 Okay, so I guess it's time to start. So first, I'm very happy to be here and have this chance to give this class. The goal of the class is going to be to describe the construction of an invariant called contactomology to the invariant of contact manifolds. In the first lecture, though, I just want to give a brief overview of the general area of contact topology and symplactic topology through the holomorphic curves. I see there are a number of experts in the audience and I hope this isn't too boring for you, but I'll sort of give an overview and sort of say what the goal is. So let's start with a manifold, odd dimension. And a hyperplane field, so that is a co-dimension one sub-bundle of the tangent bundle, so just at every point you give a hyperplane sub-space of co-dimension one in the tangent space at that point, and they vary continuously. So, Equivalently, or almost equivalently, to specifying a hyperplane field, you can specify a non-vanishing one form called lambda, and then C is just the kernel of lambda. So it's convenient to have both of these descriptions. Up to. Yeah, up to scaling, yeah. Yes. Right, so we say C defines a affiliation or is integrable if and only if d lambda restricted to C is zero. So what this means geometrically is that through every point there's a co-dimension one sub-manifold, and C is just the tangent space to that sub-manifold. So, or locally, C defines or gives charts for why we're, I'm sorry, you have one distinguished coordinate. So, C is called a contact structure, or is contact structure. And we can think of this as being, since since the farthest from being affiliation is possible. So, it's often called maximally non-integrable if and only if d lambda restricted to C is a non-degenerate pairing. So, another way of saying that, what it's usually said is that lambda wedge d lambda to be n minus one is non-zero. So, for example, r2n plus one, there's this so-called standard contact structure, which is the kernel of dz plus ri squared d theta i. Another example on r3, there's a so-called over twisted contact structure, which is given by kernel of cosine r dz plus. Yeah, so it's polar coordinates in r2, the n and then z here. r sine r d theta. So, it's a rather non-trivial theorem of Benneken that in dimension three, these are not the same. There's no diffeomorphism of r3 taking the standard one to the over twisted one. So now, just a few basic facts about contact structures. How do they behave? So, first is Darbouz's theorem says that every contact manifold is locally isomorphic to r2n plus one with the standard contact structure. So, contact structure, just like affiliation, has no local invariance. It's not like a Riemannian metric. Riemannian metric has lots of local invariance, curvature, things like that. All contact structures look locally the same, so they're only sort of interesting questions to ask our global questions, which sort of interact with the topology of why. Another basic result is that if you have a path of contact structures, deformation, there exists an isotope y to itself such that this deformation of contact structures is induced by the isotope. So, it sounds kind of strange, but let me, they're technical, but let me say what it means. So, one way of interpreting this is saying that the, or a consequence of this is that the moduli space of contact structures is discreet. So, if we take sort of the space of all contact structures on y. No, sorry, yeah. Yeah, probably. Yeah, so y-compact. Yeah, so on a compact manifold, yeah, they're discreet. So, the consequence is that this is discreet. So, in some sense, this is a combinatorial object. You can, it makes sense to ask the question, what are all contact structures on the given closed three manifold? So, Darbou's theorem was sort of the same for contact structures and for foliations. Sorry. Yeah, maybe not, I don't know whether this one can, but yeah, there's an over twisted contact structure on S3 and on anything. This one, okay. Yeah, so Grace's theorem is true for contact structures. Foliation, this is something which is, fails completely for foliations. Let me just sort of understand what this statement is saying. Let me show why it's not true for foliation. So, if we look at the torus, then we can define the foliation by picking any slope and looking at just the lines of that slope. Any slope. And if you vary the slope, lambda, this is a deformation of foliations, which definitely is not come from an isotope. The foliation encodes the dynamics of the sort of return map adding by lambda on the circle. So, some basic questions we're interested in are existence of contact structures, classification, and symplectic ability and cobertism between contact structures. So, I'll define this precisely later. So, these questions are existence and classification are completely answered by Gromov's H principle on open manifolds. So, we say a manifold is open if and only if every component either has non-empty boundary or is non-compact. Y2 and B open. So, then the following map is a homotopy equivalence. So, we look at the space of all contact structures. And we can map this to the space of just all hyperplane distributions. In dimension three, this is essentially enough up to specifying some orientation. In higher dimensions, we also need to specify some almost complex structure. See, again with some twisted by some orientation business. But this map is a homotopy equivalence. Okay, so why is this sort of called an answer to this question? This space looks perhaps just as complicated as this one. The answer is that this space is described completely homotopy theoretically. This is, there's some Grasmanian of co-dimension one subspaces of the tangent bundle equipped with some extra structure. And this is just the space of sections of that vibration over the manifold. So, it also has a sufficient specification to exactly contribute to your example of this thing. This one. Yeah, it's open manifold. But this doesn't open. Right, so here we're putting the topology of uniform convergence on compact subsets. So, if you're in the same connected component of this you're not necessarily diffeomorphic. Okay, classification up to homotopy in this sense. Okay, so these questions are mostly interesting and are mostly focused on closed manifolds. Okay, so on closed manifolds there's a version of this statement although it's more involved. So, Eliasberg-1989 and Borman-Eliasberg-Murphy very recently in 2014 is that, so if you look at all contact structures on a closed manifold, so say Y closed, then you can, there's a canonical sort of partition of this set into two different spaces called tight. And over twisted. And again, there's this. And this is a homotopy equivalence in the set of over twisted. It was given some extra marking of the sort of over twisted region. So, right, the point of this is that there's some class of contact structures which are, somehow the contact condition isn't adding anything extra. It doesn't add any extra information to the, just the plain homotopy theoretic information contained already. And there's this other, so over twisted ones are sort of completely flexible. Tight contact structures on the other hand are more mysterious. So, these questions are mostly interesting for tight contact structures. They'll turn out to be easy answers for over twisted ones by these results. So, I believe an open question whether irreducible free manifold have a tight contact structure. And there are examples of non-irreducible free manifolds with no tight contact structures by Atnar and Honda. For example, the 235-point-crate homology sphere and connect sum, it's reverse. It doesn't have any tight contact structure. So, there's a conjecture by Ali Ashberg that every close three manifold has only finitely many tight contact structures. This turns out to be not exactly true as stated, but it's almost a way to correct it to be a true statement. So, there's a sort of course classification due to Cologne, Giroud and Honda 2009, which states that on any, if you have an irreducible free manifold and a toroidal free manifold. So, for instance, any hyperbolic free manifold or any free manifold whose thirst and geometrization decomposition consists of only a single piece and is not one of the easier to understand pieces. Only has finitely many tight contact structures. And the assumption of a toroidal, there could be, Giroud has found examples of toroidal irreducible free manifolds with infinitely many tight contact structures. But the number of homotopy classes of playing fields you get this way is finite. Okay, so this is somehow a brief, very brief overview of some basic results and contact geometry, and hopefully shows that there's some more interesting questions to be asked, especially in higher dimensions. About? That's my question. Yeah. Is it a condition that, or at least for the structures, there should contain some, like, maybe a group of molds or at least some things that can be corrected? Yeah, yeah, so it just, yeah. So, there's some standard contact form you can write down on the ball and it's over twisted if and only if it has that inside. In fact, you can state it even more abstractly just for any contact structure on the ball whatsoever it has that inside if and only if it's over twisted. Let's talk a little bit about symplectic geometry. So, if you have a manifold X, even dimension, so W is a two form, is called symplectic if it's closed and non-degenerate. So, the natural notion of cobertism between contact manifolds or one natural notion of cobertism is that of symplectic cobertism, which is as follows. So, X omega is a cobertism from Y plus C plus to Y minus C minus if, so the boundary X is Y plus this joint union Y minus and when you restrict the symplectic form to the boundary this is D lambda for contact forms on Y plus minus and there's also an orientation or orientation condition which I won't write just saying that symplectic form gives X an orientation that gives its boundary an orientation a contact structure also gives a contact form gives an orientation to Y plus and Y minus it should be the same on Y plus and Y minus. One should notice that this definition is not symmetric with respect to Y plus or Y minus. You can, cobertism like this is often drawn like this a negative end and a positive end. If you have a cobertism from Y plus to Y minus it doesn't necessarily go the other way. What is the trivial cobertism from a manifold to itself? I call it zero one with the symplectic form. So if you take, so this is a cobertism sort of looks compact. We have these boundary components. You can also attach sort of infinite ends on both sides and that's often a convenient geometric operation to do. So you can often sometimes attach Y across if it's zero and same. So just some more terminology. We say Y C is fillable if and only if there exists a cobertism from this to the empty set. So a typical example is like the sphere with the standard contact structure. So if we take S2N minus one the unit ball in CN then it has a contact distribution given by the complex tangencies to the real hypersurface. In general if you have a real hypersurface in CN it has a levy distribution the complex tangencies to the hypersurface that's co-dimension one. Hyperplane distribution and it's contact if and only if the surface, if the hypersurface is strictly pseudo convex just equivalent definitions. We want to add to the definition of complex structure co-orientation because it will be invaluation. So because this cobertism it says symmetry will change co-orientation. Yeah, okay, so this, I guess when you talk about cobertism you usually want co-oriented, guys. Here this opposite co-orientation will be a change in co-orientation. If you have complex structure in the sphere if you change co-orientation it will be as co-orientation. Will it be itself? I don't know. Okay, so I want to now introduce pseudo holomorphic curves and this is a very fundamental tool in symplactic and contact topology introduced by Gromov. So fix a symplactic manifold actually for this first sentence we need this symplactic structure. What I want is an almost complex structure and now what we want to consider is maps for a Riemann surface into X which are holomorphic in the sense that the differential is complex linear. So if you have a map from a Riemann surface to X this is called holomorphic if normally differential is J linear. Okay, so to study these holomorphic curves we usually consider modulized spaces of all holomorphic curves in the given manifold. So for instance we can look at the modulized space of stable maps introduced by Konsevich which is the set of pairs consisting of the nodal Riemann surface and a map from C to X such that let's see the holomorphic we also say fix the homology class and it should be finite automorphism group. So one fundamental property of this modulized space is that if J is tamed by a symplactic form omega which means that omega is positive on complex lines then this space of stable maps are really any reasonable modulized space of holomorphic curves is compact. So this is one fundamental property of this space. Another fundamental comes from the structure of this J holomorphic curve equation. This is some an elliptic equation and what this tells you is that for under some generative assumptions modulized spaces such as this is a manifold of a dimension which you can calculate using the index theorem to be so given these two facts the compactness and the fact that this is a manifold you can extract so-called innumerative invariance out of this. For example so you have this fundamental class and you can in the homology of the modulized space itself you can push this forward to the homology of mg bar and for instance if this is zero you're counting the number of curves. So very rich theory of holomorphic curves in symphactic geometry but I want to contact homology is one specific example. So let me just focus on this example. So certain sort of important precursor to contact homology is Hofer's work on the on the 3D Weinstein conjecture. So let me say what that is. If you have a contact manifold and let's also pick a contact form. So then out of a contact form there's an associated vector field called the the ray vector field. So our lambda ray vector field is defined by the properties that lambda of the ray vector field is one and the lambda ray vector field is zero. So this defines some flow on the manifold and this flow is a flow by contact morphisms. It preserves the contact structure. So there's a conjecture of Weinstein I guess 1979 on any closed manifold. Every ray vector field has a closed orbit and of course once you ask this question you can ask many other questions such as what is the growth of the number of ray orbits? Do I have more than one? Things like this. There's any sort of dynamical question about this this flow of positive entropy, things like this. Now you might ask why is the contact condition important? So there's the cypher conjecture that any vector field on S3 has a closed orbit. This is false by counter example it's found by Cooper Berg and in fact there are volume preserving vector fields on three manifolds with no periodic orbits. So there's some subtle geometry of the contact structure which is conjecturally versus a closed orbit. These are somehow very special vector fields. So what Hofer noticed in his work on the 3D Weinstein conjecture is that holomorphic curves can actually be used to construct periodic orbits of ray vector fields. So let's consider a symplectic cohort is done. So this is x omega and recall that the top end is some contact manifold and so is the bottom end. So now we're going to attach these cones this y cross y plus cross zero infinity and y minus cross infinity zero. And we're going to look at holomorphic curves in this in this cobertism. So you can't just look at let me specify what almost complex structure to use. So we choose a J on this completed cobertism in the interior I just want it to be tamed tamed by omega condition. But in the ends require some conditions. So one it should be there's this translation action in any end should be our invariant ds so s is the r coordinate we want this to equal the ray vector field r lambda and 3 if you restrict it to the contact structure this should be compatible with d lambda. So there's some technical conditions you need. So let's suppose we're looking at holomorphic curves with respect to a normal complex structure satisfying these conditions. So it can happen. So this is not a compact manifold so gromov compactness doesn't apply. Maybe fix the homology class of the curve it might be non-compact. The moduli space might be non-compact. But so let's see how can it be non-compact. So these level sets are pseudo-convex so like in this example of s2n-1 in fact any contact manifold cross r these level sets are pseudo-convex with respect to almost complex structure satisfying these conditions. So what that means is that there a holomorphic curve can't escape to infinity. It's not possible to be tangent like this by a maximum principle. What can happen though is a holomorphic curve can escape to minus infinity so you can have some sequence of holomorphic curves which look like this on your surface and it sort of sends some long finger down the negative end of the co-portism and you can assign to such a sequence of holomorphic curves a sort of a limit in the sense that what you get is you have your co-portism with some curve and this cylinder and it turns out to be asymptotic to a ray of orbit on the negative end you can sort of see from the definition of the almost complex structure that if you have a closed orbit of the ray vector field and cross that with r you got to see the holomorphic cylinder in the sympathization in r cross y and the cylinder is sort of asymptotic to that trivial cylinder over the ray of orbit so in 1993 Hofer used this proved a compactness theorem like this showed that a degeneration of holomorphic curves and sympathizations allows you to find a ray of orbit like this to prove the Weinstein conjecture in a number of cases so the 3D Weinstein conjecture holds for mainly over twisted contact structures also any contact structure on the 3 sphere so actually by a theorem of Aliashberg there's only one tight contact structure on S3 so there's to get 2 out of 1 you only need to analyze one more example and 3 any manifold which is not irreducible so in these 3 examples you can produce a sequence of holomorphic curves which has to be non-compact and therefore escape down the negative end of y cross r so yeah so in 2007 Tobs proved the Weinstein conjecture for 3 manifolds in general the way he proved this was also using holomorphic curves but using a holomorphic curve invariant known as known as ECH or embedded contact hemology to the hatchings and Tobs and in particular so what is this this is an invariant of contact manifolds which involves counting the embedded holomorphic curves in r cross y and so there's an isomorphism between the embedded contact hemology and the monopole cyber-guiton and so basically some non-triviality results which are known for cyber-guiton hemology shows that ECH is non-zero gives this conjecture actually I'm stating it a little bit inverting the history Tobs proved this equivalence after he proved the Weinstein conjecture but somehow this the work for this was the first step in proving this equivalence in full so in higher dimensions the Weinstein conjecture is completely open it's known in a number of examples in a number of special cases I'm not sure yeah so it's yeah they want to prove stability of the solar system because they hope that the planets don't just collide it to the sun in a thousand years so I think that's maybe that's the motivation for this conjecture so this is I guess now time to tell you what contact hemology is so this is an invariant due to Aliashberg given to all Hofer the paper published around 2000 but I think the idea somehow goes back to almost immediately after Hofer's first work on this so let's fix a contact manifold Y and let's also fix a contact form and J will be I'll just write admissible almost complex structure on R cross Y so what that means is these three technical conditions so it should be R invariant should send ds to the red field and it's a restriction to the contact distribution should be compatible so now we're going to say P is the set of periodic orbits now what this definition is going to be is we're going to make some chain complex generated by the periodic orbits yeah yeah so let's assume non degenerate so we're going to define a chain complex generators are all the periodic orbits in differential defined in terms of through the holomorphic curves and this is going to turn out to be an invariant of the contact structure so it won't depend on the form or the almost complex structure just the contact structure and a corollary of just the definition of the fact that the definition makes sense is going to be that if we can calculate this invariant and show it's non-trivial we get the Weinstein conjecture because there are no periodic orbits contact technology is just trivial because there's no chain complex so the definition is the following so we'll look at the free algebra generated by this set P well actually it's not all the elements of P it's some subset so there's some issue about pathology theory which issue about ray orbits being multiple covered and so there's a natural partition of the set of ray orbits traditionally the two sets are called the good ray orbits and the bad ray orbits and this chain complex is just generated by the good ray orbits the other comment to make here is the free super commutative algebra generated by P good so these ray orbits have an additional discrete invariant they're either even or odd and so this algebra A B is B A times sure super commutative odd elements and they commute okay so far we haven't used holomorphic curves holomorphic curves are going to define the differential of this complex so for gamma plus in P good I need to tell you what the differential of this is the differential is going to depend on the choice of almost complex structure so I'm going to take the sum of overall finite subsets of good ray orbits we're going to look at number of elements in this modular space so the holomorphic curves in R cross Y where the top end is asymptotic to gamma plus and the negative ends are asymptotic to gamma minus and the index of the curve is one times gamma minus so in this case index is virtual dimension plus one because we're mounting out by R yeah genus zero yeah so there's in this paper we have to be given to Alhofer to define a more general theory where they sort of count all holomorphic curves in here the sum is finite yeah so this is an algebra I told you what the differential of a generator is you extend it to satisfy the Leibniz rule so this becomes differential graded algebra so now so this is the definition as it's presented by Aliasper-Giventhalhofer so the what I want to talk about the goal of the course is to tell you how to define these moduli that counts the numbers in this definition so up here when I said under certain assumptions this moduli space is an orbifold of this fixed dimension basically those assumptions almost often fail in this example nevertheless one still expects that these moduli spaces to carry a rich and imperative structure so I want to tell you how to count these moduli spaces so this is if you want to look on the archive the result is that such as in contact homology such moduli counts exist making this satisfy d squared equals 0 so that when you take the homology contact homology of yxc is an invariant of yxc that doesn't depend on the choice of contact form or the choice of almost complex structure why is this called homology I mean it's the homology of some chain complex but where did this chain complex come from so here's one one motivation so we can look at this space of all maps from s1 to y often just called the loop space in fact I want to quotient by so it just looks at loops in y up to reparameterization up to rotation now there's a map to r given by given loop gamma to the integral of the contact form if you have a contact form you can look at this this contact homology is very roughly speaking a version of this homology on this space with this on with respect to this function so it's somehow the space of disjoint unions of loops so if you the straightforward calculation shows that the critical points of this functional are the ray orbits periodic orbits and the gradient flow formally speaking is exactly the pseudo holomorphic curve equation so this is I mean if we just took the free q vector space generated by the periodic orbits this would contact homology would actually be more homology for this function so one can study Morse theory on the loop space and serve much more simple examples like this than this if you say take at this level of precision it doesn't matter much because this critical points will be in objectivity function yeah okay yeah yes let's divide by so a sort of more common form of doing Morse theory on the loop space is to just look at the Dirichlet energy of a loop and one can do Morse theory this is sort of very classical the technique of studying the loop space is to study sort of Morse homology of loop space with this functional so but so let me say why this is why these two somehow very different so the critical points of this functional are closed jd6 so maybe I want to mod out my rotation here the critical points of these are closed jd6 but Morse is true so when you have a Morse function the critical point you look at the Hessian so at a given closed jd6 here the Hessian is some self-adjoint operator it has infinitely many positive eigenvalues and only finitely many negative eigenvalues and somehow very much related to this fact is that the downward gradient flow of this function called e Dirichlet energy is a parabolic PDE it's basically the heat equation nonlinear version of it and what this means is that the initial in particular the initial value problem is well posed if you just pick up any sort of random loop there's a well defined downward gradient flow of this energy starting at that loop this is not the case for this flow here so the downward gradient flow of this function which is usually a for the action functional is it's a holomorphic curve equation which is elliptic and as a result there's no well defined downward flow so if you start at just a random loop here there's no well defined downward flow and so what this means is that so the consequence is that when you do this isn't a priori and in fact it's not at all doesn't calculate the homology of this loop space we're just trying to study the topology of the loop space this sort of version of Morse homology works and you can get existence theorems for closed geodesics on manifolds by saying something about the topology of the loop space and using this version of Morse theory because this actually gives you some sort of solid composition of the loop space this doesn't in fact it depends on the contact structure you choose otherwise it will be sort of boring and variant so I think I think it should be but somehow this would be I haven't seen yeah I mean still it's been done like there's a Havana stable homotopy type you know yeah there should be some some stable homotopy something you can get out of counting the mod-size spaces of higher virtual dimension job not just the ones of dimension zero so somehow the reason this works is that to do Morse theory you don't need the downward gradient flow to be well defined or to be well posed all you need is the problem of sort of specifying a critical point to the top and specifying a critical point to the bottom and the space of flows to give you a well defined naturalized space and this is well defined for elliptic PTEs like this so so even though the downward flow isn't well defined we can still sort of do a version of Morse theory so if you try to sort of take this definition and do exactly this you get something called cylindrical contact technology so in some cases what you can do is take the complex which is just sort of the free vector space generated by Q by P and take a differential which just counts the good so you just count holomorphic cylinders index one in R cross Y does it mean that in this situation the output could be not empty set here the output here it could be we do count holomorphic planes also yeah it seems to be not really a sub complex in this case oh yeah so this is somehow this is a different which exists in less generality so I'm going to say why this is this doesn't work in general so this is somehow the most direct idea of taking Morse theory on the loop space and turning it into a homology theory but why doesn't this work so to get a homology theory you need the square of the differential to be zero so in all these types of Morse flow theories one needs to show this and the reason it's true is the following so if you look at the square of the differential applied to Y plus what is this well it just counts of two-story curves starting at Y plus going to some any curve here and then ending at Y minus gamma minus where these both are index one curves so why should this be zero well in favor of what you want to say is that this is the boundary of the moduli space of trajectories just starting at gamma plus and ending at gamma minus of index two so this moduli space is generically one-dimensional and it gives a cobertism from this moduli space to the empty set in particular it follows that this coefficient is zero now the problem is so this the boundary of this moduli space has curves like this but it doesn't only have curves like this it also have a splitting which looks like this and so you no longer get d squared equals zero you get d squared equals something else so you can define some sort of series of operations on this vector space where d zero gamma counts moduli space is where you have no negative ends d one counts moduli space is where you have one negative end d two counts moduli space is where you have two negative ends etc so what this says here somehow d one squared is d one squared plus d zero d two so if you're if you assume say there are no contractable ray orbits then this degeneration can't happen and cylindrical contact homology is well-defined but in general we have to sort of count curves with one positive end in arbitrary numbers of negative ends to get a well-defined homology theory now the reason we can restrict to just one positive end is because of this maximum principle I talked about earlier so if you have a copartism like this curves can't escape towards the positive end so if you start with one positive end you can never degenerate to have more positive ends but you can always sort of add negative ends so I have to include all of them so embedded contact homology is defined in dimension three and in dimension three sort of r cross y has dimension four and so the condition of being embedded so you have a sort of well-defined intersection theory between holomorphic curves in four dimensions and you can look at embedded curves and that gives you a very subtle restriction on the set of curves you consider and this gives you a different homology theory which is isomorphic cyberquid unfortunately it doesn't it's not really clear how to generalize the higher dimensions and there's there's no corresponding non-triviality result for contact homology there's some sample applications of this theory so first obviously what I said earlier if this is if this is not q then then the Weinstein conjecture yx obviously they're things like distinguishing different contact structures another one is certifying tightness so it's a fact that contact homology of any over twisted manifold is zero so why is this true so in particular if you can calculate contact homology for something and it's not zero then it's not over twisted so why is this true in in any over twisted manifold you can find the contact form and j such that there exists a rape orbit bounding exactly one homomorphic plane in the simpletization so it looks like this so the construction of this so this is Aliashberg in dimension three bourgeois either Krueger in dimension higher dimensions and what this tells you is that the differential of this guy is zero so one is a boundary in this chain complex so one equals zero in contact homology so somehow all you need is this one low plane which exists locally in standard over twisted ball and contact homology of anything over twisted vanishes so this is actually a common thing which happens this should be expected in simplectic contact geometry there's some structures or some class of problems which are very flexible over twisted contact structures is one example of that they're classified completely by homotopy theoretic data and for these sorts of flexible problems homomorphic curves generally tell you nothing or the curve invariance are usually trivial rigidly results in contact and selected topology are often proved using homomorphic curves and finding obstructions using homomorphic curves so as a result you can combine these two results and conclude that the Weinstein conjecture holds for anything over twisted so this is actually do the Hofer 93 for three manifolds and Alper's Hofer 2009 so Weinstein conjecture for over twisted even PS over twisted so another thing you can say with this is obstructing fillability so if you have a contact you have a simplectic cobertism from y plus to y minus you can count homomorphic curves in the cobertism with one positive end with a very morbid of y plus and of genus zero and a bunch of negative ends with a very morbid of y minus and you get so called cobertism BAP on contact technology from y plus to y minus and you'll notice something interesting happens if it is over twisted because then y plus over twisted so if the positive end has zero contact homology then so does the negative end because this is a unit of ring that has to send one to one in particular if you're over twisted you're not fillable because that would be a cobertism from your over twisted manifold to the empty set in the contact homology the empty set is q so it's an interesting question whether contact homology detects over twistedness it would be amazing if vanishing contact homology implied that you're over twisted if you change co-orientation or twist it you'll be game again or twist it here so it means that you can it's going to be filled from other side which doesn't fall from this argument yet you get from zero to zero they can definitely but over twisted can definitely be capped yeah yeah I mean assuming that a cap exists like topologically but yeah the over twisted can be capped no I think right Alasberg Murphy have this pre-print like constructing making cobertism symplectic I think they show the negative end if you have a cobertism like this and you have sort of a contact structure here y- c- and if you have a formal symplectic structure here almost contact structure here then you can make everything symplectic and contact as long as the top boundary is non-empty and the negative boundary is over twisted so then you can then you can just choose the positive boundary to like be a sphere and cap it with cp cpn yeah definitely for tightness this is a subtle thing but if you want to change the co-orientation changing the sign of lambda you also change the sign of the differential of lambda so you change this almost complex structure on the side so it's not really the same almost so I don't know this is an interesting question and the corollary of this would be that if the positive end of a cobertism is over twisted then the negative end is over twisted they're widely open right now it's known in dimension 3 if this is a Weinstein cobertism but this is a very recent theorem of Andy Wand that Legendre and Surgery in dimension 3 preserves tightness and it would follow if we knew this but this seems this would be this is a much harder statement apparently and completely open in higher dimensions so there's there's a related and variant just an aside if you have a Legendre in R2N plus 1 there's a related and variant Legendre in contact homology and there's there's a related notion of a Legendre in being loose analog of over twisted in dimensions at least 5 so you can and for loose Legendre the Legendre in contact homology is 0 just like this so you can ask the same question and it's false so very recent examples of Tobias Ekholm found Legendre in R2N plus 1 at least 5 which are not loose but the Legendre in contact homology is 0 so this is probably false in higher dimensions but no examples known right now so finally let me give some examples of calculations of contact homology so the easiest case is when you can find the contact form where all rape orbits have even grading because then this chain complex is concentrated in degree, the differential is degree 1 so it has to vanish so this is, you can actually do this in some cases so Ustilowski found all these even contact forms on certain risk-grown spheres so links of isolated type or surface singularities and the corollary of this is that there are infinitely many tight contact structures on S4K plus 1 because contact homology they're different of course this also implies the Weinstein conjecture for these contact structures it implies a certain growth rate of the number of rape orbits for these contact structures everything all of these sort of applications are say something here Abramakarini found even contact structures on S2 cross S3 and again also distinguished using contact homology and many others so there are many other calculations of contact homology a lot of them happening sort of before a rigorous definition of contact homology was in place so I believe most of them should work with this definition I'm good to give there are lots of them and I haven't read them so I think 99% of the what's done should be completely valid I'll stay the more precise version of the result in a second but it says in particular that if all the holomorphic curves are cut out transversely then the counts are what you think they are if you have transversality you can calculate using those ok let me just mention one more application symplectic embedding capacities so if you have some open symplectic manifold say some subsets omega 1 and omega 2 inside of r2n the standard symplectic structure interesting question to ask is is there a symplectic embedding of one into the other so the famous grommos non-squeezing theorem says that if you embed want to embed say ball of radius r into a cylinder then r is less than or equal to r so this is the first evidence that this is an interesting question you can you can find a so there's an obvious obstruction the volume this has finite volume for any finite r and this has infinite volume but yet there's some non-trivial constraint here you can use contact tomology to get other non-trivial constraints so in the following way in fact a lot of they're already embedded capacities which come from embedded contact tomology are known and been used a lot I think nobody's really studied contact tomology capacities that much I know so contact tomology capacities so if you have an embedding say omega 1 into omega 2 you get omega 2 minus omega 1 is a cobertism from boundary of omega 2 the positive end to boundary of omega 1 and so on contact tomology you get a cobertism map from contact tomology of the boundary the positive the main to the sorry yeah so I guess yeah so let's assume these have contact type boundary and so each of these groups has a filtration by the action that's this Morse function not this Morse function the Morse function I wrote down earlier the integral of the contact form and this map has to decrease action by some Stokes theorem argument and if on the other hand say if these are like say convex domains then you know the contact tomology of the three sphere and you know that this map has to be an isomorphism so on the one hand it's an isomorphism on the other hand it decreases action so if you can somehow calculate the action filtration on contact tomology of the boundaries of these domains that gives you obstructions to embedding one into the other so so you can say so for any any element of the contact tomology of say S3 you get something called a capacity which say C sub y of omega which is just the minimum action such that this element is represented by some ray of orbits of action less than a and this number is sort of monotonic with respect to symphactic embeddings so that's all the applications let me just sort of conclude by sort of staling a more precise version of the theorem that I want to spend some time proving in the rest of the class so for any y lambda j contact manifold of the contact form and an almost complex structure closed manifold there exists a set data just functorial in this input data along with for every element of that set a collection of numbers we just think of the cardinality of this moduli space such that when you form the contact tomology differential using these moduli counts we get these critical zero so I need an extra condition I need two extra conditions because as stated this theorem is vacuous so first this set better be non-empty it's still vacuous because I could take all of the moduli counts to be zero and let's satisfy these critical zero so the other condition I'm going to impose is just that if this moduli space is if it's cut out transversely then the virtual counts are equal to the usual counts that you expect so cut out transversely is this technical assumption you need to show that it's a manifold of the expected dimension for each theta so when you have a bunch of non-transverse if you have a single non-transverse moduli space then it's virtual fundamental class in homology is well defined but if you have a bunch of moduli spaces and they have boundary and somehow the boundary of a given moduli space is a product of a bunch of other moduli spaces and you can't sort of say what these numbers are canonically you have to make some choices perturbation choices and these choices affect the answer you get so this theta is somehow the set of all perturbations you could make to make things transverse and then this is the resulting numbers if something's already transverse you don't need to perturb it and even if you did you'd still get the same answer so that's what this is saying so there are corresponding theorems for symplectic cobertisms and for families of symplectic cobertisms and these are necessary to show that this is an invariant but maybe I'll just state this one because the others are essentially the same so next time I'll define the moduli spaces in more detail and I'll start the proof of this construction of fundamental cycles