 I think I'll begin by reviewing the problem I want to talk about, reviewing the setup that I explained last time. So we're going to fix a contact manifold y, odd dimensional. We're going to fix a contact form lambda. So that just means the kernel of lambda is xy. And in fact, xy is going to be co-oriented, which means we actually care about the sign of lambda. And contact homology is going to be a form of Morse Fleur homology for the action functional from the space of loops in y, so this associates to a given loop in the growl of the contact form. Now the critical points of this action functional a are precisely the periodic orbits of the ray vector field r lambda. So recall that if we have a contact form lambda, then we get this vector field r lambda, the ray vector field. And it's defined by the properties that lambda of r lambda is 1 and d lambda contracted with r lambda is 0. Now since we want to do Morse Fleur homology for this action functional, and in particular, we want some sort of reasonable Morse complex, we need to assume that all the critical points are non-degenerate. So this set, we'll call the set p. We're going to assume elements of p are non-degenerate. Now, there's an actual partition of p into two sets called good ray orbits and bad ray orbits. So ray of orbit gamma is called good if and only if the action of z mod, the covering multiplicity on the orientation line of gamma is trivial. So I'll give some definition of the orientation line later. But for the present purposes, this is what the definition is. d gamma, this is the covering multiplicity of gamma. So here we're looking at unparameterized ray of orbits, but they can be multiply covered. So if you go around n times the covering multiplicity is n. To any ray of orbit, you can associate so-called orientation line of gamma. It's a free z module of rank 1. That is, it's just a copy of the integers, without a canonical choice of generator. And a choice of generator is to be thought of as a choice of orientation for that ray of orbit. The analog in usual Morse theory is that to a critical point, you usually associate the orientation line of the negative eigenspace of a Hessian. Now, here the Hessian at a given ray of orbit, the Hessian of this action functional has infinite dimensional negative and positive and negative eigenspaces. But nevertheless, there's a well-defined way of regularizing the determinant to get a well-defined orientation line here. And this construction of the orientation line is functorial in the ray of orbit. In particular, if you have something which is default, multiply covered, then the rotation by 2 pi over d induces an action on the orientation line. And the ray of orbit is called good if the action is trivial. So clearly, the only ray of orbits which have the possibility of being bad are ones which are even multiple covers. So we write p is p good union p bad. So now, the Morse complex we want to consider, at least as a vector space, is the free super commutative q algebra generated by these ray of orbits, the good ray of orbits rather. So super commutative has the usual meaning A, B is B, A, except they anticommute when both are odd. So every ray of orbit, in addition to having an orientation line, also has a parity, either even or odd. Now, what's the differential on this complex? So to define the differential, we need to fix almost complex structure, j r cross y with the following properties. j should be r invariant, j. So use s as the coordinate on r, j of ds should be r lambda. And finally, j should send the contact plane to itself and be compatible with d lambda. So the condition of the fact that c is a contact form means that d lambda is a symplectic form on c. And so you can have a notion of a compatible almost complex structure. So really, all you have to choose is the restriction of j to c, and then these two properties define it uniquely on the symplecticization. So now, given an almost complex structure like this, we can define the differential. So the differential, the sort of Morse for differential on this complex, counting gradient trajectories of A, that's what we think of it as counting, is defined as follows. So it depends on j. So this is an algebra, and the differential is going to be a derivation that satisfies the Leibniz rule. So in particular, d AB is dA times B plus A times dB with a sine. So to specify the differential, I just have to tell you what it is on generators. It's just a free algebra. So this will be a sum, overall, subsets of good ray orbits of the following. So you look at the space of holomorphic curves. Looks like this, index 1 in r cross gamma, where the positive end is asymptotic to the orbit gamma plus, and the negative ends are asymptotic to orbits gamma minus. And this should be, and this is the coefficient of gamma minus. Gamma minus is a set of ray orbits. And this is just the product in this free algebra. Now, for certain reasons, I also have to divide by the covering multiplicity of gamma plus. Because the modulite space is going to consider it will have asymptotic markers. If you know what that means, this is why it's 1 over d gamma plus. So the reason we need to work over q instead of over z is, for one reason, these coefficients, these modulite spaces can also, in general, are sufficiently stacky, have sufficient orbital structure that the counts you get are usually only rational numbers. So now our goal is to define these numbers, the modulite counts, such that we get a well-defined homology theory. So what I've said so far is due to Aliashburg given to Alhofer, this is their definition of contact homology. And what I want to do is describe how to make this definition complete by defining these numbers here. So let me just state it as a theorem. So for all y lambda j as above, there exists a set, theta, theta, which is a functor of this triple. And for all elements of this set, numbers such that d squared equals 0. And moreover, this virtual modulite count is the count you expect if m bar is cut out transversally. So a non-empty set. Now you may know this in this theorem. I didn't say anything about the result or the homology you get being independent of the choice of lambda j or, in particular, little theta. So there's an analogous theorem for the cobertism map for an exact symplectic cobertism between two contact manifolds. And there's an analogous theorem for counting holomorphic curves in a one-parameter deformation of exact symplectic cobertisms from one contact manifold to another. And taking together all of these counting results, you use them together to show that the homology you get, contact homology of y, is just an invariant of the contact structure. It doesn't depend on lambda j or theta. But that's not part of this statement. It's something that follows from the statements for counting curves in cobertisms and families of cobertisms also. OK, so I should now tell you sort of a bit more specifically what compactifications I'm using of this moduli space. So the only curves I've drawn are one level, one story curves like this in r cross y. But in general, together that's only the top stratum of this compactified moduli space. And in order to define these virtual fundamental cycles, you really need to understand the structure of the full compactification. So first, just to recall the usual Morse theory of some function on a manifold, we have some moduli space of trajectories from p to q. For p and q critical points, this is just the set of gradient flow lines. And maybe I should not use gamma. So let me draw this like this, gradient trajectory from p plus to p minus. And so the space is not in general compact. Compactification consists of the moduli space of all possibly broken gradient trajectories. Start from p plus, you end at p minus. But in the middle, maybe you pass through a bunch of other critical points. So the analog and contact technology is the following. So the top stratum of the moduli space is going to be the set of u from punctured Riemann surface of genus 0 to r cross y. Use asymptite to gamma plus at puncture, asymptite to gamma minus at the punctures, genus of c to 0, and u is j holomorphic. And we quotient by r. And moreover, I think it's always a good idea to specify a homology class of the curve. So I'll just write this like this, gamma plus, gamma minus. Now to compactify, we have to consider multi-story curves. But now the notion of what a multi-story curve is is a little bit more complicated than in the usual Morse theory case. Because somehow the set of connected components of the building is no longer linearly ordered because there are multiple negative ends. So the strata of the compactification are indexed by trees. So let S be the set of non-empty directed trees, T which are connected such that every vertex has a unique incoming edge along with labels corresponding to the asymptotics of the curve. So the edges should be labeled rape orbits. And the vertices should be labeled homology classes. So I wrote h2 of y, but it's really some homology relative to the rape orbits. So a typical tree looks something like this. Draw all the edges. I will never again draw the directions on the edges because I'll just draw the tree such that all the edges are oriented down. Now all the edges are labeled with rape orbits. And there's some internal rape orbit here. And then the homology classes for each of the vertices. And this corresponds to a holomorphic curve, a holomorphic building which looks like this. So each of these curves is separately equipped with a map to r cross y up to translation. So each component mapping to r cross y up to translation. Now this set S has a little bit more structure, namely, there's an operation on trees, namely contraction of edges. So if I have a tree like this, I can contract any interior edge. Here the only option is this edge and get a new tree. So when you say vertex, so you only consider vertices in giant size or vertices in general? Yeah, so I allow these dangling edges so that they don't have an incoming vertex or they don't have an outgoing vertex. So we can regard S as category where morphisms are contractions of edges. So now for any tree, any T and S, the modular space of holomorphic curves modeled on T be the set of, so maybe I should give a precise definition of what this actually is. So to say what this is, let me actually add one more piece of data here. It's really better to say that we choose a choice of base points for the ray of orbits on the input and output edges. So we choose a base point for gamma plus, gamma 1 minus, gamma 2 minus, gamma 3 minus. We don't choose a base point for gamma. Then the notion of morphism between trees includes some data of how to match up the base points. So now what's a building modeled on T? So given a T, given a tree T, you can form a curve, which is whose dual graph is T. So I'll call that curve C sub T. We map it to r cross gamma, j holomorphic. Let me just say, so it should be asymptotic to the chosen ray of orbits, gamma e for edges e in the correct homology classes, beta v. We also want to choose, so at input and output edges, we specify asymptotic markers, which are mapped by a u to the base points. And at the interior edges, that is, every edge, which is not an input or an output edge, we specify a matching between the tangent spheres at these two points in CT. So what that means is just if this ray of orbit is multiply covered, you have to specify a matching between the positive end and the negative end so that you know how to glue the curve to resolve it. So CT is like a disconnected curve, sir? Yeah. Yeah, you just, yeah. And at the input and output edges, these ray of orbits are all equipped with base points. And again, if any of them are multiply covered, there are multiple asymptotic markers you might choose, which are mapped to the base point. So that's the definition. OK, so these are technical conditions. I'm listing all of the specific things, all of the precise definition for the experts. But what you should think of this as is just holomorphic buildings whose dual graph is the treaty. So now? Come on, sorry. Am I correct in understanding that each one of these components is going to stretch all the way from the top of our crossway to the bottom and then start again? Yeah. Yeah, so I guess I wrote it this way for convenience. But it's really best to think of each component mapping to a separate copy of our crossway. And the map is only well-defined up to translation on each factor. So now we can define the compactified moduli space. m bar of t be the union of m of t prime for all trees mapping to t. Holomorphism, do you add to the energy crisis? Yes, yeah, the labels of the beta 1 and beta 2 get added if we contract gamma. So these contractions of edges, if I contract gamma, corresponds to gluing the holomorphic curve at this rate of orbit and getting a single curve mapping to our crossway. So now there's a natural Gromov topology on this compactified space. It's sort of restricted to every piece here. It's the topology, you think, just uniform convergence. On compact subsets. Since the J-holomorphic curve equation is elliptic, these curves are all smooth, and they satisfy even a priori estimates on their smoothness. And as a result, there's really only one reasonable topology you might put on this space. You can say topology of C0 convergence, topology of C infinity convergence, they give you the same topology. To give a topology on the compactified modular space, well, you do what I said. So the neighborhood of a curve like this is given by, well, you can take a C0 or a neighborhood of each of the components, but you're also allowed to glue the curve at every interior edge. So now it's a theorem of Bourgeois-Eliashberg for Hofer-Wazatsky-Zender that the space M bar T is compact. Now let me say a little bit more about the structure of this space, how the strata fit together. Give me an illustration here. So let's consider, so here's a tree. It has two possible morphisms where you contract a single edge. And it has, and then you can contract everything like this. So the way that this, yeah, the plane, yeah, no negative, yeah. So then the corresponding structure of the compactified modular space looks like this. So this corresponds to a whole curve looking like this. Have a plane here, curve like this. Now if you have a curve like this, there are two gluing parameters. You can glue here, and you can glue here. And these are independent operations, both take the gluing parameter. The space of gluing parameters at each ray of orbit is the non-negular real numbers, zero being no gluing. So then if we go here and glue along this ray of orbit, we get something that looks like that. We could glue everything, or we could glue just this. So this is what the compactified modular space will look like if it's transverse near a curve like this. So every tree has an index, and the dimension is the index minus one. So there are two definitions. One is the index of the linearized operator, some Fredholm operator. And by the index theorem, you can calculate it as sort of a difference of certain mass law for Conley's Ender indices of the ray of orbits on the boundary. There is some topological quantity, which depends on the tree. In fact, in the case, say like C1 equals zero and we're dealing with contractable ray of orbits, you can just assign each ray of orbit a number. And then the index of a tree is just the difference between the top and the sum of the bottom. So I need to actually slightly correct this definition. And the reason is the following. If you're looking at the monosci space of cylinders and you want to compactify this, one thing that could happen, or one possible limit you have is a curve with three negative ends on the top and then two holomorphic planes. So this corresponds to this tree and this corresponds to this tree. Now, the stratum of the compactified modular space of curves corresponding to this tree has a stratum where you have curves that look like this. And there's no label of these two components. But if you look at curves corresponding to this tree, these two planes are distinguished. One of them corresponds to one vertex, one of them corresponds to the other. So to solve this, just mod out by the automorphism group of t, t. That's what actually appears. Another thing to say about this compactification is that it's a little bit different from something which is usually considered. So this is what another compactification or larger compactification could look like. So sometimes you consider this curve and this curve. There's a sort of standard SFT compactification where you consider this curve and this curve to be lying in the same copy of r cross y. And as a result, there's a notion of the relative vertical position of these two curves. And so this stratum now becomes co-dimension 1. This is a blow up of this compactification. So this compactification is not so nice from the perspective of proving the master equation, d squared equals 0, because there's this extra stratum here, which is co-dimension 1. And it, a priori, has some contribution. And you have to argue that it doesn't contribute. Whereas in this compactification, where it's co-dimension 2, it's clear by definition. You don't get any contribution. And this is mostly just a technical point, though. I strongly believe that these two compactifications contain essentially the same information. Now these modular spaces have a little bit more structure. So S has what I'm going to call concatenations, which is some sort of vaguely symmetric monoidal structure. Oh, not exactly. OK, let's forget about the word symmetric, vaguely monoidal structure. So all I'm saying here is that you have a collection of trees. And you are given some data of how to identify some pairs of input edges with output edges. You can form a new tree. So what this corresponds to, so let's abstractly know that like this, if you have a collection of trees, the TI and the matching data, you can get concatenation, which I'll denote by this. On the level of moduli spaces, there's a map, obvious map from the product of the moduli spaces for TI to the moduli space for the concatenation. Now this map is not quite anisomorphism because at each edge, like this, where you glued one tree to another, in this moduli space, there's a choice of asymptotic markers at this rape orbit. And in this space, there's no choice of asymptotic markers. There's a matching of the two ends, but not the choice of asymptotic marker. And so if the rape orbit is multiply covered, this map is not 1 to 1. It is though, it's an isomorphism once you quotient by this ambiguity though. So if we quotient, there's an action by z mod d gamma, where gamma ranges over, I'll just save it junction edges, the edges that were glued. So S is the set of trees, and it has these two structures. One is it's a category. We have morphisms between trees. And the other is this monoidal structure. There's a way of concatenating trees to get new ones. And these two structures encode just basically all of the structure that's here, how the moduli spaces fit together. And this structure will come up again later, sort of fundamental to the proving the master equations. OK, so let me now say a little bit about the basic analysis, which. So can you explain again the difference between the stalkeration? It looks like you're gluing on both gauges. Yeah, so one basically just corresponds to like stacking them on top of each other. And the other corresponds to gluing. One is like I take a tree. One operation is just used to have a single tree, and you pick some subset of edges to contract them. The other operation is you have a bunch of trees, and you pick some pairs of edges, and you stack them on top of each other without doing any contractions. They are somewhat similar if you decided that you only care about trees with a single vertex. This would not be an unreasonable thing to decide. You can decide you only care about trees with a single vertex. Then these two operations just become one operation. If you have a way of stacking single vertex trees, then you contract all the edges, and you give another single vertex tree. And you could work like this, too. I think it's convenient to separate the two operations, but they are certainly closely related. What I want to talk about next is just their basic standard analysis for how these, which allows one to get a handle on what these modular spaces actually look like. So a good reference is, oh, for most aspects of holomorphic curves and simple geometry, this is a good reference, but particular for this, you can look at McDuff Salmon for more details. They don't treat the case of contact homology with these ends, but let's fix compact remand surface C and some punctures. And now I want to give a way of describing this modular space of buildings of a given type. So to do that, we're going to look at some spaces of maps. So this is the space of smooth maps u from C to r cross y. So u should be a class w kp. Locally, that is, it has k derivatives, which lie in lp. And moreover, it should approach trivial cylinders over the rape orbits at the punctures, exponentially guided by delta. So delta greater than 0. So what that should mean is that u of s t minus sb gamma of t, this difference times e to the delta s should also be in w kp of infinity cross s1. So this is some cylindrical coordinates near these punctures, p plus minus. The gamma also depends on plus minus. So this is a smooth Banach manifold, which is a manifold of all maps. And the modular space of solutions of the Delbert equation lies inside this space. It's just a set of holomorphic maps. And more precisely, it can be described as follows. So there's w kp delta. And over it, there's this bundle whose fiber over a given map u is w k minus 1 p delta C. 0, 1 form is valued in the pullback of tangent bundle of r cross y. Now to every, so this is a smooth Banach bundle. So the infinite dimensional space of all maps and infinite dimensional bundle over it. Each map, so there's a section here. Namely, to each map, we can associate del bar u. So if u is a map, del bar u is a section of this bundle over C. So this map sends u to del bar u. Let's just call this section. So then the modular space of this tree, where gamma plus is on top, gamma minus is on bottom, is exactly this resimage of 0. And let's quotient by r. S is the del bar section. Yeah. N and B, they are just green. Right, yeah. So B is just a real number. And L is the length of this Ray-Borovic gamma. So if you have a Ray-Borovic of length gamma, this is called the trivial cylinder over gamma. So it just looks like this in r cross y. And it's j-holomorphic in particular. And a general j-holomorphic map, which has finite hover energy, will be asymptotic to the trivial cylinder at either plus or minus infinity. So a basic property of this section is that its derivative is a Fred Holm operator. So let me just, again, review some of the analysis, the theory, which is relevant for modular space of holomorphic curves. So a linear map A from e to f or e and f or Banach spaces is called Fred Holm. But only if kernel is finite dimensional. The image of A is closed. And the co-kernel is finite dimensional. And in this case, there are two things you can define to invariance of A. One is the index, which is just the kernel minus dimension of the co-kernel. In the case that e and f are finite dimensional, this is simply the dimension of e minus the dimension of f. So one can think of the index as simply a particular way of using A to regularize the undefined difference between the dimension of e minus the dimension of f. You can also define, given A, you can define the orientation line of A. I don't. So let me call. Yeah, so the orientation line of A. So the determinant of A is the top wedge power of the kernel tensored with top wedge power of the co-kernel dual. And absolute values means take the orientation line. So this is just a real vector space. Absolute value means take the orientation line. So the orientation line of a vector space is the homology of the sphere. So it's isomorphic to z, and choosing a generator is the same thing as choosing an orientation on v. Moreover, this is z mod 2 graded. And it's parity being the dimension of v. So the example we really care about here is the following. So if e and f are bundles over a compact manifold, m and l is elliptic operator of order n, then this map l from k p sections of e to k minus n p sections of f is fret home. So it's important here that p is not 1 or infinity for this fret home statement to be true. In fact, you could really just work with p equals 2. And then things are slightly easier. And let's take k at least 2 to be safe also. So this ds, the derivative of this section s, is called the linearized operator. And it is an elliptic operator. So this is where this fret home statement comes from. It doesn't follow exactly from what I wrote here, because c is not compact. It has these cylindrical ends. So you need a slight extension of this result due to Lockhart-McGoen. And this fret home statement for the operator with ends, linearized operator with ends, uses crucially the fact that the boundary periodic orbits are non-degenerate. This is crucial. Great. So now we can define the index of t to be the index of this operator ds. So it turns out that both of these are invariant under deformations of operator A. Your fret home operator, if you have a family of fret home operators, the index is constant. And the orientation line forms a local system over this family of operators. Right, so a little bit of. So if you have a section of a bundle, its differential is only well-defined on the zero set, otherwise it depends on the choice of connection. So up here, when I say ds is fret home, this statement only makes sense over a point in the modular space. It turns out, though, there's a nice family of connections you can put on this Banach bundle, namely connections which are pulled back from a connection on the tangent bundle of r cross y. And if you use those connections, ds is fret home for all s. I mean, sorry, for all maps, you, regardless of whether they're holomorphic or not, again, because it's an elliptic operator. And this allows you to define the index for any tree, regardless of whether the moduli space is empty or disconnected or anything. So this is well-defined, even though just looking at it sort of a priori, it might depend on the moduli space being some property of the moduli space. So this is the index. And there's also the orientation line, which I'll write like this, which is the orientation line of the section. So in many cases, for example, if the first train class is 0 and only considering contractable orbits, the index is simply or can be expressed as the difference between the index of the gamma pluses and the index of the gamma minuses. You can define indices of ray orbits with this property. And you can define orientation lines of ray orbits. Oh, actually, this you can do in favorable cases. This you can always do. You always get orientation lines for individual ray orbits. What is the orientation line of the ray orbit? So you look at a holomorphic plane, positively asymptotic to the ray orbit, and take its orientation line. And this is, by definition, the orientation line of the ray orbit. Now you might worry there are lots of choices of planes, or maybe gamma isn't contractable, so there isn't any choice of plane. But the thing is you just do this abstractly. So you have the trivial cylinder, and that gives you like, so you at least have a positive end asymptotic to gamma. There's this well-defined notion of that. And you simply extend that vector bundle to C and put some Cauchy-Riemann operator there, and take the determinant line. And that's your orientation line of gamma. So the only choice there, up to deformation, is the choice of relative train class. And you can argue that it's independent of that choice, up to canonical isomorphism. From this definition, there's this functorial action of Z mod the covering multiplicity that I was talking about earlier, just corresponding to rotating the orbit by 2 pi over d. Now the real thing we want to, we're interested though, is what's the structure of these modular spaces? So to do that, we need the implicit function theorem in the setting of an infinite dimensional setting. So if we have a map, S from e to f, it's a smooth map of bionic spaces. Oh, you don't really need smooths. It's just Ck for some reasonable k, such as maybe 1. And let's suppose that the differential at 0 from e to f is surjective. So then there is a local diffeomorphism between the kernel of dS and the 0 set of S, near 0 and e. So the surjectivity of this linearized operator is a very important property. We would be very happy if this always held. And I'll denote by m bar of t reg, the locus where this linearized operator is surjective. The corollary now of this implicit function theorem is that at least if I don't compactify, I just look at one particular stratum. This is a smooth manifold of dimension, the index of t, minus the number of vertices of t. Each vertex of t gives you an r action that you have to quotient by. Now the remark, which is obvious for the experts, but Bear's pointing out is that this main result about defining contact homology holds if these moduli spaces are all cut out transversely. So the real content is how to perturb the moduli spaces abstractly so that they become transverse. In many cases in symplectic geometry, if you choose a generic, almost complex structure, if your input data is generic, then your moduli spaces are all transverse. Therefore, by the implicit function theorem, they're all manifolds of the expected dimension. And this allows you to define your homology theory by simply counting dimension zero moduli spaces, oriented appropriately. In the context of this remark, the fact that you don't have quite the same compactification, it plays no role because it happens only for two dimensional spaces that you don't have to. Yeah, so the difference in compactification is only seen at, if everything is transverse, then the difference is only seen for index two and higher. So if I come to the homology, you come on the one-to-one spaces of dimension zero? Yes. Is there any interesting structure if you come to other dimensions? Like, for example, you can sum it up pretty easily, yeah? Let's see, so the higher dimensional ones, if you have like smooth manifolds, one thing you can do is that the fundamental class in homology pushed forward to a point is trivial. But their fundamental class in smooth bordism pushed forward to a point is not trivial. Or maybe you can frame that, maybe framed bordism. And if you get these bordism classes, if you can construct these bordism classes, you can do the Coen-Jones-Seagull construction to get not just homology groups, but actually a stable homotopy type, which is an invariant. Or maybe less than a stable homotopy type, but something in between. So that would be one interesting thing to do. Does it look like a graded stable homotopy type, because there's also this index? Yeah, probably, yes. Great. I mean, graded, I think, so I mean the grading just, I think, corresponds to like the, like the grading of homotopy groups of the stable homotopy type. But I think the, like, the higher dimensional homotopy type, I would have thought that was the dimension of the homotopy space that comes with it. Whereas you're getting this grading just using zero in the modular space. Let's see. So, yeah, I mean, I think that the case to keep in mind is like finite dimensional Morse theory. So if you, so suppose we have a Morse function on a finite dimensional manifold. And the principle is that if the Morse structure, we endow the modular spaces with the more, the more that we can recover of M. So if we, if we simply endow them with no structure, we can get the homology of M with Z2 coefficients. If we endow them with orientations, we can get the homology of M with Z coefficients. If we endow them with framings, then we can recover the stable homotopy type of M. There's some, there's some, fairly elaborate construction using the framed board some classes of, or the relative framed board is some classes of all of these modular spaces to fit together to build some of the stable homotopy type of M. And in this finite dimensional Morse case, yeah, you can, I think, you don't, you even, you get it on the nose. You don't get it sort of speak up to, up to suspension. You get it on the nose. Yeah, in the case of contact homology, I don't know whether one expects to be able to get a framing. It's probably a bit too much. But one could, so stable homotopy type is probably not possible or not expected to be possible, but maybe something more than homology group, some sort of modules over some interesting spectrum. So finally, let me discuss the extra structure we need to endow the modulized spaces with in order to define the virtual fundamental cycle. So when your modulized spaces are all transverse, their topological structure is enough to say what their virtual fundamental cycle should be. In the general case, your modulized space is simply some horrible compact house topological space. And as a topological space, you don't, it doesn't say anything about what the, what the virtual fundamental cycle should be or what these modulized counts should be. So what we have to do is figure out what sort of additional structure we need to remember in order to be able to define that. Now the basic idea is the following. So we have some horrible modulized space called x and over some given open subset of the modulized space, u, we embed this, we embed u into some thickened version of x. This is as the zero set of a function to a finite dimensional vector space. X alpha is a manifold. So if we denote by d, I'll write the virtual dimension of x, but I mean this index, the expected dimension of x, that is the index of this corresponding linearized operator, then x alpha is a manifold of dimension d plus, plus dimension e alpha. So this, a diagram like this is called a finite dimensional reduction. And it tells you what the modulized space or sort of what the virtual fundamental cycle should be locally in the sense that if you curve the section s alpha to be transverse to zero, you get some perturbation of this small open subset of x. And if your perturbation of s alpha is transverse to zero, this perturbation of x is a manifold of dimension d. And that's, it represents the virtual fundamental cycle locally over u. And to recover the global virtual fundamental cycle of x, what we have to do is sort of glue these local cycles together given some appropriately compatible atlas of charts of this form. This idea of defining the virtual fundamental cycle in this way has a, is very old. It's been around, applied many, to many problems in this area. Fukaya Ono introduced the theory of Karinishi structures to encode one sort of notion of an atlas of this type. This was, and concurrently there are other works by other authors doing essentially the same thing in related contexts. Also Fukaya Ono continuing with Karinishi structures. There's also been recent work of Mekda Ferheim on this with the notion of Karinishi atlas. And also Spivak Resaf Gowal and Joyce on derived manifolds. Derived manifold is sort of an, here's the way I say it. A maximal atlas of smooth charts is to a manifold as an atlas of this type is to a derived manifold. So derived manifold is some sort of intrinsic notion of a space equipped with charts like this. What I'm going to describe is something called implicit atlas which is sort of another way of formalizing a collection of charts like this which are sufficiently compatible so that they allow us to construct a virtual fundamental cycle. So before I give you, so I think today I probably won't give you the definition of an implicit atlas, rather I will construct one on the modular spaces of holomorphic curves and this will somehow motivate what the axioms are. So maybe actually before I construct one on the modular space of holomorphic curves let me construct one on a much simpler space. So let's just take a vector bundle. So B is a finite dimensional manifold, F is a vector bundle. And let's take a section S, not necessarily transverse to zero. So X is this zero set. Clearly it's virtual fundamental cycle or what I mean by that is just what you get if you perturb S to be transverse and that gives you a cycle. But suppose we're just giving this setup S not necessarily transverse to zero. What is the atlas I want to associate to this? So let's A be the set of all pairs, V alpha, I guess I mean triples, E alpha, lambda alpha, where V alpha is some open subset of V. E alpha is some vector space and lambda alpha is a map from E alpha to the restriction of F, the linear map. And then going to let X alpha be the set of pairs little b and x and E alpha, so four some alpha and A. I'm going to define X alpha to be the locus where little b is in this open set corresponding to alpha and S alpha, no S, sorry, S of b plus lambda alpha of E alpha is zero. So what happens here? If I just, if I remove this condition and just set E alpha to be zero I'm just getting the definition of X. So to pass from X to X alpha what do we do? We add, so first we restrict b to lie in this open set. We localize. We add an extra degree of freedom of sort of an element of this vector space and then we modify the equation like this. So there's an obvious map from X alpha to E alpha simply forgetting everything except little E alpha. And if you look at the zero set of this map this embeds inside X. In fact it's exactly X intersect this open subset V alpha just by definition. So this is exactly a chart of this form. And now you can also observe that if I pick E alpha big enough and lambda alpha sort of non-trivial enough so it maps, say, surjectively onto the fiber of F then it's very easy to make this space cut out transversely. The linearized operator is just the usual linear, say the linearized operator at a point of X intersect V alpha is the usual linearized operator of the section F plus lambda alpha. So if you make lambda alpha surjective this is definitely surjective. So X alpha is a subset of V alpha times E alpha, right? It's a sub, yeah, yeah. So when you say X alpha equals to zero is X intersect V alpha do you mean? Yeah. No, I guess it's correct. Sorry, you mean this? Yes. So this is not quite in... So if I set... So if I take the zero set of S alpha that's equivalent to setting E alpha equals zero here so if I just ignore this, this term goes away. So it becomes the set of B with S of B equals zero and in V alpha which is exactly this intersection. Yeah, so it's intersection of B cross zero if you want to. So we have a bunch of charts like this indexed by the set A. In fact, we have charts indexed by the collection of finite subsets of A. So we're generally for finite subsets I contained in A X sub I be the set of B and B little E alpha and big E alpha for all alpha in the set such that B is in V alpha for all alpha in I and this equation is satisfied. You have sort of analog of this picture for these sets as well. If I take I to be the empty set I'm just getting exactly X the original modular space I care about. And these are, what I'll call thickened modular spaces are, let me write this, they'll be transverse in large portions of X their transverse low side will cover X and X I has a map to the E alpha for every alpha in I and if I take the zero set of one of those maps I'm getting say X sub I minus alpha that's how the charts fit together. So maybe let me actually give you a sketch of the definition of an implicit atlas before I define one on modular spaces for contact technology. So this is implicit atlas on a space X okay this will be sort of a simplified version in particular I'll save a, it's implicit atlas with trivial isotropy. Consists, so X is always here compact Hausdorff consists of an index set together with some data. So first I want these spaces E alpha for alpha and A these are finite dimensional spaces. Second I want spaces X sub I for finite I say just Hausdorff topological spaces and moreover X sub the empty set should equal X. The implicit atlas is going to consist of a bunch of data and the only way that this data actually interacts with the original modular space X is through this isomorphism. Thirdly maps S alpha from X I to E alpha for alpha and I and I should have maps Cij your isomorphisms from right so this is for I contained in J right look at sort of just those S alpha for alpha in J and not in I acting on X J look at the zero set that should be identified with an open subset and so these are called the thickened modular spaces and also going to specify open subsets X I reg. So there are a bunch of axioms about how these pieces of data fit together but they're mostly tautological and sort of things like Cij composed of C jk should be equal to Cik things like this. So I defined the X I's in this particular case the S alphas are the tautological projections and it's it's clear that if I take X J and require some of the alphas to be zero I'm just getting an open subset of X I so the structure is all apparent here X I reg denotes the locus where X I's cut out transversely so one of the very important axioms is that if I look at the portion of X which is covered by the regular locus in some thickening we want it to be all of X top a priori this is just some open subset of X we want it to be all of X sort of just saying that there are enough charts here so maybe it's better if I write like this so this a priori here is a subset of X and I mean it's equal as a subset of X so it's empty set in this abstract definition the regular part is defined as the really as the place where S alphas transverse to zero or is it an abstraction of this? So in that because X of I is not a manifold in the definition or is it? Yeah so this is just some space in the abstract definition it's just some abstract open set X in this in implicit atlas the X I's are just spaces they don't sort of have enough structure for you to tell where they're transverse and where they're not that's basically why I have to include it as a piece of extra data in the definition of an atlas so in practice when you define an atlas you define X I reg to be the locus where it's transverse but you cannot recover X I reg from the rest of the data here it's part of what you have to specify in particular you can always shrink X I reg if you wanted to make a I mean there are various transversality conditions you might want to apply for instance if you have like a curve with lots of negative ends there are two notions of transversality you might have for this curve one is the linearized operator as a map from this fixed curve is surjective and one is that the linearized operator of this as a map but also allowing sort of deformations of the complex structure you could say that's surjective that's a weaker condition and you can define X I reg using either one and it just gives you a slightly different I mean all of the rest of the atlas is the same it's just you have a more strict definition of what being transverse means great so let's define one of these on a multi-spaces for contact tomology so let's fix some tree and let's define A of T this will be the index set for the atlas so it will be the set of quadruples so R alpha is a non-negative integer D alpha will be an R invariant co-dimension to sub-manifold with boundary closed co-dimension to sub-manifold with boundary henceforth I'm just going to refer to D alpha as a divisor but this is what I mean E alpha is a vector space and lambda alpha is a map linear map from E alpha two smooth functions on valued in vertical 0 1 forms so here this is the universal curve m0 and is the deline from unproved modular space of nodal genus stable nodal genus 0 Riemann surfaces with n marked points and C0 and is the universal curve over that secretly this is just m0 and plus 1 and the map is forgetting the last these sections also should be R invariant D alpha is a closed and R invariant closed, I mean just a closed subset I'll allow it to have boundary it's a closed subset, sub-manifold with boundary R invariant that's not a compaction yeah, I mean you I want to use D alpha to puncture holomorphic curves and make them intersect by taking the intersections with D alpha but the holomorphic curve is only defined up to translation so there's no hope of getting anything well-defined unless we make D alpha R invariant think of it as a compact subset of Y and you cross with R oh yeah E plus minus so E plus minus of T is like the input and output edges so everything except the interior you'll notice what I've written here is a finite set rather than a positive integer but I mean I just mean it's convenient to look at modulized spaces of curves where the points are labeled by some given finite set instead of the integers from 1 through N so this is the finite set and then this is the vertical 0 1 forms on this curve family this should be supported away from the nodes of the fibers of C what is the C-draw and the N-draw? so this is the delineum modulized space of stable nodal genus zero Riemann surfaces with N mark points and this is the universal family we just secretly M0 N plus 1 for getting though so now we can define this modulized space I'll just go ahead and directly define it for any I so it's the set again maps along with E alpha big E alpha I want you to be transverse to D alpha with exactly R alpha intersections which stabilize C sub T now from this you get this map classifying C together right so C is equipped already with its marking by the input edge of T and the output edges of T those are marked punctures on C and we add these R alpha intersections as marked points and if those make C stable then there's this classifying map and since we have this classifying map we can now write down this equation so you'll notice this is very similar to the implicit atlas I defined earlier the analog of the set V alpha is the locus of say smooth maps which are transverse to D alpha with exactly R alpha intersections which stabilize C as before they're natural forgetful maps from M of T I to E alpha and if you take the zero set of one of them you're just getting back M of T sub I minus alpha right so I just defined the non-compactified version but as as before the compactified version was defined as follows so there are a bunch of axioms of an implicit atlas which you have to verify for this construction but the vast majority of them are basically tautological given the construction so I didn't write down the full definition I probably will write it down next time but one thing to keep in mind is that in practice most of the axioms you don't have to verify there's this sort of universal construction which works for basically any reasonable modularized space of holomorphic curves I mean you can see that in this construction I didn't really use anything specific to contact homology we just picked we just looked at divisors intersecting our curves the only real thing you have to show is that every curve in the modularized space for contact homology has some divisor intersecting it transversely destabilize it so once you verify this property this general procedure for defining an implicit atlas gives you one so one thing you have to check here is the regular loci in the thickened modularized space is cover the original modularized space and as I said what this comes down to is showing that every curve in contact in this modularized space has some divisor which intersects it transversely once you choose your divisor to stabilize the domain it's straight forward to choose e-alpha and lambda-alpha to surject onto the co-kernel at that operator at that point basically because once you stabilize the curve you have these absolute coordinates on the curve it's embedded canonically inside the universal family and so you by choosing lambda-alpha appropriately you can really hit any element in the this space of sections that you want what if you stabilize some of the points intersecting with the alpha there are not other but the positive and negative functions that make decisions ok so I've simplified things slightly but yes let me so there's actually this group the symmetric group on our alpha letters and you're right there's this gamma-alpha ambiguity so what's the actual definition so this needs an action of gamma-alpha the vector space and this lambda-alpha should be gamma-alpha equivariant and so now actually what you have to do is take phi-alpha to be part of the data and then your phi-alpha should be induced by labeling these intersections with one through our alpha and now you see there's this action of gamma-alphas on this modular space by changing the marking here and by acting on e-alpha yeah so this is what you actually did so this definition gives atlas's a of t on m bar of t now it should be clear that this is somehow not enough to give us what we want because if we just sort of define the virtual fundamental class of each modular space individually it's sort of defined in homology relative to the boundary and this isn't enough to give out a number for the ones which are expected zero-dimensional since they may have non-empty boundary there's no map from this relative to z so instead we somehow need to make some coherent choices of virtual fundamental cycles over the whole system of modular spaces and to do this we need the atlas's to somehow be compatible between the different modular spaces so to get such atlas's let me let a bar of t be the union of a of t prime connected connected subtrees so for instance if this is your tree you have so what's a subtree? a subtree means you pick some subset of vertices such that the resulting and take all the edges instant to them and you require that the resulting subtree be connected so there's a subtree with a single vertex there are three subtrees with one vertex there are two subtrees with two vertices you're not allowed to choose these two vertices because you don't get a connected tree and there's one subtree with all vertices you're not allowed to choose no vertices because that's also not connected so I claim that the above construction gives atlas's with index at a bar of t on m bar of t now why is that? so what do I do if I have a thickening datum for the whole tree that's this construction here now if I have a thickening datum just for the thickening datum is an element of the set a of t if I have a thickening datum for some subtree t prime you just impose this condition over the corresponding subcurve you require that that subcurve correspond to the subtree t prime be stabilized by the intersections with the divisor and then this this element lambda alpha of ealpha only makes sense over that subcurve you define it to be 0 also and now finally we can say how the implicit atlas are compatible so these atlas's are compatible as follows so for one for any morphism t to t prime there's this inclusion of m bar of t into m bar of t prime up to quotienting by the automorphism group of t over t prime which let me forget about for right now there's an atlas on this space a bar of t an atlas on this space a bar of t prime now every subtree of t prime pulls back to a subtree of t simply taking the inverse image therefore you get an inclusion here there's an atlas on m bar of t given by just taking the sub atlas corresponding to the just using these thickening datums the subset of the index set moreover you can take this space with this atlas and just take the restriction of this atlas to this subspace just by looking at the corresponding stratum in every thickened moduli space and these are cononically the same thing and second for any concatenation there's a map from this product of moduli spaces to moduli spaces of concatenation in the first point is there anything to check or is it that the constructionist cannot go enough that this is totally it's just tautological, yeah there isn't really anything to check I'm just sort of formalizing the compatibility properties that we need to make the construction work so this is an isomorphism up to this quotient thing by the z mod d for the junction edges there's an atlas here which is the disjoint union of these a bar of ti there's an atlas here which is a bar of so now again any subtree of a given ti pushes forward to a subtree of the concatenation so there's an inclusion here again this is compatible this subatlas of this coincides with the product atlas on this product of spaces okay so now I'm a bit over time next time I will talk about I'll give a precise definition of an implicit atlas and I'll talk about what algebraic topology one needs to extract the virtual fundamental cycle of a space given an implicit atlas on the space