 Thank you for the invitation to talk in this very nice summer school. It's a pity that I cannot enjoy a lot more than today of this thing. But it's really nicely organized and I really have the feeling that I'm on vacation. So what I'm going to talk about today is some joint work with Alex Wancia. And so indeed, the idea is to get an invariant for contact manifolds. And if you are working with holomorphic curves, well, probably the most natural thing that you would like to do if you are really given a contact manifold and not a symplectic manifold is to use something like contact homology. And so what I would like to start discussing is the simplest version of this invariant, which is the cylindrical version of contact homology. So assume that your manifold, contact manifold is y xi and denotes the dimension of the contact manifold by 2 and minus 1. And as we usually do in this series, we assume the contact structure to be co-oriented so that it is expressed as the kernel of one form. And so I will assume that this one form is non-degenerate, which you can always achieve by a small perturbation. And so this means that if you have a closed wave orbit, that is a point P that lies so that the wave flow after time t for some positive t gets back to P, then if you look at the differential of this wave flow at time t restricted to the contact hyperplane at point P, then this has no eigenvalue equal to 1. And whenever you are in such situations, modulo some choices of trivialization of the contact structure along the trajectory starting and ending at P, then you can associate to such an orbit the connolly-zender index or integral-valued index for such orbits. So that's one of the ingredients, the choice of non-degenerate contact forms that is necessary to set up this theory. And the second ingredient lives at the level of the simpletization of our contact manifold. So this is just the product of y with the real line equipped with the symplectic form d of e to the t alpha, where t is a coordinate along this real line. So that's the simpletization. And on this thing you can choose j to be a compatible almost complex structure with the following properties. The first property is that j preserves the contact distribution and on it, and that's the meaning of the word compatible here, this is compatible with the symplectic structure d alpha. Second, it maps the vertical direction d over dt along the real line to the wave fields of our contact form. And finally, j is invariant under translation along the real line. So these are the two ingredients you need to set up this homological theory. And once you've made this choice, you can consider now modulated spaces and denote by m gamma plus gamma minus the modulated space of g-holomorphic cylinders, which are maps with two components, e, u, defined on the cylinder, and we value in r times y. So a will be the height function and u will be the mapping in the contact manifold. So it just means that the differential of this intertwines the obvious complex structure on the cylinder with the almost complex structure j on our simpletization and with the following asymptotic properties corresponding to this data here in the definition of the modulated space. So such that when you take the limit for s going to plus or minus infinity, so here s stands for the coordinate in the line direction on the cylinder, of a of s and it's called theta, the angular coordinate on the cylinder. This is plus or minus infinity and the limit as s goes to plus or minus infinity of u of s theta is going to be given by the omit gamma plus or minus and then I need to put some factor here. Sorry, no plus minus, it's really plus. So just to take into account the fact that this theta has period one on the circle by choice, by convention and omit, of course, can have different periods for the wave field. And so this is a modulated space in the sense that we are putting together solutions of the Cauchy-Riemann PDE with these asymptotic conditions but modulo by holomorphic we parametrizations of our cylinders and translation in the r direction in the target. Since J is invariant under this direction if you have a solution and you shift it you get again a solution. So that's the modulo space that are essential to contact homology, at least in cylindrical version and now big if, if transversality holds for the elements of this modulo space then we know that the dimension of this modulo space will be given by the threshold index of this elliptic problem and its dimension then is given by the difference of the kernels in the indices of the asymptotes minus one, minus one coming from the quotient by the r translation. We lose one degree of freedom that way. Okay, now let me put aside this important restriction. For now I'll come back to it in a minute and let's continue with the construction. So now that we have this analytical ingredient we can construct some algebraic objects in the chain complex. So we can define the contact complex or its cylindrical incarnation. So star will stand for grading these things and this will be the z-module generated by all good closed-wave orbits. What good means is that for every periodic trajectory of the wave field there exists the shortest trajectory of the same image. Now you can compute the connolly-zendo index for the trajectory and its simple counterpart and it's good if and only if the properties of the connolly-zendo indices coincide. It's purely numerical if you wish. Easy to check condition on the periodic orbits of your wave vector fields. And this is graded by not exactly the connolly-zendo index but you shift it by constant n minus three. Remember that n appears in the dimension of your contact manifold and the reason for this shift will become apparent in a minute where conditions will be spelled out and that are more natural to express once the shift like this is made. Now this is still potentially huge and probably too large for your purposes so it's sometimes good to restrict yourself to a smaller object and that's something we typically do. So I pick a free homotopic class of loops in Y and I can restrict myself to those wave orbits that are in the class A. So it's something already potentially smaller depending on the topology of my manifold. So that's just incorporating the wave dynamics and now for the homotopic cylinders where you can define a differential, by defining d of some orbits that I write gamma plus in analogy with the definition of my modular space to be the sum over all orbits that I denote gamma minus with the property that the grading of gamma minus is one less than the grading of gamma plus. You count elements in your modular space. So here this is not really the count of elements in the set. It's an algebraic count. You can actually associate a sign to elements in modular space via some coherent orientation construction but I won't get into that. So you count with signs the elements of this modular space and you multiply with the corresponding bottom generator and you see the fact that you restrict yourself to free homotopic class is not an issue because definitely if gamma minus appears in the differential of gamma plus then they are homotopic because there will be a cylinder between them. So you get a differential here and the hoop is that you can get a homology theory so that the square of this thing would be equal to zero but the study of the generation of homomorphic curves shows that this is not always the case. dc squared is not always going to be equal to zero and that's because cylinders in our modular space if the modular space has dimension one so again under transversality assumptions can also degenerate not only as two consecutive cylinders which would be counted by the square of this differential but it can also degenerate into another configuration which corresponds to a pair of pants and a homomorphic plane and here this is a vertical cylinder over or close to a orbit and so this configuration lives into two separate copies of r times y. This piece and that piece have each index one so if you don't do this art one installation business the index would be equal to one so here it's the condenser index of the top minus the condenser indices of the... sorry, the grading on top minus the sum of the gradings on the bottom and here the grading on the top would be the dimension and of course this vertical cylinder is isolated and so this is index zero and so if that happens then definitely dc squared counts the generations into pairs of cylinders but it's not equal to zero it's equal to the count of these objects and so if we want to get a differential we need to prevent that sort of phenomenon from happening and one way of preventing this phenomenon from happening is to prevent this orbit where the breaking occurs from existing if I call this orbit gamma tilde then clearly due to the fact that the index of this element here is one the contact homology grading of gamma tilde is equal to one and of course since it bounds a plane it is a contracted orbit so this means that if we are under these ideal transversality conditions we have the following result for this construction so if again transversality holds I'll spell this out in a minute and if you assume enough to avoid such phenomena and this means that you ask that the contact complex in the Zebo-Romotopy class vanishes when the grading is equal to one Zebo-1 then dc squared is equal to Zebo and the cylindrical contact homology in class A which is defined as the homology of that complex is an invariant of our initial contact manifold so here this condition that the complex is trivial in grading one corresponds precisely to the assumption I made here so that's to guarantee that dc squared is indeed equal to Zebo so you can define this homology and these two conditions are necessary to show that this homology is an invariant okay so that's a very nice tool to study contact manifolds but the big if is this assumption because it is very very rarely satisfied and well I cannot say exactly under which conditions it is satisfied but at least I can show you a situation which is I think the only situation where we could make a statement in finitely many words where we can be sure that this holds and you will see that it's pretty restrictive so transversality rarely holds and this is for example the case if first there are no contractible closed-wave orbits at all independently of the grading so this means that simply connected manifolds cannot work with and second you need to choose the class A the free homotopy class A that we respect ourselves to construct the complex so that it contains only simple closed-wave orbits that is injective orbits you cannot have multiplicity greater than one when it does hold so you need both conditions to be satisfied and that's an example where it does hold but I cannot give you very general conditions more general than this for which it holds maybe in some very specific cases using some ad hoc argument depending on which specific manifold you can hook up something but okay that's more or less the best we can do so that's indeed extremely restrictive and the reason it works is that well if you have these two conditions you never of course have to worry about this kind of situation independently of transversality because you don't have planes for one and second all the cylinders you are going to work with since you have simple asymptotes so injective asymptotes the holomorphic cylinders will also be somehow injective near the ends and by a result of a dragnet then it is insufficient to be able to choose your almost complex structure as above in a generic way so that transversality does hold so that's basically the one case where you can do something with this invariant but it leaves out a lot of interesting situations Can you give an example of the examples? Sorry? Give you an example of the example and then what is it actually? An example of the example, yes if you take the unit good engine bundle of some hyperbolic manifold then the wave flow coincides with geodesic flow and then you can choose some simple homotopic class that is we're going to represent it by geodesic and you don't have contractable geodesics so that's one big class of manifolds you can find a few others I mean the flat case works as well it should work on torus it works as well but well that would be my main class for example I'm afraid Sorry, what is the second star setting and then I just can't read it So the class A the free homotopic class A in which we decide to respect ourselves contains only simple closed wave orbits so that's cheap you can always choose this class like this but of course the difficulty is that maybe this small piece of the invariant would be too small for whatever you want to do with it so that's also possible restriction Okay, so that's quite frustrating that we cannot make these things work at least without polypholes Is there a ray orbit in this example that you don't want to arise? Is that a hyperbolic or a elliptic? I assume in dimension three If you're in dimension three this is a hyperbolic orbit, isn't it? Yes, thank you Yes Okay, so let's see what else we could do in order to salvage the situation and still have something that is pretty similar to this construction but that will work in much greater generality than these very heavy restrictions and one way to do that is to adopt some symplectic point of view and work with S1-equivariant symplectic homology Okay, so the big difference between contact homology and symplectic homology is that now we are really in Fleur's theory and for Fleur's theory you need an extra ingredient which is a Hamiltonian function So let's consider first an easy model of a Hamiltonian function small h defines our symplectic manifolds namely the simpletization of y and assume that this h behaves as in the following schematic picture I'm going to draw a sketch of the graph of h as a function it will depend only on the t-coordinate here and instead of writing t as the variable I'll write the variable to be exponential of t so it means that it takes value from 0 to infinity and this small h I will give it some asymptotic conditions I ask for that very large values of parameter this function is a piece of line with big slopes which will be something of the form capital T times p to the t plus some constant beta, whatever and on the other hand near the origin I ask that it starts with a very small slope so it's also linear there with a small slope and then in between you just interpolate it to the t it's e to the t, indeed I can even get you through to my face myself please, don't worry okay, so now if you choose such a Hamiltonian function well you have to look at what the Hamiltonian vector field associated with this is doing because it's really the meaningful orbits in fluid theory and this Hamiltonian vector field is going to be minus the derivative of this function each with respect to not t but this other variable e to the t multiplied with the wave field so that's what you easily find out and so this means that one periodic orbit of the Hamiltonian vector field are in bijective correspondence with closed wave orbits but with a constraint that's the period of this orbit cannot take values that are obtained by this derivative which is between something very small so you can assume that there is no period of a closed wave orbit in zero if you take the slope and t, so these are closed wave orbits with periods in interval zero t and in order to avoid unpleasantnesses I will assume that this final slope capital t is chosen generically so that it is not a period for the wave field definitely you will not have closed orbits in this region so all the periodic orbits will actually be contained in compact region of your simpletization so that's a very nice Hamiltonian, easy to understand but you cannot quite work with such a Hamiltonian to do a standard fluid theory because this vector field is independent of time and therefore if you work with a parametrized one periodic orbit of this vector field then as soon as you have one you have a whole family of it because you can reparameterize you can shift the parametrization and in flow theory you work with parametrized orbits and so you are going to use another Hamiltonian capital H which depends on time, so this periodic time s1 and of course on our simpletization and this capital H which is going to be a suitable theta dependent totubation of this small H so what do I mean by suitable well I would like to understand in fact the correspondence and continue to be able to understand the correspondence between one periodic orbits the total vector field and my closed way orbits what happens is that in order to perturb what you will typically need to do is to use this theta dependence you can view this as a function on the circle and along each closed way orbits you are going to introduce a perturbation by a function on the circle and if you choose this to be a most function then each critical point of the most function will give rise to a one periodic orbit of the perturbed Hamiltonian field and so in order to avoid spreading zillions of periodic trajectories after the perturbation you are going to choose the minimal number of critical points for a function the most functional circle namely two so the suitable perturbation is chosen so that one periodic orbits of or time dependent Hamiltonian vector field for capital H are actually given by a pair of orbits that I denote by gamma check and gamma hat for each closed way orbits gamma with periods between Z1 and Z2 so that's really going to be the interesting guys so two generator or two orbits to Hamiltonian orbits per closed way orbits that we had in the in the contact picture okay so now that we have introduced this extra ingredient then we can make a definition for symplectic invariant now if we were simply doing ordinary flow theory you would need to work with a symplectic complex which is the Z-module generated by all these one periodic orbits of the time dependent Hamiltonian vector field this is again going to be graded by a conlysander index and for consistency with the previous definition I will ask that this is graded by conlysander index plus the same shift and minus we as before this differs a little bit from the usual conventions in literature that are meant for symplectic homology here I'm twisting the picture to make it look like the contact the contact invariant okay so that would be the complex we would use for the ordinary flow theory but that's not what we are doing we are going to do some S1 equivalent version of that second layer is to define an S1 equivalent version of that of the flow complex so this is going to be based on a different Z-module which is obtained by taking the usual construction, the usual Z-module, tensed over the coefficients Z with a polynomial ring in one formal variable U which has to be equal to 2 seems like a kind of arbitrary definition at this point no no no they are not forgotten it's just that when I say generated by one periodic orbit of these things this includes both gamma checks and gamma hats for all possible gammas oh yeah the idea is that yes the check is like an error going down so it's one index one less and hat is like an error going up index one more the index differ by one it's the most index on the on the circle I'm not going to use the check and hats notation anymore because I don't really need to so I just did that doesn't really matter this context depends on T so T so the T is the variable in the r direction yes capital T yes yes yes of course you are limited to periodic orbits with a period up to T so indeed that's it's a choice at some point we need to remove that choice later in the construction but at this point yes it depends on T on the choice of having two many things it's only at the end of the construction that you will end up with an invariant and then on this the module I need to define something that will play the role of a differential so I denote this by ds1 and this ds1 well I need to say what it does on each of these factors here and you can decompose this as a sum for K from 0 to infinity index K corresponds to the effect on the polynomial ring because it will lower the power of U by K now if you have a polynomial with degree less than K this actually kills the polynomial so it means that even though the sum is infinite it acts on a given element in here you only have finitely many terms contributing so you don't have any convergence issue it's just formal notation on any given element there are finitely many terms and then on this factor you act by some map and I will denote for now this map as phi K and we need more time to define what is this phi K because it will be counting solutions of some modulate problem so that's the general form of the differential what did you say that it's actually a finite sum because by definition convention the effect of this lowering the degree of U by K has the effect on a polynomial of degree less than K of killing the polynomial to start slowly and for those who are not quite familiar with flow theory let me say what phi zero is going to count so phi zero is going to count solutions of the flow equation and so these are maps U tilde I don't need to distinguish between the r and the and the y factors anymore defined on the cylinder in in our symplectic manifold such that d U tilde over ds plus j of d U tilde over d theta minus that's the effect of the Hamiltonian now the Hamiltonian vector fields along our cylinder is equal to zero and U tilde of s and theta converges when s goes to plus or minus infinity to some one periodic orbit of the Hamiltonian field and here I'm putting a tilde because it's not a web orbit where I used to make its gammas but it's a Hamiltonian orbit that's the only meaning of this tilde so that's the flow equation now notice of course that this equation is theta dependent due to the Hamiltonian vector field and so this means that there is no reason that we should keep exactly the same data capital J as we used to have in the context situation where we had one almost complex structure we can have an s1 family of almost complex structure we can choose this J to be theta dependent as well it's always already a theta here so that's one big difference between the context construction explained before and this construction is that now you can choose you have a wider choice of geometric data because you can choose your almost complex structure to depend on the variable theta and that's actually quite important because even whatever the orbits are actually doing in the manifold since geometric data depends on theta orbits are as good as simple somehow for transversality purposes and so this means that in practical situations we will be able to choose generic theta dependent J to achieve transversality for such modular spaces that's a big advantage compared to the previous construction and the index difference is 1 the index yes so here so yes the guys that accounted by phi zero corresponds to the fact that gamma plus minus gamma minus minus one is equal to zero right the same same dimension yes so you're just adding another parameter to give you more control of your channel yes it's actually enlarge the space of geometric data you're working with so that with this wider choice you have enough to get generosity okay so that was just phi zero now I have to repeat that infinitely many times to give you the differential and more briefly phi k will count not just maps u tilde but tuples composed of such maps but also of parameters tau bar and l bar the bar just stands there to say that these are vectors in fact because tau bar will live in s one to the power k while l bar will live in r plus to the power k minus one so there are components for each of these two pieces of data and these things will be solutions of again some flow type equation writing down the same type of equation but now the difference is that both my almost complex structure and my Hamiltonian vector field also depends in some way on the variable s the real direction along the cylinder and of course I need to tell you how these more general ingredients are defined out of the original Hamiltonian h and the original almost complex structure jc and I need some room to explain this little s is the variable in here one of the variables for your cylinders so j of s theta will be constructed from the original j that depends only on theta from that j here by setting theta equal to some function beta of s and theta and similarly this Hamiltonian up there that depends both on s and on theta so the theta dependence was implicit but it was always there of course will be original Hamiltonian but with variable theta being set equal to some function beta of s theta and so now what I have to tell you is what is this function is this function beta of s theta how it is defined over the cylinder and so to do this let me do schematically how it depends on s so that's my real line with the s variable ok so for s less than 0 you take beta equal simply to theta so that's the old equation you set theta equal to itself and it's like the ordinary flow equation now after one things are changing and they will be changing until you hit the first component of this L vector and in this region it will be theta minus tau 1 so basically you are shifting the angle by some other angle tau 1 which is the first component of this parameter up there and here in between you have some interpolation between L1 and L1 plus 1 you will leave this constant region to do some interpolation with the next one where you shift by tau 2 and you will keep this constant value until you hit L1 plus L2 again you have interpolation here here for duration 1 you have again interpolation and you go on like this until you hit the last special moment which is L1 plus Lk minus 1 and for duration 1 you have interpolation and after that you have the final value theta minus tau and the last one is tauk and so that's what the map beta is doing so actually if you forget momentarily about this interpolation region you are just simply rotating the cylinder by different angles tau using some finitely many regions of different sizes interpolate between these different torsions for the flow equation does the tau increase on the dark end? so what we are going to do now is to count triplets of solutions so tau can take any value just looking for tau so that there exists a solution U looking for tau and for L so that this twisted equation this complicated S-dependent equation has a solution and so this means that if you really look at the threshold index of that and then you freeze temporarily tau and L if you freeze tau and L and you look at this equation you will typically look for solutions that have a negative index and the index will freeze and go back up to 0 only because of the presence and tau and L because you have the freedom of varying tau and L to search for solutions that will eventually find some solutions and that's the guys that you are going to count in the map Phi K so really Phi K count cylinders but the cylinders themselves you think that alone has negative index unlike what happened before when it had index 1 because each time you increase K by 1 you are introducing two real parameters 1x tau, 1x tau L and so the index is going to be lowered by 1 right so these the quantity U till does have index well before it used to be 1 but now you subtract 2K and so this means that Phi K will have grading so the effect of Phi K of the grading will be the opposite of that 2K minus 1 but then U shifts down by 2K so the end if you shift by 1 and so indeed all terms and therefore this differential has the effect of lowering the degree by 1 so everything is as it should be ok so so that's the definition of this S1 equivalent differential so it looks a little bit strange and coming out of blue I mean this this definition with multiple twists so if I want to say in a nutshell what is this DS1 well DS1 if you only start writing the first few terms of this infinite expansion this is going to be first Phi 0 but Phi 0 it's the usual flow differential it comes the usual things plus U inverse multiplied by Phi 1 but Phi 1 will have no L variables there will be only one twist between Theta and Theta minus some tau a single twist and this Phi 1 is typically denoted by Delta and this we are familiar with S1 equivalent in theory is the BV operator and then you have higher higher terms in U inverse that will come in and the reason for the presence of these terms you might say ok that's if you know about S1 equivalent theories incorporating the BV operator should be enough somehow to have an S1 equivalent theory well the thing is that these two terms alone do not square to 0 so the presence of these terms is to guarantee that the square of this thing is equal to 0 ok so that's the definition of this S1 equivalent complex and so now what's the what's the result about about this well the results obtained jointly with Alex Wancha is that if we assume that the usual symplectic complex vanishes if the degree does not exceed 1 so that's slightly stronger than asking that the the contact complex was 0 when the degree was minus 1, 0 and 1 because you ask that for all negative degrees but in many many classes of examples it's not really an additional restriction so that's that's still a relatively nice assumption so if you have that then then this S1 equivalent differential squares indeed to 0 and the S1 equivalent homology I need to denote a plus because that's the notation I mean there is another guy that has no plus and that's something different but in this context I need to put a plus which is defined of course by taking the homology of this complex so this of course will remove a lot of the dependence on the choice of G of the Hamiltonian H, capital H perturbing the small Hamiltonian H but indeed the small Hamiltonian H does depend on capital T so I need now to remove the capital T dependence and it turns out that this collection of homologies fits into a directed system with respect to this parameter T and one can take the direct limit when capital T goes to infinity so that we consider all close to a bobbits in the theory and the result of this is going to be an invariant of our initial contact manifold and the big gain compared to the previous story is that the assumption is just an assumption on the dynamics is satisfied a lot more often than the very restrictive conditions that are necessarily here for 20% that's a big gain so we get now a device that we can use instead of linearized contact homology for this for contact manifold I think this may be the first time a plus has appeared in your notation yes here yes I should have it doesn't matter I mean okay it's just that if I don't work with simpletization but if I work with a symplectic manifold with convex boundary then there are two objects S1 without plus and S1 with plus and somehow this corresponds to the plus version because the feeling and generators that could be inside the feeling in the interior of the feeling are not contained but it just fits with these other cases but if you concentrate purely on simpletization right yes for example yes you can do that well the price to pay of course is that all the action if you like is not contained in a compact region because you go up to infinity but you can also do this sort of thing I mean these have been done in the standard flow theory by various sources right so you don't need to take the limit anymore because because then you are done immediately yes there is a little bit of extra care of complexity to take care of the fact that you are not in compact region etc but then you don't have to do that anymore so you know just matter of taste not really crucial okay so now maybe let me explain quickly why this assumption is sufficient to guarantee that the theory works and that we have no 20th century issues what is really the key argument in this story well a priori flow type cylinders and by this I mean flow cylinders of that nature but also the more general type of flow equation satisfying this twisted equation right flow type cylinders can also if you look at them of in one parameter families is typically what you are looking at when you want to prove that this square is equal to 0 well they can also degenerate instead of generating into two consecutive cylinders they could also degenerate as a pair of pens in a plane and the way it happens is a little bit different here because we are working with how internal trajectories that are in a compact region of our simpletization but it might be that there is some arm that develops that goes down down down down and in the end it hits the negative end minus infinity of the simpletization and then you need to look down in another copy of that simpletization and what you recover is a genuine g-holomorphic plane or g-theta holomorphic plane for some fixed value of theta and which value of theta depends on where the gradient blow up occurred so whenever you have a picture like this then the bottom part is as bad as this used to be in the contact chromology situation for 20th society purpose so that really cannot hope to do anything good with that however the top part is much better because you always have these asymptotes here where g and h both depend on theta and therefore there is room enough to perturb this and achieve 20th society here so we can achieve transversality for the top part not necessarily for the bottom part but you can for the top part so now let's look at this orbit gamma tilde where the breaking occurs right? we want this breaking not to occur well given our assumption this orbit has no choice but to have gradient strictly bigger than one since the gradient is bigger than one then well if you look at the the dimension of a modulate space right now it's one before you break but after you break the top piece the dimension of the top modulate space is going to be minus this extra negative puncture but that's negative now and since you have transversality then you know that the top subject cannot exist otherwise you would have a contradiction and so that's how this assumption is sufficient to guarantee the existence of this theory because transversality can work in the top part that's really the key reason for this to work okay so that's a nice alternative to linearized contact homology actually it's more than that it's more than that because our initial motivation with Alex Wancia was to understand the relationship between symplectic and contact point of view for holomorphic curve invariance and well if both invariance are defined and this means that you have to assume strong transversality assumptions for contact homology so if transversality holds for the cylindrical contact homology and of course the assumptions there also holds SC is trivial if the degree less or equal than 1 then in fact the two objects on one hand the cylindrical version restricted say to free homotopy class A and the symplectic version now again I can play the same game but restrict myself to free homotopy class A since I'm working with cylinders of course the complex will split according to this free homotopy class well these two things will actually be isomorphic but careful they will not be isomorphic as Z-modules but over Q coefficients so the torsion principle may not agree but over rational coefficients these two things are actually the same whenever both are defined of course this is not often defined without heavy machinery like polypholes but ok the meaning of this theorem is that's really philosophically this should be thought as a substitute for that object in all situations and since it is defined a lot more generally than all applications that you have in mind that you could accomplish with cylindrical contact homology you can then do them completely rigorously for contact manifolds with that object instead that's the idea and the intuition of why these works only over rational coefficients and it doesn't work over Z so with the following we can make a comparison between this complicated symplectic contact theory and something a lot simpler in algebraic topology where you have a topological space X with a circle action and we consume further that the circle action has no fixed points so by this I mean that there is no point in X which is fixed by the whole circle there are roots of unity of the neutral element that fixes some point of X that can still occur but no point of X is fixed by the full circle that's what I mean no fixed point for the full S1 then if you have such a space with an S1 action and you want to cook up some homological invariance there are two possibilities either if you want really to take into account this S1 action you take the quotient and then you take the homology of the quotient but of course this object if X wasn't manifold this is not going to be a manifold anymore so it behaves not as well as X itself so the homology conceivably also does not satisfy very nice properties either or you keep the S1 action in store to define a more adapted homology which is an S1 equivalent homology for this topological space and you get something that is typically better behaved and the thing is that this homology of quotient corresponds philosophically to the cylindrical contact homology while the S1 equivalent counterpart corresponds to SHS1 plus and now on this side here on the left using a pure algebraic topology argument under this assumption for the circle action then these coincide over Q but not over Z I have a comment to ask you I thought your SHS1 had some new coefficients so it was really over a ring Z where you've enjoyed U so I don't think you have to tensile the cylinder homology by Q or the parsley as U I mean what happens about U? U is simply hanging around and the generators in these things are just tensor products of orbits and powers of U but you don't consider it to be a ring as a Z module or as Q vector space after a tensor with Q but I just keep you itself and for these constructions I don't use U or its structure to get something additional you could do that but here I'm not doing that there is U and the generators are just orbits with extra power of U but that's just what it is I think SHS1 is not even invariant over Z I think there is no definition of currently existing corresponding to H star of X1 I mean we have counter examples to show that it's not invariant over Z oh yes I pretty much can believe that but if you please with extremely absolutely all the orbits are simple in those cases yes then it's defined etc but in more generality it's not even clear indeed that this is properly defined over Z but while this other part here is defined and is invariant over Z so that's an extra advantage to this version of the invariant it's the Z features of it polyfold or not then you get something well defined over Z in this analogy what is X in space of loops yes yes so so what X should be conjecturally is a space of loops that are positively transverse to the construct structure that's what it should be but that's a conjecture now if you work with co-tension bundles or unit co-tension bundles then you can prove such statements with a loop space on the base then it's a theorem but in more generality I mean that's this positively transverse loops play the natural role but we don't know how to prove this and so I'm already over time so I just want to finish by saying that indeed applications that were already done or thought of using cylindrical contact tomology can indeed obtain via this invariant the price to pay for using the right hand side instead of the left hand side is that it's more complicated to compute it's not purely geomorphic curves and you have all these extra things guys like this to work with but it is possible to do that and so some applications of this S1 equivalent simplex homology have been obtained by Jean Gut was a joint PhD student of Alex Wancha and myself for example he shows that the calculation of this invariant coincide with this one for the viscose spheres considered by Ustilovsky to show that there are infinitely many contact structures on 4k plus 1 dimensional spheres and also obtain some results about the multiplicity of simple closed revobits on contact manifolds using this object thank you please comment is that the invariant being invariant on the right side the contact here upper board yes you said it's a contact invariant which is independent so why is it independent there is no feeling it's a simpletization I never talked about any feeling there never was a feeling so it's a symplectic point of view on a contact or purely contact situation there is no feeling no no this construction works without a feeling of course you can place symplectic homology and that's what's usually done with the symplectic manifolds with convex boundary blah blah blah but here I'm just explaining that if you don't have that, if you simply have a contact manifold provided you make a suitable assumption namely this one you can work with a simpletization alone any other questions you have something for a very nice talk so this assumption an SC star is equal to zero yes I guess it depends on the manifold of course I can show you examples of manifolds where this is not satisfied but in many nice cases it is satisfied it's going to work for already wide classes of contact manifolds so there are definitely many applications you can derive out of this compared to what you had here