 to have the s coordinate we want the limit as s goes to the plus or minus infinity of the projection of u and the contact manifold component to be a reparameterization of the rave orbit at gamma plus if we go to positive infinity and gamma minus if we go to minus infinity and then we want the limit as s goes to plus or minus infinity of the projection to the r component of u to be plus or minus infinity so you want to think like if my contact manifold was something stupid like a circle then your counting curves like this gamma plus gamma minus and then there's also going to be an r action that technically you have to mod out by coming from translations on the target and so when we go to look at the differential there will be this translation action but a lot of times I'll ignore it when it comes to talking about index calculations and stuff. Okay so. Is anything other than the gradient dependent on your assumption that c1 equals 0? No. So I can just drop that and you have your z2 gradient? Yeah I mean eventually I have to yeah you can drop that and use the z2 gradient of course when I define a dynamic convex contact form you'll need some sort of way of making sense of the gradient on contractable orbits. So the gradient is given by the Conly's Ender index on our chain complex we have that the virtual dimension of this moduli space is given by gamma plus minus the gradient of gamma minus and here the gradient is an SFT grading so this is going to be Cz of gamma plus n minus 3 so if the dimension of our contact manifold is 3 then you get this is just Cz of gamma minus 1. So then I can tell you what the chain complex is. So this is going to be generated by only good Ray-Borbets so we take the set of all Ray-Borbets and then we throw out bad Ray-Borbets because they're bad and so to define a bad Ray-Borbets a little bit annoying but so the Cz index of a Ray-Borbets can either have the same parity under iteration of your orbit or it can flip flop between being even and odd sorry odd and even if it flip flops then you throw out the even multiple covers so bad Ray-Borbets even covers of flip the Cz guys if you want the real definition or you want an example so in dimension 3 bad Ray-Borbets are even multiple covers of negative hyperbolic guys meaning that they have the eigenvalue of the linearized return map if we restrict it to the contact distribution is negative and real. Is there a reason for your V there for which the Cz VQ like in your top line? It's not a V this is the vector space generated by all the Ray-Borbets and you throw out all the bad Ray-Borbets. Yeah sorry he just thinks the stars never mind. Is there another way for why are bad Ray-Borbets bad? So if you keep them in the chain complex even if you had all the transversality in the world you'd never be able to prove invariance but secretly they you can't orient curves that limit on bad Ray-Borbets unless you have some sort of a parametrization you're using of your bad Ray-Borbets and so there's a when you try to assign orientations you can get both positive and negative orientations assigned to the same cylinder and this causes a headache and in orbithold Morse theory you can kind of think of the bad Ray-Borbets as being like the bad points whose like gradient flow lines get reversed underneath whatever action you're modding out by. Any other questions? It's mobilized spaces. Yeah so that is kind of incorporated by the fact that this is the projection onto the contact manifold as a reparametrization of gamma plus minus so that already includes that. And then the differential. So there's actually two equivalent ways over Q of defining the contact differential. So either we can define it to so I'll just write this down and then I'll say. So either you can define it by looking at the multiplicity of the top Ray-Borbit or you can define it by encoding the multiplicity of the bottom Ray-Borbit. So and this epsilon U is just supposed to be some choice of coherent orientations. M of U is the degree of the covering map of the curve if your curve U is not somewhere injective and the multiplicity of your Ray-Borbit just means you've had to iterate some simple orbit some number of times and the multiplicity will be the number of times you've iterated a simple Ray-Borbit to get to X. And so it might look like these coefficients are defined are not integral but they are in fact integral. And so you can see that because if you have a k to one fold cover where you have X is gamma let's say plus to the kp and Y is gamma minus to the kp this would be a k to one fold cover of a cylinder gamma plus to the p gamma minus to the q and so the multiplicity of X is kp the multiplicity of Y is kp the multiplicity of this curve U is k and k obviously divides both kp and kq. I'll just write against that M of X equals one if X is a simple Ray-Borbit so maybe simple hasn't been defined but that just means that the map X which is r mod tz into your contact manifold is injective and M of U equals one if U is somewhere injective. So that looks like beautiful theory but it's not really so beautiful. So some of the issues that are going to arise is that we're working with multiple covers. Is there a question? So can you explain the difference between the two? One encodes the multiplicity of the top Ray-Borbit and one encodes the multiplicity of the bottom Ray-Borbit of your cylinder. And it turns out that their equivalent sort of if you had all the transfer salinity in the world that would mean that your differentials were defined and you actually got a homology and then on the chain level you could just divide and multiply out by multiple seed of so Ray-Borbit's the pass from one chain complex to the other chain complex. So homologies are isometric under this? If you live in magical transfer salinity. Yes but they're equivalent. So why is this theory possibly not so correct? So the first issue is that we're working with multiple covered curves and Chris spent many lectures telling us as well as Dusa and others telling us about the horrors of multiple covered curves. And then also I think Michael mentioned this a few times but you can look at multiple covered curves and they can end up having negative index even though they live in even though they have to be there and you can't just use a generic perturbation of J to get rid of them and so this means that like compactness issues are going to suck. Well maybe they don't suck, maybe they're beautiful but they make the theory harder. Alright so everyone's favorite congerum from 2000 which is in a paper by Ali Ashbur Gimental and Hofer says that assume there are no contractable rape orbits of index minus one zero or one so this is this grading that's the conglisander index plus and minus three. And, okay, it also started by saying what Y, C, V, A non-degenerate contact manifold, then the differential is well defined, of course no one really said which differential they were using which adds another layer of confusion but that's okay. J is generic. Okay, that you actually get a homology so DQ squared is zero and this homology which depended on your contact form, your choice of J was actually an invariant of the contact distribution so EG independent of the choice of lambda and almost complex structure J. I think should be fair, they didn't quite say that because they said that some abstract perturbations of the modulate spaces would be needed but they wouldn't go into the details of that. Because this one is too much of a fault. Well, this is, well it doesn't even appear in this form in their paper, it appears as a series of like lemmas or propositions. I mean they said at the beginning this is all suiting some perturbations of the name. Well at the beginning it said that this is all like conjectural blah blah blah so this is why it's a congerum so it's a conjecture of A plus. So the conjecture is that the statement can be corrected to the other side. Well the other problem is that some people then said you don't need abstract perturbations and this works anyways. Is anyone happy? No? No one's happy? Okay. So what is this beautiful compactness that I alluded to? So this is Bourgeois, Lyoszberg, Hofer, Vizatsky and Zander. This is a result from 2003. Again I'll paraphrase this result. So there's no bubbling like in Hamiltonian Flora theory but we do have breaking of curves into buildings. So Michael drew some beautiful buildings of asymptotically cylindrical pseudo-holomorphic curves. And so the heuristic reason for why we have this breaking is because so there is a maximum principle in the R component of the symplectization but you can grow minimums and so if you try to compactify the space you're going to be allowed to develop new punctures at the bottom end. So you had some index two cylinder. You could conceivably start to have the picture people draw so you have some sort of a minimum developing in the R direction and so then if you want to compactify the space you could pop off a plane and this would be allowable elements of your compactification if you cannot sort of exclude it for other reasons. And then this makes us sad. And cylinders in your compactified modulized space of index two curves. I have a dumb question. Why is breaking and not bubbling? How much? Because you have cylindrical end which is as a real bubbling parameter instead of a complex bubbling parameter. It breaks like a trajectory of more stereo or core theory and not like a bubbling chroma or a bubbling, okay, that's a bubbling parameter. So I'm not going to draw a complicated sequence of buildings but you can imagine that sort of, well, Michael was talking about how you can have branched covers of trivial cylinders. Those guys can have index zero so you could be in some situation where x is gamma squared, z is gamma and then the cylinder could break to be gamma squared, gamma, gamma, gamma and this could be a holomorphic plane of index two. This guy has index zero and this guy has index zero. So this would be an allowable compactification. And in dimension three there is really nice iteration formulas for the Conly's Ender Index plus automatic transversality so I'll explain in a minute how you can salvage this theory in dimension three but in higher dimensions you can have cylinders of index one and you take their multiple cover and they suddenly have negative index. And then that means that you, in other elements of your compactified modular space, are going to have to include things like a chain of cylinders where you just have to end up being able to add to two at the end of the day. So that makes things complicated. And the, you know, philosophy with a lot of this stuff is well if we only had some more injective curves you just pick j generically so all of my moduli spaces have to have positive virtual dimension because we're working in a simpletization and have this r action and the only way that you can add to two with positive numbers is one plus one. And this original assumption about no contractable orbits of index minus one zero one come because this sort of building is not excluded by the assumptions but you could have some building where you have one, one and one. So this is like why the assumptions in the congerum exist. And then the zero and minus one sort of come up when you try to prove invariance and homotopy depending on what level of transversality you want to assume. So let me see. So it turns out that in dimension three you can get automatic transversality and index calculations to work favorably. And so in my thesis I found not a very large class of contact forms but the class of some contact forms where you could actually define contact tomology. So sort of the definition I came up with is we say a contact form is dynamically separated provided the conles-eander index of gamma is between three and five for gamma simple and contractable. And every time you iterate gamma you're just increasing the conles-eander index by four. So this holds and then plus you can do this if you, so I guess I should say if gamma is non-contractable have analogous definition but you have to keep track of free homotopy classes of ray orbits and it just becomes a mess to write down so I'm not going to. Is there a question in the front? Okay. And then the theorem that I proved is that if, so this condition, this is usually true up to large action. And action is the integral along gamma of some ray of orbit. So by up to large action I mean for all ray of orbits of action less than or equal to something you have this dynamically separated property and these guys all for the most part arises there's a natural class of perturbations you can put on pre-quantization spaces so this is where these guys come from. So if my three lambda is non-degenerate dynamically separated then and J generic then the cylindrical contact tomology is defined and sometimes up to like some large action sort of. So if you don't have this dynamically separated for all of your ray of orbits but only up to some large action then you can only define cylindrical contact tomology up to that large action and index and then there's a way you can take direct limits to actually sort of compute it. Variance under the choice of J and dynamically separated contact form. So this is not so awesome because it excludes a lot of examples like dynamically like certain dynamically convex things like ellipsoids so you team up with someone that knows more than you and then you prove a theorem. Six things is annoying. So a dynamically convex contact manifold is something that Hopper, Vazatsky and Zander studied in the 90s and what Katrin knew as HWZ papers when she was a graduate student. So a dynamically so Y3 lambda is dynamically convex provided the first churn class of the contact distribution restricted to pi2 of Y is 0 and the Conley-Zander index of gamma is greater than or equal to 3 for all contractable ray of orbits. And so an example is any strictly convex hyper surface in R4 with the standard symplectic structure which is transverse to the radial vector field. So this is a nice class of contact manifolds because I erased the theorem but it's basically without sort of a theory of abstract perturbations. It's going to be the class of contact manifolds so that all the contractable ray of orbits give rise to planes that all have to have virtual dimension at least two. So this would be sort of like a nice class of manifolds you'd like to prove. So the termology is defined for an invariance in dimension 3. So you also dropped the assumption in C1 if there were no contract. So we'll make condition A C1 of pi2 is 0 or B. So what Michael and I proved is that if Y lambda is dynamically convex, J is generic, then the only buildings of index 2 in R cross M are either just a cylinder, a once broken cylinder or if you, I should also write this, so we're going to assume that Connelly's Andrer index of gamma is strictly greater than 3 for gamma non-simple and contractable and I'll explain why in just a second or you have a pair of pants, a plane and a trivial cylinder so gamma squared gamma. So this is 0, 0, 2. So we have this extra assumption that, there should be gamma d, d minus 1. So we have this assumption that the Connelly's Andrer index is strictly greater than 3 for gamma non-simple and contractable. We expect that it's removable but we have to do a little bit more estimating things to make sure that something bad doesn't happen in the next blackboard but sort of what we proved is that the only buildings of index 2 are either an unbroken cylinder, a once broken cylinder or this pair of pants configuration. If you assume, if you don't have this assumption, then your pair of pants configuration can have like this gamma could be gamma to the d1, this could be gamma to the d2 and then this is just a d1 plus d2 fold cover of a trivial cylinder. And the second part of our lemma is that no index 2 cylinder can limit to the third configuration. So we didn't prove that there are no buildings of this type, there certainly are, but we just showed that there's no index 2 cylinder that could limit to it, which means that as a corollary the cylindrical contact tomology differential is well defined and it squares to 0. And this is innocent flexion, so the reason why we don't have invariance yet is because this sort of analogous lemma in cubortisms for sure does not hold. Any questions? So a priori, what are the possible configurations for that third one? I mean, I'm looking at this and you're saying if it can't be the case that a cylinder of index 2 limits on third configuration, I'm looking at that thinking, okay, well, if in some world I could perturb and actually glue that thing together in some sense, then it would be an index 2 cylinder. So therefore, if you, like, what kind of perturbing do you want to do? So if you were doing obstruction bundling gluing and you decided to try to see if you could do this, the number of ways you could glue it together would be 0. Actually, just to, that's not how you prove it, right? No, no, no. It's a proof by contradiction, sorry. So we use, yeah, you use intersection theory. You say, like, suppose you have this curve living into this, then you look at some intersection theory properties and prove that you get a contradiction. Therefore your index 2 cylinder can't limit to a building of that type. True proofs are actually the same if you look deeply enough. Yeah, and since obstruction bundle gluing goes back to winding numbers and, um, actually, I have, well, can't the obstruction bundle gluing, can it only tell you that the sign counter phrase to glue is 0, or is this really telling you there's no way to glue it? I think in this case you'll actually see if the obstruction section never manages to look for that. In where do you use the index 2 at the point of the dynamic convex? Sorry? I know that the proof has this intersection theory stuff. Oh, the intersection theory stuff is to prove that no index 2 cylinder can limit to the third configuration. And where does that be used, the fact that it's a dynamic convex? Oh, so we use the fact that it's dynamically convex to prove that the only buildings of index 2 are of those three types. And then you also, I mean, yeah. And then you're also using automatic transversality and some index stuff, improving that those are the only three types of buildings, and that's also what's getting you the transversality you need to say that the differential is well-defined. So we also prove that all index 1 and index 2 cylinders and subluxizations are cut out transversely and, um. So do you need the dynamic convex for that? So you don't need dynamically convex for that, but you're going to have a problem with looking at your compactified, modulized space of index 2 cylinders. Like then I'm going to have way more things, and I'm not going to be able to prove that d squared is zero. Let's say you can know that you only have these three kinds. You can only get that if you're dynamically convex. You all have bad vision, the only people in the front. But what are the indices of the curves in the third building? Zero, zero, and two? Yeah. I didn't want to put the zeros here and here, because then the people in the front row would ask why those curves had genus and I was counting cylinders. Yeah, no, no, it's good. I just need glasses. The top level there is a branch cover of a trivial cylinder. He's a trivial, trivial cylinder. Everyone happy? If you're not happy, I'll get out of my hook. I actually really like holding the hook, but I think that makes for a slightly scarier talk. I was going to move on to the non-equivariant part of the talk, but do people have any other questions about what goes into the Waman corollary? I guess I should maybe upgrade it to a theorem. So two, we're stuck on invariants, so we need to do something. So we'll try to define, so we'll define stuff non-equivariant. So we'll define a non-equivariant version. If I stuck, you mean data approachable? Oh, yeah. Okay. So to clarify, this normal approach looks bad because you can't prove this sort of lemma in co-bordisms. You also don't have the sort of automatic, I mean, you have some automatic transversality in co-bordisms, but it's not good enough, and you're not going to be able to prove the chain homotopy equation. So invariance is not really going to go so well. So what we can do is we can define a non-equivariant version. And when we set up a non-equivariant version, it's kind of analogous in Hamiltonian-Fleur theory to when you pick a time-dependent Hamiltonian. So all of your curves end up being somewhere injective. So we're going to do something similar in the contact world. There's still going to be some things we have to work out, but they're doable. And then it turns out that there's a way to take an S1-equivariant version of this non-equivariant picture, prove that it's isomorphic to classical cylindrical contact tomology, and in a very roundabout way, get invariance for the classical cylindrical contact tomology. OK. So this non-equivariant theory is kind of look like the positive part of symplectic homology, whereas this cylindrical theory, if it was defined, it would look like the S1-equivariant version of the positive part of symplectic homology. But we don't want to use symplectic homology because we'd like to define something directly in terms of the RABE data and in terms of asymptotically cylindrical pseudoholomorphic curves as opposed to looking at the Fleur equation. OK. So we're going to take J here. Instead of using 1J for all time, we're going to take a domain-dependent J. So JT for T and S1 is going to be an almost complex structure. And we still want for each time T that it goes to the RABE vector field, and we want for each time T that it's D e to the tau lambda compatible. And then we're going to be counting, well, we're not going to be quite counting pseudoholomorphic curves where I just throw this T dependence in because that's not going to automatically get us a non-equivariant version. We're going to have to be a little bit more careful than that. But the moduli space we're going to start to want to look at and put point constraints on is going to be this guy where we're using a domain-dependent family of almost complex structures. OK. So you just put a T in your J-homomorphic curve equation. OK. And this T depends. Oh, shit. Wait. Sorry. My R coordinate should have been S and my S1 coordinate should have been T. Now it makes a lot more sense. OK. Maybe this is stupid, but what happens if you make lambda time dependent? Well, so this is forcing in some sense. You can't make your lambda time dependent. This is like the closest you can get to forcing your periodic orbits to be time dependent. I can't make lambda time dependent. It's a content. How would you make it time dependent? It's just a contact form. I could take a time, a group of context, and give them the same context. Well, I would rather take a loop of... We're not looking at period one orbits. We're looking at arbitrary period orbits. Oh, yeah. That's the other thing. Sorry. I mean, the point is now all the curves that were multiply covered are no longer going to be multiply covered for a generic family of almost complex structures that depends on the domain. So, the good news for a generic family, mj gamma plus, gamma minus is a manifold of dimension equal to the Conley-Zehnder index of gamma plus minus the Conley-Zehnder index of gamma minus. And I broke some symmetry, so it's got an extra dimension on it. All right. And so you're going to be working in a situation where the... So there's a paper by Fleur Hofer and Zalaman that gives you transversality in the Hamiltonian Fleur case. And so this is sort of the... You can't pick a time-dependent Hamiltonian, but you can pick a time-dependent, almost complex structure in your J-homomorphic curve equation. And so that's how... So all your curves are going to be somewhere injective and ta-da. Okay. But you're not going to be able to define the differential sort of in this naive way here. So let me tell you how to define the differential. Okay. So to define a differential, we're going to need to throw in some point constraints. And this is kind of how we're forcing a parametrization of our ray of orbits in terms of time. So here's my curve. I'm going to be able to look at U of r cross 0. And this is going to pick out some line on my cylinder. Okay. And you can use this to define evaluation maps from our modulized space into the image of gamma plus minus, which we define to be the limit as s goes to plus or minus infinity of the projection of the curve when we look at time 0. Okay. So this is supposed to be like when t equals 0. And this is some point that's in my ray of orbit gamma plus. And you can do the usual thing where you can map this evaluation, map the sense to an evaluation map after you quotient out by r. Because even if we do define a differential, we're still going to have to mod out by r where this r is coming from the translation and the simpletization direction. Okay. Great. And what do I want to say? Okay. So chain complex, I no longer have to throw out all these bad ray of orbits, but I am going to have to have twice the number of ray of orbits in it as before. So you can kind of think of this as a Morse BOT version where your Morse BOT manifolds through your ray of orbits and you put a height function on s1. And so that gives you two critical points. One critical point has index 0, one has index 1. So we're just going to kind of, that's not really what we're doing, but it's the vestigial motivation for defining our non-equivariant chain complex to be generated. So it has two generators associated to each ray of orbit. So gamma check and gamma hat, gamma is any ray of orbit. And then I'm just going to arbitrarily shift the grading up on my gamma hats by one. So we have that gamma, the image of gamma, okay, I'm just going to say that gamma conleysander index of gamma check is equal to gamma, conleysander index of gamma hat is equal to the grading, I reversed this. So I should have written the inequalities the other way, but the grading on gamma, what am I doing? Okay, I'm getting confused with hats and checks. Okay, so the grading of gamma hat is going to be equal to the conleysander, the grading of gamma check is going to be equal to the conleysander index. The grading of gamma hat is just going to be the conleysander index shifted up by one, and conleysander index of gamma hat and gamma check doesn't make sense because it's the same underlying ray of orbit gamma. So this should be the correct formula. So you just think that I have gamma check and gamma hat are formal variables associated to the ray of orbit gamma. Okay, yeah, this is any ray of orbits, you allow also the battery of orbits. So it's just like in symplectic homology you would also see the battery of orbits. And this is kind of, so this non-equivariant picture is sort of not original, but it is an adaptation from stuff by Berjewil, Yashberg, and Ekholm, and also Berjewan Wansha. All right, and now my differential is going to have a block decomposition. And I'm going to have sort of stuff that goes from a check, ray of orbits to check ray of orbits. So there's four different ways, sort of you can go from, let me just write this out and then we can talk about it. So hat, this is just z generated by this guy. Let me see, check, let's see. So, okay. All right, and I promise that you'll see this evaluation map and I just wrote down and show up. Uh-huh. So we think of it as like a triumph that you no longer have to try out for bad ray of orbits? Is that something desired or is it just a quirk of this? I mean, so it is desirable and it turns out that this S1-equivariant version, like once we figure out what the S1-equivariant version of this non-equivariant theory is, it's actually defined over z. And so it's going to have more information coming from bad ray of orbits because it's defined over z. And it turns out that if you tensor that theory with q, you get the usual cylindrical contact tomology theory. So it's a quirk, but I guess it's good because you get more information. Okay, so what do each of these terms count? So d check, alpha check, beta check. So here you would want the degree index between alpha and beta to be 1. So this will count cylinders where you are going to, right, so you want to, you're going to fix a point on the underlying simple ray of orbit associated to alpha. So this is some point on the image of, and alpha bar is a simple ray of orbit. So alpha, and then we want, so this is going to be curves u in this moduli space going from alpha to beta, mod r, with the positive evaluation map of your curve u being equal to p alpha bar. And we're also going to have to count Morse bot buildings with a cyclic ordering condition. So the evaluation map constraint is supposed to be this sort of point where p alpha bar it needs to match up with pi y of u of s0. And then we'll count these guys, and then we'll also have some buildings that we have to count where you've got a cyclic ordering condition, meaning that you want f minus of the top curve, f plus of the bottom curve, and whatever point you're picking on this intermediate curve, let's call this guy gamma, to be ordered this way as you go around gamma. So you're going to need to orient your, like a direction for how you're going around your ray of orbit. And then the check of alpha check, beta check is going to be something analogous, except instead of having the positive evaluation constraint, you're going to have the negative evaluation constraint, and then negative evaluation constraint, d plus alpha check, beta hat, this has no constraint. So this looks like the too bad map that you see in the Birch-Wile-Yashberg-Eckholm formulation. So if we look at d plus of alpha check, alpha, sorry, alpha hat, alpha check, you get two points if alpha is bad, and zero if alpha is good, and d minus alpha check, alpha hat, this has constraints at the top and bottom. And this non-equivariant theory is not really so easy to count because not only are you sort of counting cylinders that either have one point constraint, two point constraints, or zero point constraints, you also need to count more spot buildings that have the cyclic ordering condition, and these more spot buildings, they're not going to be of infinite length because you only have a finite number of ray of orbits in each conical azander index, but they are going to be some sequence of buildings that you have to figure out how to count. But it turns out that if, so I say, okay, so theorem is that, oh, and I should have this d minus here, so this also, in addition to counting u and m alpha beta with f plus and of minus constraints and more spot buildings, it also will count ways to glue an index 3 curve using obstruction bundle gluing that looks like a branch cover of a cylinder. So, this is zero, two, and one. Okay. What is the j on the cap there? The, it's a two. No, it's a j, I mean, so you fix a family j of t on the remains. Yeah. So what would that be on the cap this class? So it's a bubbling off at a point? Yeah. Yeah. So some j would bump it off? Yeah. Is it clear why you can only have one plane of all this? Yeah, index calculations. Is the reason that you define this chain problem is to see that the kind of penance means you can do isotropy? Yeah. So your curves are always going to be somewhere injective, so you don't have the isotropy anymore to deal with. So for y3 lambda dynamically convex or y2n minus 1 lambda, well, no, I don't want to say that. This non-equivirian contact homology and j generic is a chain complex, and the homology of it is independent. This is a family of j's of your choice of family of j's. And the dynamically convex lambda. Let me tell you how to relate this back to cylindrical contact homology in a way that will make everyone much happier than they are now. So if there is sufficient automatic transversality, then as needed to define the classical cylindrical contact homology differential, then we can use for some generic choice of j, we can use this j to look at this non-equivirian contact homology. And what's nice is that it turns out, so in this case, d check looks like the standard cylindrical contact homology differential. And d hat looks like the other cylindrical contact homology differential, and you end up having opposite orientation, but that's not so important. But it turns out that you can show that, so if y lambda is non-degenerate, dynamically convex j generic, then for d0 equal to dq minus d, well, I did not use good notation for this. So the first differential, this is supposed to be the second differential that I told you, d plus, d minus, and d1 equal to 0, 0 multiplicity of gamma 0. I'll explain what that is in just a sec. You can actually get a chain complex by taking the non-equivirian chain complex, tensoring it with the formal power series of variable 2 that has degree 2. So here u is a formal variable of degree 2 with dz being equal to d0 plus d1, and this is j, that this is a chain complex. So this was kind of a lie. So you still will be able to see your bad orbits, and you're not going to have the problems that you had before with orienting them because you have point constraints running around that are going to allow you to deal with orientations. So I'm doing a bad job explaining this because there are a lot of details I'm omitting. But then, so the degrees of d0 are coming from a complex where the bad points are at the same time. Oh, okay. This shouldn't be, I should have just written, this isn't technically this differential, it should be, I should have written something like this is counting, my notation is really awful. Yeah, we'll just use d hat and d check, but they have a more familiar expression in terms of multiple cities of ray orbits and multiple cities of your pseudoholomorphic curves. So what I should have written here is that if j is, if we can replace j with a single j, then d check counts u and mj alpha beta, you still are going to have your checks and hats with a point constraint at the top and you'll get that d check of alpha check is going to be the sum over the multiplicity of alpha divided by the multiplicity of your curvy u that connects them and now my font has become microscopic beta with the orientation. So the chain complex is still this thing that we're using and then the differentials just have more familiar expressions in terms of the previous differential, but that was a very unclear and helpful way that I explained that. Can I say one thing? I mean the d hat and d check, they're the same as the differentials before when you're going between good orbits. Yeah, and you can show that when they do encounter bed ray orbits nothing horrible happens. Yeah, I should finish the statement of the theorem and the homology Z of this chain complex is independent of lambda and j and a proposition is that CH Z tensored with q is isomorphic to CH q and so this is how you end up getting a variance of cylindrical contact homology. Okay, and I will end. I mean if you don't have something that's dynamically convex then you're going to have to be working with like full SFT or symplectic homology. Yeah, so if you take the connect sum of two tight spheres you're no longer dynamically convex. So this sort of result doesn't work for connect sums, but I mean cylindrical contact homology was never built to being in variance of things that weren't dynamically convex in dimension three. So there was a result that Viterbo mentioned in his talk where he said that if you have this assumption about conical Zander indices then the positive part of symplectic homology is an invariant of the contact structure. It's exactly this dynamical convexity condition and it has an analog in higher dimensions. What's your favorite manifold way to compute it? The sphere. So you can also compute it for like lens spaces and like pre-quantization spaces. What's the answer? Oh, so, well I guess I can just say it in words. For the three sphere you have Q in every even dimension that's greater than or equal to two. And for lens spaces you also only have for ln plus 1n you have n copies of Q in dimension zero and n plus 1 copies of Q in every even dimension that's greater than or equal to two. Any more conversation requests? Any other questions? Right, so that's what you're doing but you've kind of been able to cheat a little bit. So normally your differential would have like an infinite number of terms in it but because we could cheat and use our use a time independent almost complex structure and still be able to define the complex that's why we were able to drop the extra terms but that's pretty much exactly what's going on. Suppose you took two dynamically convex just go back to the convex sum idea. You took two dynamically convex and then you did a convex sum but you did a really, really, really small convex sum where you have very precise control over the sort of, you have very precise control over this hyperbolic orbit and the finite energy planes that would be sort of introduced. Is there any chance that these ideas could extend in that case? I mean maybe but it's my understanding that even when you do a convex sum you always end up introducing a new... You do....rave orbit. But you know exactly where it is and you know pretty quickly you have pretty much......generated longer ones. Yeah you could use action filtration arguments anyway. No because the...my answer is no. So I mean it's conceivable that we could define some sort of linearized contact tomology if we...there's another type of building that we would have to consider and we'd have to be able to understand how to glue it or exclude curves limiting on it and you'd have to take some sort of like augmentation so... I'm going to disagree with you on this one. You think that... We might be able to do it. Well that's...yeah I was saying you might be able to... It's not linearized which is super cool. It might be possible but it's a harder project. Anything is possible. Okay I think Helmut's ready to clap for me and get coffee. Yes so... Yeah why not? Have a good one.
 Thanks for the introduction. I'd like to thank the organizers for allowing me to talk.  Okay, so a lot of this will be joined with Michael and it was helpful he came with his  obstruction, bumble gluing and intersection theory from ECH. So he was kind enough to lend  them to the contact tomology. Alright, so we're going to start off with a contact manifold.  For most of the talk it will be three-dimensional, but in some parts I might get to tell you  about what we can do in higher dimensions. This is going to be non-degenerate contact  and it's going to be co-oriented meaning that there's a global one form that defines the  contact structure and for grading reasons let's pretend or assume that the first turn class  of the contact distribution vanishes and then R is going to be my rave vector field. Okay,  and then I'm going to have J is an almost complex structure on r cross m and I'll take  d e to the tau lambda. So here tau is going to be my coordinate on r. It's going to be  the translation invariant so I have d tau r is going to be equal to r and I want J restricted  to, I'll just say J is d e to the tau lambda compatible. Okay, then we want to count cylindrical  curves so crystallized asymptotically cylindrical curves are curves the domain is just going  to be a cylinder so it's not going to be super exciting but they're still fun to look at.  So I'll denote the space of, so let's gamma plus gamma minus be periodic rave orbits.  Mj gamma plus gamma minus is going to be the set of J holomorphic curves from a cylinder  r cross s1 j0 into the simpletization  with this d lambda r invariant almost complex structure and then we want du of J to be J  of du so we want them to be pseudo holomorphic and then we want the limit so if we take r  to have the t coordinate s1 to have the s coordinate we want the limit as s goes to  the plus or minus infinity of the projection of u and the contact manifold component to  be a reparameterization of the rave orbit at gamma plus if we go to positive infinity  and gamma minus if we go to minus infinity and then we want the limit as s goes to plus  or minus infinity of the projection to the r component of u to be plus or minus infinity  so you want to think like if my contact manifold was something stupid like a circle then your  counting curves like this gamma plus gamma minus and then there's also going to be an  r action that technically you have to mod out by coming from translations on the target  and so when we go to look at the differential there will be this translation action but  a lot of times I'll ignore it when it comes to talking about index calculations and stuff.  Okay so.  Is anything other than the gradient dependent on your assumption that c1 equals 0?  No.  So I can just drop that and you have your z2 gradient?  Yeah I mean eventually I have to yeah you can drop that and use the z2 gradient of course  when I define a dynamic convex contact form you'll need some sort of way of making sense  of the gradient on contractable orbits.  So the gradient is given by the Conly's Ender index on our chain complex we have that the  virtual dimension of this moduli space is given by gamma plus minus the gradient of  gamma minus and here the gradient is an SFT grading so this is going to be Cz of gamma  plus n minus 3 so if the dimension of our contact manifold is 3 then you get this is  just Cz of gamma minus 1.  So then I can tell you what the chain complex is.  So this is going to be generated by only good Ray-Borbets so we take the set of all Ray-Borbets  and then we throw out bad Ray-Borbets because they're bad and so to define a bad Ray-Borbets  a little bit annoying but so the Cz index of a Ray-Borbets can either have the same  parity under iteration of your orbit  or it can flip flop between being even and odd sorry odd and even if it flip flops then  you throw out the even multiple covers so bad Ray-Borbets even covers of flip the Cz  guys if you want the real definition or you want an example so in dimension 3 bad Ray-Borbets  are even multiple covers of negative hyperbolic guys meaning that they have the eigenvalue  of the linearized return map if we restrict it to the contact distribution is negative  and real.  Is there a reason for your V there for which the Cz VQ like in your top line?  It's not a V this is the vector space generated by all the Ray-Borbets and you throw out all  the bad Ray-Borbets.  Yeah sorry he just thinks the stars never mind.  Is there another way for why are bad Ray-Borbets bad?  So if you keep them in the chain complex even if you had all the transversality in the world  you'd never be able to prove invariance but secretly they you can't orient curves that  limit on bad Ray-Borbets unless you have some sort of a parametrization you're using of  your bad Ray-Borbets and so there's a when you try to assign orientations you can get  both positive and negative orientations assigned to the same cylinder and this causes a headache  and in orbithold Morse theory you can kind of think of the bad Ray-Borbets as being like  the bad points whose like gradient flow lines get reversed underneath whatever action you're  modding out by.  Any other questions?  It's mobilized spaces.  Yeah so that is kind of incorporated by the fact that this is the projection onto the  contact manifold as a reparametrization of gamma plus minus so that already includes  that.  And then the differential.  So there's actually two equivalent ways over Q of defining the contact differential.  So either we can define it to so I'll just write this down and then I'll say.  So either you can define it by looking at the multiplicity of the top Ray-Borbit or you  can define it by encoding the multiplicity of the bottom Ray-Borbit.  So and this epsilon U is just supposed to be some choice of coherent orientations.  M of U is the degree of the covering map of the curve if your curve U is not somewhere  injective and the multiplicity of your Ray-Borbit just means you've had to iterate some simple  orbit some number of times and the multiplicity will be the number of times you've iterated  a simple Ray-Borbit to get to X.  And so it might look like these coefficients are defined are not integral but they are  in fact integral.  And so you can see that because if you have a k to one fold cover where you have X is  gamma let's say plus to the kp and Y is gamma minus to the kp this would be a k to one fold  cover of a cylinder gamma plus to the p gamma minus to the q and so the multiplicity of X  is kp the multiplicity of Y is kp the multiplicity of this curve U is k and k obviously divides  both kp and kq.  I'll just write against that M of X equals one if X is a simple Ray-Borbit so maybe simple  hasn't been defined but that just means that the map X which is r mod tz into your contact  manifold is injective and M of U equals one if U is somewhere injective.  So that looks like beautiful theory but it's not really so beautiful.  So some of the issues that are going to arise is that we're working with multiple covers.  Is there a question?  So can you explain the difference between the two?  One encodes the multiplicity of the top Ray-Borbit and one encodes the multiplicity of the bottom  Ray-Borbit of your cylinder.  And it turns out that their equivalent sort of if you had all the transfer salinity in  the world that would mean that your differentials were defined and you actually got a homology  and then on the chain level you could just divide and multiply out by multiple seed of  so Ray-Borbit's the pass from one chain complex to the other chain complex.  So homologies are isometric under this?  If you live in magical transfer salinity.  Yes but they're equivalent.  So why is this theory possibly not so correct?  So the first issue is that we're working with multiple covered curves and Chris spent many  lectures telling us as well as Dusa and others telling us about the horrors of multiple covered  curves.  And then also I think Michael mentioned this a few times but you can look at multiple covered  curves and they can end up having negative index even though they live in even though  they have to be there and you can't just use a generic perturbation of J to get rid of  them and so this means that like compactness issues are going to suck.  Well maybe they don't suck, maybe they're beautiful but they make the theory harder.  Alright so everyone's favorite congerum from 2000 which is in a paper by Ali Ashbur  Gimental and Hofer says that assume there are no contractable rape orbits of index minus  one zero or one so this is this grading that's the conglisander index plus and minus three.  And, okay, it also started by saying what Y, C, V, A non-degenerate contact manifold,  then the differential is well defined, of course no one really said which differential  they were using which adds another layer of confusion but that's okay.  J is generic.  Okay, that you actually get a homology so DQ squared is zero and this homology which  depended on your contact form, your choice of J was actually an invariant of the contact  distribution so EG independent of the choice of lambda and almost complex structure J.  I think should be fair, they didn't quite say that because they said that some abstract  perturbations of the modulate spaces would be needed but they wouldn't go into the details  of that.  Because this one is too much of a fault.  Well, this is, well it doesn't even appear in this form in their paper, it appears as  a series of like lemmas or propositions.  I mean they said at the beginning this is all suiting some perturbations of the name.  Well at the beginning it said that this is all like conjectural blah blah blah so this  is why it's a congerum so it's a conjecture of A plus.  So the conjecture is that the statement can be corrected to the other side.  Well the other problem is that some people then said you don't need abstract perturbations  and this works anyways.  Is anyone happy?  No?  No one's happy?  Okay.  So what is this beautiful compactness that I alluded to?  So this is Bourgeois, Lyoszberg, Hofer, Vizatsky and Zander.  This is a result from 2003.  Again I'll paraphrase this result.  So there's no bubbling like in Hamiltonian Flora theory but we do have breaking of curves  into buildings.  So Michael drew some beautiful buildings of asymptotically cylindrical pseudo-holomorphic  curves.  And so the heuristic reason for why we have this breaking is because so there is a maximum  principle in the R component of the symplectization but you can grow minimums and so if you try  to compactify the space you're going to be allowed to develop new punctures at the bottom  end.  So you had some index two cylinder.  You could conceivably start to have the picture people draw so you have some sort of a minimum  developing in the R direction and so then if you want to compactify the space you could  pop off a plane and this would be allowable elements of your compactification if you cannot  sort of exclude it for other reasons.  And then this makes us sad.  And cylinders in your compactified modulized space of index two curves.  I have a dumb question.  Why is breaking and not bubbling?  How much?  Because you have cylindrical end which is as a real bubbling parameter instead of a complex  bubbling parameter.  It breaks like a trajectory of more stereo or core theory and not like a bubbling chroma  or a bubbling, okay, that's a bubbling parameter.  So I'm not going to draw a complicated sequence of buildings but you can imagine that sort  of, well, Michael was talking about how you can have branched covers of trivial cylinders.  Those guys can have index zero so you could be in some situation where x is gamma squared,  z is gamma and then the cylinder could break to be gamma squared, gamma, gamma, gamma and  this could be a holomorphic plane of index two.  This guy has index zero and this guy has index zero.  So this would be an allowable compactification.  And in dimension three there is really nice iteration formulas for the Conly's Ender  Index plus automatic transversality so I'll explain in a minute how you can salvage this  theory in dimension three but in higher dimensions you can have cylinders of index one and you  take their multiple cover and they suddenly have negative index.  And then that means that you, in other elements of your compactified modular space, are going  to have to include things like a chain of cylinders where you just have to end up being  able to add to two at the end of the day.  So that makes things complicated.  And the, you know, philosophy with a lot of this stuff is well if we only had some more  injective curves you just pick j generically so all of my moduli spaces have to have positive  virtual dimension because we're working in a simpletization and have this r action and  the only way that you can add to two with positive numbers is one plus one.  And this original assumption about no contractable orbits of index minus one zero one come because  this sort of building is not excluded by the assumptions but you could have some building  where you have one, one and one.  So this is like why the assumptions in the congerum exist.  And then the zero and minus one sort of come up when you try to prove invariance and homotopy  depending on what level of transversality you want to assume.  So let me see.  So it turns out that in dimension three you can get automatic transversality and index  calculations to work favorably.  And so in my thesis I found not a very large class of contact forms but the class of some  contact forms where you could actually define contact tomology.  So sort of the definition I came up with is we say a contact form is dynamically separated  provided the conles-eander index of gamma is between three and five for gamma simple and  contractable.  And every time you iterate gamma you're just increasing the conles-eander index by four.  So this holds and then plus you can do this if you, so I guess I should say if gamma is  non-contractable have analogous definition but you have to keep track of free homotopy  classes of ray orbits and it just becomes a mess to write down so I'm not going to.  Is there a question in the front?  Okay.  And then the theorem that I proved is that if, so this condition, this is usually true  up to large action.  And action is the integral along gamma of some ray of orbit.  So by up to large action I mean for all ray of orbits of action less than or equal to  something you have this dynamically separated property and these guys all for the most part  arises there's a natural class of perturbations you can put on pre-quantization spaces so  this is where these guys come from.  So if my three lambda is non-degenerate dynamically separated then and J generic then the cylindrical  contact tomology is defined and sometimes up to like some large action sort of.  So if you don't have this dynamically separated for all of your ray of orbits but only up  to some large action then you can only define cylindrical contact tomology up to that large  action and index and then there's a way you can take direct limits to actually sort of  compute it.  Variance under the choice of J and dynamically separated contact form.  So this is not so awesome because it excludes a lot of examples like dynamically like certain  dynamically convex things like ellipsoids so you team up with someone that knows more  than you and then you prove a theorem.  Six things is annoying.  So a dynamically convex contact manifold is something that Hopper, Vazatsky and Zander  studied in the 90s and what Katrin knew as HWZ papers when she was a graduate student.  So a dynamically so Y3 lambda is dynamically convex provided the first churn class of the  contact distribution restricted to pi2 of Y is 0 and the Conley-Zander index of gamma  is greater than or equal to 3 for all contractable ray of orbits.  And so an example is any strictly convex hyper surface in R4 with the standard symplectic  structure which is transverse to the radial vector field.  So this is a nice class of contact manifolds because I erased the theorem but it's basically  without sort of a theory of abstract perturbations.  It's going to be the class of contact manifolds so that all the contractable ray of orbits  give rise to planes that all have to have virtual dimension at least two.  So this would be sort of like a nice class of manifolds you'd like to prove.  So the termology is defined for an invariance in dimension 3.  So you also dropped the assumption in C1 if there were no contract.  So we'll make condition A C1 of pi2 is 0 or B.  So what Michael and I proved is that if Y lambda is dynamically convex, J is generic,  then the only buildings of index 2 in R cross M are either just a cylinder, a once broken  cylinder or if you, I should also write this, so we're going to assume that Connelly's  Andrer index of gamma is strictly greater than 3 for gamma non-simple and contractable  and I'll explain why in just a second or you have a pair of pants, a plane and a trivial  cylinder so gamma squared gamma.  So this is 0, 0, 2.  So we have this extra assumption that, there should be gamma d, d minus 1.  So we have this assumption that the Connelly's Andrer index is strictly greater than 3 for  gamma non-simple and contractable.  We expect that it's removable but we have to do a little bit more estimating things  to make sure that something bad doesn't happen in the next blackboard but sort of what we  proved is that the only buildings of index 2 are either an unbroken cylinder, a once  broken cylinder or this pair of pants configuration.  If you assume, if you don't have this assumption, then your pair of pants configuration can  have like this gamma could be gamma to the d1, this could be gamma to the d2 and then  this is just a d1 plus d2 fold cover of a trivial cylinder.  And the second part of our lemma is that no index 2 cylinder can limit to the third configuration.  So we didn't prove that there are no buildings of this type, there certainly are, but we  just showed that there's no index 2 cylinder that could limit to it, which means that as  a corollary the cylindrical contact tomology differential is well defined and it squares  to 0.  And this is innocent flexion, so the reason why we don't have invariance yet is because  this sort of analogous lemma in cubortisms for sure does not hold.  Any questions?  So a priori, what are the possible configurations for that third one?  I mean, I'm looking at this and you're saying if it can't be the case that a cylinder of  index 2 limits on third configuration, I'm looking at that thinking, okay, well, if  in some world I could perturb and actually glue that thing together in some sense, then  it would be an index 2 cylinder.  So therefore, if you, like, what kind of perturbing do you want to do?  So if you were doing obstruction bundling gluing and you decided to try to see if you  could do this, the number of ways you could glue it together would be 0.  Actually, just to, that's not how you prove it, right?  No, no, no.  It's a proof by contradiction, sorry.  So we use, yeah, you use intersection theory.  You say, like, suppose you have this curve living into this, then you look at some intersection  theory properties and prove that you get a contradiction.  Therefore your index 2 cylinder can't limit to a building of that type.  True proofs are actually the same if you look deeply enough.  Yeah, and since obstruction bundle gluing goes back to winding numbers and, um, actually,  I have, well, can't the obstruction bundle gluing, can it only tell you that the sign  counter phrase to glue is 0, or is this really telling you there's no way to glue it?  I think in this case you'll actually see if the obstruction section never manages to  look for that.  In where do you use the index 2 at the point of the dynamic convex?  Sorry?  I know that the proof has this intersection theory stuff.  Oh, the intersection theory stuff is to prove that no index 2 cylinder can limit to the  third configuration.  And where does that be used, the fact that it's a dynamic convex?  Oh, so we use the fact that it's dynamically convex to prove that the only buildings of  index 2 are of those three types.  And then you also, I mean, yeah.  And then you're also using automatic transversality and some index stuff, improving that those  are the only three types of buildings, and that's also what's getting you the transversality  you need to say that the differential is well-defined.  So we also prove that all index 1 and index 2 cylinders and subluxizations are cut out  transversely and, um.  So do you need the dynamic convex for that?  So you don't need dynamically convex for that, but you're going to have a problem with looking  at your compactified, modulized space of index 2 cylinders.  Like then I'm going to have way more things, and I'm not going to be able to prove that  d squared is zero.  Let's say you can know that you only have these three kinds.  You can only get that if you're dynamically convex.  You all have bad vision, the only people in the front.  But what are the indices of the curves in the third building?  Zero, zero, and two?  Yeah.  I didn't want to put the zeros here and here, because then the people in the front row would  ask why those curves had genus and I was counting cylinders.  Yeah, no, no, it's good.  I just need glasses.  The top level there is a branch cover of a trivial cylinder.  He's a trivial, trivial cylinder.  Everyone happy?  If you're not happy, I'll get out of my hook.  I actually really like holding the hook, but I think that makes for a slightly scarier talk.  I was going to move on to the non-equivariant part of the talk, but do people have any other  questions about what goes into the Waman corollary?  I guess I should maybe upgrade it to a theorem.  So two, we're stuck on invariants, so we need to do something.  So we'll try to define, so we'll define stuff non-equivariant.  So we'll define a non-equivariant version.  If I stuck, you mean data approachable?  Oh, yeah.  Okay.  So to clarify, this normal approach looks bad because you can't prove this sort of lemma  in co-bordisms.  You also don't have the sort of automatic, I mean, you have some automatic transversality  in co-bordisms, but it's not good enough, and you're not going to be able to prove the  chain homotopy equation.  So invariance is not really going to go so well.  So what we can do is we can define a non-equivariant version.  And when we set up a non-equivariant version, it's kind of analogous in Hamiltonian-Fleur  theory to when you pick a time-dependent Hamiltonian.  So all of your curves end up being somewhere injective.  So we're going to do something similar in the contact world.  There's still going to be some things we have to work out, but they're doable.  And then it turns out that there's a way to take an S1-equivariant version of this non-equivariant  picture, prove that it's isomorphic to classical cylindrical contact tomology, and in a very  roundabout way, get invariance for the classical cylindrical contact tomology.  OK.  So this non-equivariant theory is kind of look like the positive part of symplectic  homology, whereas this cylindrical theory, if it was defined, it would look like the  S1-equivariant version of the positive part of symplectic homology.  But we don't want to use symplectic homology because we'd like to define something directly  in terms of the RABE data and in terms of asymptotically cylindrical pseudoholomorphic  curves as opposed to looking at the Fleur equation.  OK.  So we're going to take J here.  Instead of using 1J for all time, we're going to take a domain-dependent J. So JT for T  and S1 is going to be an almost complex structure.  And we still want for each time T that it goes to the RABE vector field, and we want  for each time T that it's D e to the tau lambda compatible.  And then we're going to be counting, well, we're not going to be quite counting pseudoholomorphic  curves where I just throw this T dependence in because that's not going to automatically  get us a non-equivariant version.  We're going to have to be a little bit more careful than that.  But the moduli space we're going to start to want to look at and put point constraints  on is going to be this guy where we're using a domain-dependent family of almost complex  structures.  OK.  So you just put a T in your J-homomorphic curve equation.  OK.  And this T depends.  Oh, shit.  Wait.  Sorry.  My R coordinate should have been S and my S1 coordinate should have been T. Now it makes  a lot more sense.  OK.  Maybe this is stupid, but what happens if you make lambda time dependent?  Well, so this is forcing in some sense.  You can't make your lambda time dependent.  This is like the closest you can get to forcing your periodic orbits to be time dependent.  I can't make lambda time dependent.  It's a content.  How would you make it time dependent?  It's just a contact form.  I could take a time, a group of context, and give them the same context.  Well, I would rather take a loop of...  We're not looking at period one orbits.  We're looking at arbitrary period orbits.  Oh, yeah.  That's the other thing.  Sorry.  I mean, the point is now all the curves that were multiply covered are no longer going  to be multiply covered for a generic family of almost complex structures that depends  on the domain.  So, the good news for a generic family, mj gamma plus, gamma minus is a manifold of dimension  equal to the Conley-Zehnder index of gamma plus minus the Conley-Zehnder index of gamma  minus.  And I broke some symmetry, so it's got an extra dimension on it.  All right.  And so you're going to be working in a situation where the...  So there's a paper by Fleur Hofer and Zalaman that gives you transversality in the Hamiltonian  Fleur case.  And so this is sort of the...  You can't pick a time-dependent Hamiltonian, but you can pick a time-dependent, almost  complex structure in your J-homomorphic curve equation.  And so that's how...  So all your curves are going to be somewhere injective and ta-da.  Okay.  But you're not going to be able to define the differential sort of in this naive way  here.  So let me tell you how to define the differential.  Okay.  So to define a differential, we're going to need to throw in some point constraints.  And this is kind of how we're forcing a parametrization of our ray of orbits in terms of time.  So here's my curve.  I'm going to be able to look at U of r cross 0.  And this is going to pick out some line on my cylinder.  Okay.  And you can use this to define evaluation maps from our modulized space into the image  of gamma plus minus, which we define to be the limit as s goes to plus or minus infinity  of the projection of the curve when we look at time 0.  Okay.  So this is supposed to be like when t equals 0.  And this is some point that's in my ray of orbit gamma plus.  And you can do the usual thing where you can map this evaluation, map the sense to an evaluation  map after you quotient out by r.  Because even if we do define a differential, we're still going to have to mod out by r  where this r is coming from the translation and the simpletization direction.  Okay.  Great.  And what do I want to say?  Okay.  So chain complex, I no longer have to throw out all these bad ray of orbits, but I am  going to have to have twice the number of ray of orbits in it as before.  So you can kind of think of this as a Morse BOT version where your Morse BOT manifolds  through your ray of orbits and you put a height function on s1.  And so that gives you two critical points.  One critical point has index 0, one has index 1.  So we're just going to kind of, that's not really what we're doing, but it's the vestigial  motivation for defining our non-equivariant chain complex to be generated.  So it has two generators associated to each ray of orbit.  So gamma check and gamma hat, gamma is any ray of orbit.  And then I'm just going to arbitrarily shift the grading up on my gamma hats by one.  So we have that gamma, the image of gamma, okay, I'm just going to say that gamma conleysander  index of gamma check is equal to gamma, conleysander index of gamma hat is equal to the grading,  I reversed this.  So I should have written the inequalities the other way, but the grading on gamma, what  am I doing?  Okay, I'm getting confused with hats and checks.  Okay, so the grading of gamma hat is going to be equal to the conleysander, the grading  of gamma check is going to be equal to the conleysander index.  The grading of gamma hat is just going to be the conleysander index shifted up by one,  and conleysander index of gamma hat and gamma check doesn't make sense because it's the  same underlying ray of orbit gamma.  So this should be the correct formula.  So you just think that I have gamma check and gamma hat are formal variables associated  to the ray of orbit gamma.  Okay, yeah, this is any ray of orbits, you allow also the battery of orbits.  So it's just like in symplectic homology you would also see the battery of orbits.  And this is kind of, so this non-equivariant picture is sort of not original, but it is  an adaptation from stuff by Berjewil, Yashberg, and Ekholm, and also Berjewan Wansha.  All right, and now my differential is going to have a block decomposition.  And I'm going to have sort of stuff that goes from a check, ray of orbits to check ray of  orbits.  So there's four different ways, sort of you can go from, let me just write this out and  then we can talk about it.  So hat, this is just z generated by this guy.  Let me see, check, let's see.  So, okay.  All right, and I promise that you'll see this evaluation map and I just wrote down and show  up.  Uh-huh.  So we think of it as like a triumph that you no longer have to try out for bad ray of  orbits?  Is that something desired or is it just a quirk of this?  I mean, so it is desirable and it turns out that this S1-equivariant version, like once  we figure out what the S1-equivariant version of this non-equivariant theory is, it's actually  defined over z.  And so it's going to have more information coming from bad ray of orbits because it's  defined over z.  And it turns out that if you tensor that theory with q, you get the usual cylindrical contact  tomology theory.  So it's a quirk, but I guess it's good because you get more information.  Okay, so what do each of these terms count?  So d check, alpha check, beta check.  So here you would want the degree index between alpha and beta to be 1.  So this will count cylinders where you are going to, right, so you want to, you're going  to fix a point on the underlying simple ray of orbit associated to alpha.  So this is some point on the image of, and alpha bar is a simple ray of orbit.  So alpha, and then we want, so this is going to be curves u in this moduli space going  from alpha to beta, mod r, with the positive evaluation map of your curve u being equal  to p alpha bar.  And we're also going to have to count Morse bot buildings with a cyclic ordering condition.  So the evaluation map constraint is supposed to be this sort of point where p alpha bar  it needs to match up with pi y of u of s0.  And then we'll count these guys, and then we'll also have some buildings that we have  to count where you've got a cyclic ordering condition, meaning that you want f minus of  the top curve, f plus of the bottom curve, and whatever point you're picking on this  intermediate curve, let's call this guy gamma, to be ordered this way as you go around gamma.  So you're going to need to orient your, like a direction for how you're going around your  ray of orbit.  And then the check of alpha check, beta check is going to be something analogous, except  instead of having the positive evaluation constraint, you're going to have the negative  evaluation constraint, and then negative evaluation constraint, d plus alpha check, beta hat, this  has no constraint.  So this looks like the too bad map that you see in the Birch-Wile-Yashberg-Eckholm formulation.  So if we look at d plus of alpha check, alpha, sorry, alpha hat, alpha check, you get two  points if alpha is bad, and zero if alpha is good, and d minus alpha check, alpha hat,  this has constraints at the top and bottom.  And this non-equivariant theory is not really so easy to count because not only are you  sort of counting cylinders that either have one point constraint, two point constraints,  or zero point constraints, you also need to count more spot buildings that have the cyclic  ordering condition, and these more spot buildings, they're not going to be of infinite length  because you only have a finite number of ray of orbits in each conical azander index, but  they are going to be some sequence of buildings that you have to figure out how to count.  But it turns out that if, so I say, okay, so theorem is that, oh, and I should have  this d minus here, so this also, in addition to counting u and m alpha beta with f plus  and of minus constraints and more spot buildings, it also will count ways to glue an index 3  curve using obstruction bundle gluing that looks like a branch cover of a cylinder.  So, this is zero, two, and one.  Okay.  What is the j on the cap there?  The, it's a two.  No, it's a j, I mean, so you fix a family j of t on the remains.  Yeah.  So what would that be on the cap this class?  So it's a bubbling off at a point?  Yeah.  Yeah.  So some j would bump it off?  Yeah.  Is it clear why you can only have one plane of all this?  Yeah, index calculations.  Is the reason that you define this chain problem is to see that the kind of penance means you  can do isotropy?  Yeah.  So your curves are always going to be somewhere injective, so you don't have the isotropy  anymore to deal with.  So for y3 lambda dynamically convex or y2n minus 1 lambda, well, no, I don't want to  say that.  This non-equivirian contact homology and j generic  is a chain complex, and the homology of it is independent.  This is a family of j's of your choice of family of j's.  And the dynamically convex lambda.  Let me tell you how to relate this back to cylindrical contact homology in a way that  will make everyone much happier than they are now.  So if there is sufficient automatic transversality, then as needed to define the classical cylindrical  contact homology differential, then we can use for some generic choice of j, we can use  this j to look at this non-equivirian contact homology.  And what's nice is that it turns out, so in this case, d check looks like the standard  cylindrical contact homology differential.  And d hat looks like the other cylindrical contact homology differential, and you end  up having opposite orientation, but that's not so important.  But it turns out that you can show that, so if y lambda is non-degenerate, dynamically  convex j generic, then for d0 equal to dq minus d, well, I did not use good notation  for this.  So the first differential, this is supposed to be the second differential that I told you,  d plus, d minus, and d1 equal to 0, 0 multiplicity of gamma 0.  I'll explain what that is in just a sec.  You can actually get a chain complex by taking the non-equivirian chain complex, tensoring  it with the formal power series of variable 2 that has degree 2.  So here u is a formal variable of degree 2 with dz being equal to d0 plus d1, and this  is j, that this is a chain complex.  So this was kind of a lie.  So you still will be able to see your bad orbits, and you're not going to have the problems  that you had before with orienting them because you have point constraints running around  that are going to allow you to deal with orientations.  So I'm doing a bad job explaining this because there are a lot of details I'm omitting.  But then, so the degrees of d0 are coming from a complex where the bad points are at  the same time.  Oh, okay.  This shouldn't be, I should have just written, this isn't technically this differential,  it should be, I should have written something like this is counting, my notation is really  awful.  Yeah, we'll just use d hat and d check, but they have a more familiar expression in terms  of multiple cities of ray orbits and multiple cities of your pseudoholomorphic curves.  So what I should have written here is that if j is, if we can replace j with a single  j, then d check counts u and mj alpha beta, you still are going to have your checks and  hats with a point constraint at the top and you'll get that d check of alpha check is  going to be the sum over the multiplicity of alpha divided by the multiplicity of your  curvy u that connects them and now my font has become microscopic beta with the orientation.  So the chain complex is still this thing that we're using and then the differentials just  have more familiar expressions in terms of the previous differential, but that was a  very unclear and helpful way that I explained that.  Can I say one thing?  I mean the d hat and d check, they're the same as the differentials before when you're  going between good orbits.  Yeah, and you can show that when they do encounter bed ray orbits nothing horrible happens.  Yeah, I should finish the statement of the theorem and the homology Z of this chain complex  is independent of lambda and j and a proposition is that CH Z tensored with q is isomorphic  to CH q and so this is how you end up getting a variance of cylindrical contact homology.  Okay, and I will end.  I mean if you don't have something that's dynamically convex then you're going to have  to be working with like full SFT or symplectic homology.  Yeah, so if you take the connect sum of two tight spheres you're no longer dynamically  convex.  So this sort of result doesn't work for connect sums, but I mean cylindrical contact homology  was never built to being in variance of things that weren't dynamically convex in dimension  three.  So there was a result that Viterbo mentioned in his talk where he said that if you have  this assumption about conical Zander indices then the positive part of symplectic homology  is an invariant of the contact structure.  It's exactly this dynamical convexity condition and it has an analog in higher dimensions.  What's your favorite manifold way to compute it?  The sphere.  So you can also compute it for like lens spaces and like pre-quantization spaces.  What's the answer?  Oh, so, well I guess I can just say it in words.  For the three sphere you have Q in every even dimension that's greater than or equal to  two.  And for lens spaces you also only have for ln plus 1n you have n copies of Q in dimension  zero and n plus 1 copies of Q in every even dimension that's greater than or equal to  two.  Any more conversation requests?  Any other questions?  Right, so that's what you're doing but you've kind of been able to cheat a little bit.  So normally your differential would have like an infinite number of terms in it but because  we could cheat and use our use a time independent almost complex structure and still be able  to define the complex that's why we were able to drop the extra terms but that's pretty  much exactly what's going on.  Suppose you took two dynamically convex just go back to the convex sum idea.  You took two dynamically convex and then you did a convex sum but you did a really, really,  really small convex sum where you have very precise control over the sort of, you have  very precise control over this hyperbolic orbit and the finite energy planes that would  be sort of introduced.  Is there any chance that these ideas could extend in that case?  I mean maybe but it's my understanding that even when you do a convex sum you always end  up introducing a new...  You do. ...rave orbit.  But you know exactly where it is and you know pretty quickly you have pretty much... ...generated longer ones.  Yeah you could use action filtration arguments anyway.  No because the...my answer is no.  So I mean it's conceivable that we could define some sort of linearized contact tomology if  we...there's another type of building that we would have to consider and we'd have to  be able to understand how to glue it or exclude curves limiting on it and you'd have to take  some sort of like augmentation so...  I'm going to disagree with you on this one.  You think that...  We might be able to do it.  Well that's...yeah I was saying you might be able to...  It's not linearized which is super cool.  It might be possible but it's a harder project.  Anything is possible.  Okay I think Helmut's ready to clap for me and get coffee.  Yes so...  Yeah why not?  Have a good one.
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Thanks for the introduction. I'd like to thank the organizers for allowing me to talk. Okay, so a lot of this will be joined with Michael and it was helpful he came with his obstruction, bumble gluing and intersection theory from ECH. So he was kind enough to lend them to the contact tomology. Alright, so we're going to start off with a contact manifold. For most of the talk it will be three-dimensional, but in some parts I might get to tell you about what we can do in higher dimensions. This is going to be non-degenerate contact and it's going to be co-oriented meaning that there's a global one form that defines the contact structure and for grading reasons let's pretend or assume that the first turn class of the contact distribution vanishes and then R is going to be my rave vector field. Okay, and then I'm going to have J is an almost complex structure on r cross m and I'll take d e to the tau lambda. So here tau is going to be my coordinate on r. It's going to be the translation invariant so I have d tau r is going to be equal to r and I want J restricted to, I'll just say J is d e to the tau lambda compatible. Okay, then we want to count cylindrical curves so crystallized asymptotically cylindrical curves are curves the domain is just going to be a cylinder so it's not going to be super exciting but they're still fun to look at. So I'll denote the space of, so let's gamma plus gamma minus be periodic rave orbits. Mj gamma plus gamma minus is going to be the set of J holomorphic curves from a cylinder r cross s1 j0 into the simpletization with this d lambda r invariant almost complex structure and then we want du of J to be J of du so we want them to be pseudo holomorphic and then we want the limit so if we take r to have the t coordinate s1 to have the s coordinate we want the limit as s goes to the plus or minus infinity of the projection of u and the contact manifold component to be a reparameterization of the rave orbit at gamma plus if we go to positive infinity and gamma minus if we go to minus infinity and then we want the limit as s goes to plus or minus infinity of the projection to the r component of u to be plus or minus infinity so you want to think like if my contact manifold was something stupid like a circle then your counting curves like this gamma plus gamma minus and then there's also going to be an r action that technically you have to mod out by coming from translations on the target and so when we go to look at the differential there will be this translation action but a lot of times I'll ignore it when it comes to talking about index calculations and stuff. Okay so. Is anything other than the gradient dependent on your assumption that c1 equals 0? No. So I can just drop that and you have your z2 gradient? Yeah I mean eventually I have to yeah you can drop that and use the z2 gradient of course when I define a dynamic convex contact form you'll need some sort of way of making sense of the gradient on contractable orbits. So the gradient is given by the Conly's Ender index on our chain complex we have that the virtual dimension of this moduli space is given by gamma plus minus the gradient of gamma minus and here the gradient is an SFT grading so this is going to be Cz of gamma plus n minus 3 so if the dimension of our contact manifold is 3 then you get this is just Cz of gamma minus 1. So then I can tell you what the chain complex is. So this is going to be generated by only good Ray-Borbets so we take the set of all Ray-Borbets and then we throw out bad Ray-Borbets because they're bad and so to define a bad Ray-Borbets a little bit annoying but so the Cz index of a Ray-Borbets can either have the same parity under iteration of your orbit or it can flip flop between being even and odd sorry odd and even if it flip flops then you throw out the even multiple covers so bad Ray-Borbets even covers of flip the Cz guys if you want the real definition or you want an example so in dimension 3 bad Ray-Borbets are even multiple covers of negative hyperbolic guys meaning that they have the eigenvalue of the linearized return map if we restrict it to the contact distribution is negative and real. Is there a reason for your V there for which the Cz VQ like in your top line? It's not a V this is the vector space generated by all the Ray-Borbets and you throw out all the bad Ray-Borbets. Yeah sorry he just thinks the stars never mind. Is there another way for why are bad Ray-Borbets bad? So if you keep them in the chain complex even if you had all the transversality in the world you'd never be able to prove invariance but secretly they you can't orient curves that limit on bad Ray-Borbets unless you have some sort of a parametrization you're using of your bad Ray-Borbets and so there's a when you try to assign orientations you can get both positive and negative orientations assigned to the same cylinder and this causes a headache and in orbithold Morse theory you can kind of think of the bad Ray-Borbets as being like the bad points whose like gradient flow lines get reversed underneath whatever action you're modding out by. Any other questions? It's mobilized spaces. Yeah so that is kind of incorporated by the fact that this is the projection onto the contact manifold as a reparametrization of gamma plus minus so that already includes that. And then the differential. So there's actually two equivalent ways over Q of defining the contact differential. So either we can define it to so I'll just write this down and then I'll say. So either you can define it by looking at the multiplicity of the top Ray-Borbit or you can define it by encoding the multiplicity of the bottom Ray-Borbit. So and this epsilon U is just supposed to be some choice of coherent orientations. M of U is the degree of the covering map of the curve if your curve U is not somewhere injective and the multiplicity of your Ray-Borbit just means you've had to iterate some simple orbit some number of times and the multiplicity will be the number of times you've iterated a simple Ray-Borbit to get to X. And so it might look like these coefficients are defined are not integral but they are in fact integral. And so you can see that because if you have a k to one fold cover where you have X is gamma let's say plus to the kp and Y is gamma minus to the kp this would be a k to one fold cover of a cylinder gamma plus to the p gamma minus to the q and so the multiplicity of X is kp the multiplicity of Y is kp the multiplicity of this curve U is k and k obviously divides both kp and kq. I'll just write against that M of X equals one if X is a simple Ray-Borbit so maybe simple hasn't been defined but that just means that the map X which is r mod tz into your contact manifold is injective and M of U equals one if U is somewhere injective. So that looks like beautiful theory but it's not really so beautiful. So some of the issues that are going to arise is that we're working with multiple covers. Is there a question? So can you explain the difference between the two? One encodes the multiplicity of the top Ray-Borbit and one encodes the multiplicity of the bottom Ray-Borbit of your cylinder. And it turns out that their equivalent sort of if you had all the transfer salinity in the world that would mean that your differentials were defined and you actually got a homology and then on the chain level you could just divide and multiply out by multiple seed of so Ray-Borbit's the pass from one chain complex to the other chain complex. So homologies are isometric under this? If you live in magical transfer salinity. Yes but they're equivalent. So why is this theory possibly not so correct? So the first issue is that we're working with multiple covered curves and Chris spent many lectures telling us as well as Dusa and others telling us about the horrors of multiple covered curves. And then also I think Michael mentioned this a few times but you can look at multiple covered curves and they can end up having negative index even though they live in even though they have to be there and you can't just use a generic perturbation of J to get rid of them and so this means that like compactness issues are going to suck. Well maybe they don't suck, maybe they're beautiful but they make the theory harder. Alright so everyone's favorite congerum from 2000 which is in a paper by Ali Ashbur Gimental and Hofer says that assume there are no contractable rape orbits of index minus one zero or one so this is this grading that's the conglisander index plus and minus three. And, okay, it also started by saying what Y, C, V, A non-degenerate contact manifold, then the differential is well defined, of course no one really said which differential they were using which adds another layer of confusion but that's okay. J is generic. Okay, that you actually get a homology so DQ squared is zero and this homology which depended on your contact form, your choice of J was actually an invariant of the contact distribution so EG independent of the choice of lambda and almost complex structure J. I think should be fair, they didn't quite say that because they said that some abstract perturbations of the modulate spaces would be needed but they wouldn't go into the details of that. Because this one is too much of a fault. Well, this is, well it doesn't even appear in this form in their paper, it appears as a series of like lemmas or propositions. I mean they said at the beginning this is all suiting some perturbations of the name. Well at the beginning it said that this is all like conjectural blah blah blah so this is why it's a congerum so it's a conjecture of A plus. So the conjecture is that the statement can be corrected to the other side. Well the other problem is that some people then said you don't need abstract perturbations and this works anyways. Is anyone happy? No? No one's happy? Okay. So what is this beautiful compactness that I alluded to? So this is Bourgeois, Lyoszberg, Hofer, Vizatsky and Zander. This is a result from 2003. Again I'll paraphrase this result. So there's no bubbling like in Hamiltonian Flora theory but we do have breaking of curves into buildings. So Michael drew some beautiful buildings of asymptotically cylindrical pseudo-holomorphic curves. And so the heuristic reason for why we have this breaking is because so there is a maximum principle in the R component of the symplectization but you can grow minimums and so if you try to compactify the space you're going to be allowed to develop new punctures at the bottom end. So you had some index two cylinder. You could conceivably start to have the picture people draw so you have some sort of a minimum developing in the R direction and so then if you want to compactify the space you could pop off a plane and this would be allowable elements of your compactification if you cannot sort of exclude it for other reasons. And then this makes us sad. And cylinders in your compactified modulized space of index two curves. I have a dumb question. Why is breaking and not bubbling? How much? Because you have cylindrical end which is as a real bubbling parameter instead of a complex bubbling parameter. It breaks like a trajectory of more stereo or core theory and not like a bubbling chroma or a bubbling, okay, that's a bubbling parameter. So I'm not going to draw a complicated sequence of buildings but you can imagine that sort of, well, Michael was talking about how you can have branched covers of trivial cylinders. Those guys can have index zero so you could be in some situation where x is gamma squared, z is gamma and then the cylinder could break to be gamma squared, gamma, gamma, gamma and this could be a holomorphic plane of index two. This guy has index zero and this guy has index zero. So this would be an allowable compactification. And in dimension three there is really nice iteration formulas for the Conly's Ender Index plus automatic transversality so I'll explain in a minute how you can salvage this theory in dimension three but in higher dimensions you can have cylinders of index one and you take their multiple cover and they suddenly have negative index. And then that means that you, in other elements of your compactified modular space, are going to have to include things like a chain of cylinders where you just have to end up being able to add to two at the end of the day. So that makes things complicated. And the, you know, philosophy with a lot of this stuff is well if we only had some more injective curves you just pick j generically so all of my moduli spaces have to have positive virtual dimension because we're working in a simpletization and have this r action and the only way that you can add to two with positive numbers is one plus one. And this original assumption about no contractable orbits of index minus one zero one come because this sort of building is not excluded by the assumptions but you could have some building where you have one, one and one. So this is like why the assumptions in the congerum exist. And then the zero and minus one sort of come up when you try to prove invariance and homotopy depending on what level of transversality you want to assume. So let me see. So it turns out that in dimension three you can get automatic transversality and index calculations to work favorably. And so in my thesis I found not a very large class of contact forms but the class of some contact forms where you could actually define contact tomology. So sort of the definition I came up with is we say a contact form is dynamically separated provided the conles-eander index of gamma is between three and five for gamma simple and contractable. And every time you iterate gamma you're just increasing the conles-eander index by four. So this holds and then plus you can do this if you, so I guess I should say if gamma is non-contractable have analogous definition but you have to keep track of free homotopy classes of ray orbits and it just becomes a mess to write down so I'm not going to. Is there a question in the front? Okay. And then the theorem that I proved is that if, so this condition, this is usually true up to large action. And action is the integral along gamma of some ray of orbit. So by up to large action I mean for all ray of orbits of action less than or equal to something you have this dynamically separated property and these guys all for the most part arises there's a natural class of perturbations you can put on pre-quantization spaces so this is where these guys come from. So if my three lambda is non-degenerate dynamically separated then and J generic then the cylindrical contact tomology is defined and sometimes up to like some large action sort of. So if you don't have this dynamically separated for all of your ray of orbits but only up to some large action then you can only define cylindrical contact tomology up to that large action and index and then there's a way you can take direct limits to actually sort of compute it. Variance under the choice of J and dynamically separated contact form. So this is not so awesome because it excludes a lot of examples like dynamically like certain dynamically convex things like ellipsoids so you team up with someone that knows more than you and then you prove a theorem. Six things is annoying. So a dynamically convex contact manifold is something that Hopper, Vazatsky and Zander studied in the 90s and what Katrin knew as HWZ papers when she was a graduate student. So a dynamically so Y3 lambda is dynamically convex provided the first churn class of the contact distribution restricted to pi2 of Y is 0 and the Conley-Zander index of gamma is greater than or equal to 3 for all contractable ray of orbits. And so an example is any strictly convex hyper surface in R4 with the standard symplectic structure which is transverse to the radial vector field. So this is a nice class of contact manifolds because I erased the theorem but it's basically without sort of a theory of abstract perturbations. It's going to be the class of contact manifolds so that all the contractable ray of orbits give rise to planes that all have to have virtual dimension at least two. So this would be sort of like a nice class of manifolds you'd like to prove. So the termology is defined for an invariance in dimension 3. So you also dropped the assumption in C1 if there were no contract. So we'll make condition A C1 of pi2 is 0 or B. So what Michael and I proved is that if Y lambda is dynamically convex, J is generic, then the only buildings of index 2 in R cross M are either just a cylinder, a once broken cylinder or if you, I should also write this, so we're going to assume that Connelly's Andrer index of gamma is strictly greater than 3 for gamma non-simple and contractable and I'll explain why in just a second or you have a pair of pants, a plane and a trivial cylinder so gamma squared gamma. So this is 0, 0, 2. So we have this extra assumption that, there should be gamma d, d minus 1. So we have this assumption that the Connelly's Andrer index is strictly greater than 3 for gamma non-simple and contractable. We expect that it's removable but we have to do a little bit more estimating things to make sure that something bad doesn't happen in the next blackboard but sort of what we proved is that the only buildings of index 2 are either an unbroken cylinder, a once broken cylinder or this pair of pants configuration. If you assume, if you don't have this assumption, then your pair of pants configuration can have like this gamma could be gamma to the d1, this could be gamma to the d2 and then this is just a d1 plus d2 fold cover of a trivial cylinder. And the second part of our lemma is that no index 2 cylinder can limit to the third configuration. So we didn't prove that there are no buildings of this type, there certainly are, but we just showed that there's no index 2 cylinder that could limit to it, which means that as a corollary the cylindrical contact tomology differential is well defined and it squares to 0. And this is innocent flexion, so the reason why we don't have invariance yet is because this sort of analogous lemma in cubortisms for sure does not hold. Any questions? So a priori, what are the possible configurations for that third one? I mean, I'm looking at this and you're saying if it can't be the case that a cylinder of index 2 limits on third configuration, I'm looking at that thinking, okay, well, if in some world I could perturb and actually glue that thing together in some sense, then it would be an index 2 cylinder. So therefore, if you, like, what kind of perturbing do you want to do? So if you were doing obstruction bundling gluing and you decided to try to see if you could do this, the number of ways you could glue it together would be 0. Actually, just to, that's not how you prove it, right? No, no, no. It's a proof by contradiction, sorry. So we use, yeah, you use intersection theory. You say, like, suppose you have this curve living into this, then you look at some intersection theory properties and prove that you get a contradiction. Therefore your index 2 cylinder can't limit to a building of that type. True proofs are actually the same if you look deeply enough. Yeah, and since obstruction bundle gluing goes back to winding numbers and, um, actually, I have, well, can't the obstruction bundle gluing, can it only tell you that the sign counter phrase to glue is 0, or is this really telling you there's no way to glue it? I think in this case you'll actually see if the obstruction section never manages to look for that. In where do you use the index 2 at the point of the dynamic convex? Sorry? I know that the proof has this intersection theory stuff. Oh, the intersection theory stuff is to prove that no index 2 cylinder can limit to the third configuration. And where does that be used, the fact that it's a dynamic convex? Oh, so we use the fact that it's dynamically convex to prove that the only buildings of index 2 are of those three types. And then you also, I mean, yeah. And then you're also using automatic transversality and some index stuff, improving that those are the only three types of buildings, and that's also what's getting you the transversality you need to say that the differential is well-defined. So we also prove that all index 1 and index 2 cylinders and subluxizations are cut out transversely and, um. So do you need the dynamic convex for that? So you don't need dynamically convex for that, but you're going to have a problem with looking at your compactified, modulized space of index 2 cylinders. Like then I'm going to have way more things, and I'm not going to be able to prove that d squared is zero. Let's say you can know that you only have these three kinds. You can only get that if you're dynamically convex. You all have bad vision, the only people in the front. But what are the indices of the curves in the third building? Zero, zero, and two? Yeah. I didn't want to put the zeros here and here, because then the people in the front row would ask why those curves had genus and I was counting cylinders. Yeah, no, no, it's good. I just need glasses. The top level there is a branch cover of a trivial cylinder. He's a trivial, trivial cylinder. Everyone happy? If you're not happy, I'll get out of my hook. I actually really like holding the hook, but I think that makes for a slightly scarier talk. I was going to move on to the non-equivariant part of the talk, but do people have any other questions about what goes into the Waman corollary? I guess I should maybe upgrade it to a theorem. So two, we're stuck on invariants, so we need to do something. So we'll try to define, so we'll define stuff non-equivariant. So we'll define a non-equivariant version. If I stuck, you mean data approachable? Oh, yeah. Okay. So to clarify, this normal approach looks bad because you can't prove this sort of lemma in co-bordisms. You also don't have the sort of automatic, I mean, you have some automatic transversality in co-bordisms, but it's not good enough, and you're not going to be able to prove the chain homotopy equation. So invariance is not really going to go so well. So what we can do is we can define a non-equivariant version. And when we set up a non-equivariant version, it's kind of analogous in Hamiltonian-Fleur theory to when you pick a time-dependent Hamiltonian. So all of your curves end up being somewhere injective. So we're going to do something similar in the contact world. There's still going to be some things we have to work out, but they're doable. And then it turns out that there's a way to take an S1-equivariant version of this non-equivariant picture, prove that it's isomorphic to classical cylindrical contact tomology, and in a very roundabout way, get invariance for the classical cylindrical contact tomology. OK. So this non-equivariant theory is kind of look like the positive part of symplectic homology, whereas this cylindrical theory, if it was defined, it would look like the S1-equivariant version of the positive part of symplectic homology. But we don't want to use symplectic homology because we'd like to define something directly in terms of the RABE data and in terms of asymptotically cylindrical pseudoholomorphic curves as opposed to looking at the Fleur equation. OK. So we're going to take J here. Instead of using 1J for all time, we're going to take a domain-dependent J. So JT for T and S1 is going to be an almost complex structure. And we still want for each time T that it goes to the RABE vector field, and we want for each time T that it's D e to the tau lambda compatible. And then we're going to be counting, well, we're not going to be quite counting pseudoholomorphic curves where I just throw this T dependence in because that's not going to automatically get us a non-equivariant version. We're going to have to be a little bit more careful than that. But the moduli space we're going to start to want to look at and put point constraints on is going to be this guy where we're using a domain-dependent family of almost complex structures. OK. So you just put a T in your J-homomorphic curve equation. OK. And this T depends. Oh, shit. Wait. Sorry. My R coordinate should have been S and my S1 coordinate should have been T. Now it makes a lot more sense. OK. Maybe this is stupid, but what happens if you make lambda time dependent? Well, so this is forcing in some sense. You can't make your lambda time dependent. This is like the closest you can get to forcing your periodic orbits to be time dependent. I can't make lambda time dependent. It's a content. How would you make it time dependent? It's just a contact form. I could take a time, a group of context, and give them the same context. Well, I would rather take a loop of... We're not looking at period one orbits. We're looking at arbitrary period orbits. Oh, yeah. That's the other thing. Sorry. I mean, the point is now all the curves that were multiply covered are no longer going to be multiply covered for a generic family of almost complex structures that depends on the domain. So, the good news for a generic family, mj gamma plus, gamma minus is a manifold of dimension equal to the Conley-Zehnder index of gamma plus minus the Conley-Zehnder index of gamma minus. And I broke some symmetry, so it's got an extra dimension on it. All right. And so you're going to be working in a situation where the... So there's a paper by Fleur Hofer and Zalaman that gives you transversality in the Hamiltonian Fleur case. And so this is sort of the... You can't pick a time-dependent Hamiltonian, but you can pick a time-dependent, almost complex structure in your J-homomorphic curve equation. And so that's how... So all your curves are going to be somewhere injective and ta-da. Okay. But you're not going to be able to define the differential sort of in this naive way here. So let me tell you how to define the differential. Okay. So to define a differential, we're going to need to throw in some point constraints. And this is kind of how we're forcing a parametrization of our ray of orbits in terms of time. So here's my curve. I'm going to be able to look at U of r cross 0. And this is going to pick out some line on my cylinder. Okay. And you can use this to define evaluation maps from our modulized space into the image of gamma plus minus, which we define to be the limit as s goes to plus or minus infinity of the projection of the curve when we look at time 0. Okay. So this is supposed to be like when t equals 0. And this is some point that's in my ray of orbit gamma plus. And you can do the usual thing where you can map this evaluation, map the sense to an evaluation map after you quotient out by r. Because even if we do define a differential, we're still going to have to mod out by r where this r is coming from the translation and the simpletization direction. Okay. Great. And what do I want to say? Okay. So chain complex, I no longer have to throw out all these bad ray of orbits, but I am going to have to have twice the number of ray of orbits in it as before. So you can kind of think of this as a Morse BOT version where your Morse BOT manifolds through your ray of orbits and you put a height function on s1. And so that gives you two critical points. One critical point has index 0, one has index 1. So we're just going to kind of, that's not really what we're doing, but it's the vestigial motivation for defining our non-equivariant chain complex to be generated. So it has two generators associated to each ray of orbit. So gamma check and gamma hat, gamma is any ray of orbit. And then I'm just going to arbitrarily shift the grading up on my gamma hats by one. So we have that gamma, the image of gamma, okay, I'm just going to say that gamma conleysander index of gamma check is equal to gamma, conleysander index of gamma hat is equal to the grading, I reversed this. So I should have written the inequalities the other way, but the grading on gamma, what am I doing? Okay, I'm getting confused with hats and checks. Okay, so the grading of gamma hat is going to be equal to the conleysander, the grading of gamma check is going to be equal to the conleysander index. The grading of gamma hat is just going to be the conleysander index shifted up by one, and conleysander index of gamma hat and gamma check doesn't make sense because it's the same underlying ray of orbit gamma. So this should be the correct formula. So you just think that I have gamma check and gamma hat are formal variables associated to the ray of orbit gamma. Okay, yeah, this is any ray of orbits, you allow also the battery of orbits. So it's just like in symplectic homology you would also see the battery of orbits. And this is kind of, so this non-equivariant picture is sort of not original, but it is an adaptation from stuff by Berjewil, Yashberg, and Ekholm, and also Berjewan Wansha. All right, and now my differential is going to have a block decomposition. And I'm going to have sort of stuff that goes from a check, ray of orbits to check ray of orbits. So there's four different ways, sort of you can go from, let me just write this out and then we can talk about it. So hat, this is just z generated by this guy. Let me see, check, let's see. So, okay. All right, and I promise that you'll see this evaluation map and I just wrote down and show up. Uh-huh. So we think of it as like a triumph that you no longer have to try out for bad ray of orbits? Is that something desired or is it just a quirk of this? I mean, so it is desirable and it turns out that this S1-equivariant version, like once we figure out what the S1-equivariant version of this non-equivariant theory is, it's actually defined over z. And so it's going to have more information coming from bad ray of orbits because it's defined over z. And it turns out that if you tensor that theory with q, you get the usual cylindrical contact tomology theory. So it's a quirk, but I guess it's good because you get more information. Okay, so what do each of these terms count? So d check, alpha check, beta check. So here you would want the degree index between alpha and beta to be 1. So this will count cylinders where you are going to, right, so you want to, you're going to fix a point on the underlying simple ray of orbit associated to alpha. So this is some point on the image of, and alpha bar is a simple ray of orbit. So alpha, and then we want, so this is going to be curves u in this moduli space going from alpha to beta, mod r, with the positive evaluation map of your curve u being equal to p alpha bar. And we're also going to have to count Morse bot buildings with a cyclic ordering condition. So the evaluation map constraint is supposed to be this sort of point where p alpha bar it needs to match up with pi y of u of s0. And then we'll count these guys, and then we'll also have some buildings that we have to count where you've got a cyclic ordering condition, meaning that you want f minus of the top curve, f plus of the bottom curve, and whatever point you're picking on this intermediate curve, let's call this guy gamma, to be ordered this way as you go around gamma. So you're going to need to orient your, like a direction for how you're going around your ray of orbit. And then the check of alpha check, beta check is going to be something analogous, except instead of having the positive evaluation constraint, you're going to have the negative evaluation constraint, and then negative evaluation constraint, d plus alpha check, beta hat, this has no constraint. So this looks like the too bad map that you see in the Birch-Wile-Yashberg-Eckholm formulation. So if we look at d plus of alpha check, alpha, sorry, alpha hat, alpha check, you get two points if alpha is bad, and zero if alpha is good, and d minus alpha check, alpha hat, this has constraints at the top and bottom. And this non-equivariant theory is not really so easy to count because not only are you sort of counting cylinders that either have one point constraint, two point constraints, or zero point constraints, you also need to count more spot buildings that have the cyclic ordering condition, and these more spot buildings, they're not going to be of infinite length because you only have a finite number of ray of orbits in each conical azander index, but they are going to be some sequence of buildings that you have to figure out how to count. But it turns out that if, so I say, okay, so theorem is that, oh, and I should have this d minus here, so this also, in addition to counting u and m alpha beta with f plus and of minus constraints and more spot buildings, it also will count ways to glue an index 3 curve using obstruction bundle gluing that looks like a branch cover of a cylinder. So, this is zero, two, and one. Okay. What is the j on the cap there? The, it's a two. No, it's a j, I mean, so you fix a family j of t on the remains. Yeah. So what would that be on the cap this class? So it's a bubbling off at a point? Yeah. Yeah. So some j would bump it off? Yeah. Is it clear why you can only have one plane of all this? Yeah, index calculations. Is the reason that you define this chain problem is to see that the kind of penance means you can do isotropy? Yeah. So your curves are always going to be somewhere injective, so you don't have the isotropy anymore to deal with. So for y3 lambda dynamically convex or y2n minus 1 lambda, well, no, I don't want to say that. This non-equivirian contact homology and j generic is a chain complex, and the homology of it is independent. This is a family of j's of your choice of family of j's. And the dynamically convex lambda. Let me tell you how to relate this back to cylindrical contact homology in a way that will make everyone much happier than they are now. So if there is sufficient automatic transversality, then as needed to define the classical cylindrical contact homology differential, then we can use for some generic choice of j, we can use this j to look at this non-equivirian contact homology. And what's nice is that it turns out, so in this case, d check looks like the standard cylindrical contact homology differential. And d hat looks like the other cylindrical contact homology differential, and you end up having opposite orientation, but that's not so important. But it turns out that you can show that, so if y lambda is non-degenerate, dynamically convex j generic, then for d0 equal to dq minus d, well, I did not use good notation for this. So the first differential, this is supposed to be the second differential that I told you, d plus, d minus, and d1 equal to 0, 0 multiplicity of gamma 0. I'll explain what that is in just a sec. You can actually get a chain complex by taking the non-equivirian chain complex, tensoring it with the formal power series of variable 2 that has degree 2. So here u is a formal variable of degree 2 with dz being equal to d0 plus d1, and this is j, that this is a chain complex. So this was kind of a lie. So you still will be able to see your bad orbits, and you're not going to have the problems that you had before with orienting them because you have point constraints running around that are going to allow you to deal with orientations. So I'm doing a bad job explaining this because there are a lot of details I'm omitting. But then, so the degrees of d0 are coming from a complex where the bad points are at the same time. Oh, okay. This shouldn't be, I should have just written, this isn't technically this differential, it should be, I should have written something like this is counting, my notation is really awful. Yeah, we'll just use d hat and d check, but they have a more familiar expression in terms of multiple cities of ray orbits and multiple cities of your pseudoholomorphic curves. So what I should have written here is that if j is, if we can replace j with a single j, then d check counts u and mj alpha beta, you still are going to have your checks and hats with a point constraint at the top and you'll get that d check of alpha check is going to be the sum over the multiplicity of alpha divided by the multiplicity of your curvy u that connects them and now my font has become microscopic beta with the orientation. So the chain complex is still this thing that we're using and then the differentials just have more familiar expressions in terms of the previous differential, but that was a very unclear and helpful way that I explained that. Can I say one thing? I mean the d hat and d check, they're the same as the differentials before when you're going between good orbits. Yeah, and you can show that when they do encounter bed ray orbits nothing horrible happens. Yeah, I should finish the statement of the theorem and the homology Z of this chain complex is independent of lambda and j and a proposition is that CH Z tensored with q is isomorphic to CH q and so this is how you end up getting a variance of cylindrical contact homology. Okay, and I will end. I mean if you don't have something that's dynamically convex then you're going to have to be working with like full SFT or symplectic homology. Yeah, so if you take the connect sum of two tight spheres you're no longer dynamically convex. So this sort of result doesn't work for connect sums, but I mean cylindrical contact homology was never built to being in variance of things that weren't dynamically convex in dimension three. So there was a result that Viterbo mentioned in his talk where he said that if you have this assumption about conical Zander indices then the positive part of symplectic homology is an invariant of the contact structure. It's exactly this dynamical convexity condition and it has an analog in higher dimensions. What's your favorite manifold way to compute it? The sphere. So you can also compute it for like lens spaces and like pre-quantization spaces. What's the answer? Oh, so, well I guess I can just say it in words. For the three sphere you have Q in every even dimension that's greater than or equal to two. And for lens spaces you also only have for ln plus 1n you have n copies of Q in dimension zero and n plus 1 copies of Q in every even dimension that's greater than or equal to two. Any more conversation requests? Any other questions? Right, so that's what you're doing but you've kind of been able to cheat a little bit. So normally your differential would have like an infinite number of terms in it but because we could cheat and use our use a time independent almost complex structure and still be able to define the complex that's why we were able to drop the extra terms but that's pretty much exactly what's going on. Suppose you took two dynamically convex just go back to the convex sum idea. You took two dynamically convex and then you did a convex sum but you did a really, really, really small convex sum where you have very precise control over the sort of, you have very precise control over this hyperbolic orbit and the finite energy planes that would be sort of introduced. Is there any chance that these ideas could extend in that case? I mean maybe but it's my understanding that even when you do a convex sum you always end up introducing a new... You do....rave orbit. But you know exactly where it is and you know pretty quickly you have pretty much......generated longer ones. Yeah you could use action filtration arguments anyway. No because the...my answer is no. So I mean it's conceivable that we could define some sort of linearized contact tomology if we...there's another type of building that we would have to consider and we'd have to be able to understand how to glue it or exclude curves limiting on it and you'd have to take some sort of like augmentation so... I'm going to disagree with you on this one. You think that... We might be able to do it. Well that's...yeah I was saying you might be able to... It's not linearized which is super cool. It might be possible but it's a harder project. Anything is possible. Okay I think Helmut's ready to clap for me and get coffee. Yes so... Yeah why not? Have a good one.
with a contact manifold. For most of the talk it kind of think of the bad Ray-Borbets as being equivalent. So why is this theory possibly not so though they have to be there and you can't just of J was actually an invariant of the contact one and one. So this is like why the assumptions can get automatic transversality and index orbits of action less than or equal to something So a dynamically convex contact manifold is structure which is transverse to the radial contact manifolds so that all the contractable for an invariance in dimension 3. So you also we can define a non-equivariant version. And when S1-equivariant version of this non-equivariant picture, prove that it's isomorphic to classical almost complex structures. OK. So you just put a your lambda time dependent. This is like the actually defined over z. And so it's going to obstruction bundle gluing that looks like a your choice of family of j's. And the dynamically classical cylindrical contact homology