 OK, so let me begin by just reviewing where we are in the whole construction. So we start with some moduli space, or system of moduli space of pseudo holomorphic curves. And then on this moduli space, or collection of moduli spaces, we construct some extra data, which I'll call an implicit atlas. And then what we'd like to do is, from that implicit atlas, construct the virtual fundamental cycle on the moduli space, or on the virtual fundamental cycles on the system of moduli spaces. Now, one important aspect of these two arrows is that both are canonical. So recall last time I described a construction of implicit atlases on the moduli spaces relevant for contact tomology. But if you understand what's going on in that construction, you see that it doesn't actually use anything special about being in the situation of contact tomology. It really just applies to any sort of reasonable moduli space of holomorphic curves. And moreover, what it produces is an implicit atlas, which didn't depend on making any extra choices, as it's canonical. Now, OK, canonical, I don't mean unique. I mean, of course, you could find some other way of defining an implicit atlas. I just mean that in the process of making this definition, I didn't have to make any arbitrary choices. And similarly, there are sort of many ways you could try to go from an implicit atlas to a virtual fundamental cycle. But I'm going to describe a way which doesn't involve making any extra choices. It's canonical. So today, what I want to do is I'll give you a complete definition of an implicit atlas. And well, as you probably know, there are somehow lots of other things one can try to put here in particular Korean-ish structures, others as well, which I mentioned last time. And so I'll try to say what some, I'll point out some features of this definition of implicit atlas, which I think make it this very close to the midpoint of the shortest path between the left and the right. OK, so I'll define implicit atlas. I'll point out some of the good features of the definition, which enable these canonical constructions. And I'll tell you how to construct a virtual fundamental cycle out of an implicit atlas. Moreover, it'll be a canonical construction. So I mean, I'll literally just write down in terms of the charts of the implicit atlas a chain map, which represents the virtual fundamental cycle. So first, let me just remind you of the basic idea. We have some modularized space, x, which is up priori some horrible set, just compact house dwarf. We find dimensional reduction consists of the following. So it consists of an open subset, u alpha inside x. And it consists of a chart, which I'll call x alpha and refer to as the alpha thickened virtualized space. And an embedding of u alpha into x alpha as the zero set of some function to a vector space, a finite dimensional vector space. So u alpha equals s alpha inverse of 0. Now, the reason we want to equip x with charts like this is because they tell us what the virtual fundamental cycle should be. So namely, if you take, say, x alpha is a manifold and you perturb s alpha to be transverse to 0, then it's zero set is a manifold. So how lying close to x and that manifold, at least locally over u alpha, represents the virtual fundamental cycle of x. And now the key problem where we have to solve in the second arrow is to figure out how to glue together these local virtual fundamental cycles into a global virtual fundamental cycle over all of x, so just given this local data and certain local compatibilities between the data. So let me now define for you what implicit at this is precisely. x will always be compact Hausdorff. And a is just some set, usually an infinite set. It will index these charts. So I'm going to push to that list. A on x of virtual dimension D consists of a bunch of data, subject to a bunch of axioms. So let me just list these. So first, there are what I'll call covering groups, gamma alpha for alpha in A. These are finite groups. And just some notation for finite subset of A, rather, gamma I denote this product, just notation. In covering groups, then we have destruction spaces, E alpha for alpha in A. These are finite dimensional vector spaces. And again, EI will be the direct sum of E alpha for alpha in A. So more over, gamma alpha acts linearly on E alpha. So E alpha is a finite dimensional representation of gamma alpha. Thirdly, there are these thickened spaces, x of i. These are just Hausdorff topological spaces for finite subsets of A. And the original modular space x is identified with the thickening corresponding to the empty set. So there's going to be a bunch of data. And the only way it somehow interacts with the space x we're interested in is through this identification. So first, there are the Karinishi maps, S alpha from xi to E alpha for alpha in i continuous maps. Second, there are footprints i, j contained in xi open subsets for i contained in j contained in A. Six, there are these footprint maps, cij from zero set of sj minus i on xj to uij for gamma j equivariant. Of course, gamma i acts on this as a gamma i invariant open subset. And we just save gamma alpha for alpha in j minus i extravially here. So this is the footprint maps. Finally, last piece of data is the regular locus. So xi is some modular space. It's contrary to what I've drawn here, it's not necessarily everywhere cut out transversally. So we have to mark gamma i invariant subset where it's cut out transversally. So there are a bunch of requirements for this data, but let me just pause and somehow relate it back to this picture to make sure we're all on the same page. So we have x. So suppose we have, let's do an example, and close that with just a single chart. What does that mean? So we have space x. That is the same as x empty set. There's a space x alpha. It's a thick of modular space. And there's a map from s alpha from x alpha to e alpha. So I'll write x alpha, but maybe I should really sort of put brackets around alpha. I won't. Inside x alpha, there's this zero set, s alpha inverse of zero. And this footprint map identifies, or gives a map, from s alpha inverse of zero, it'll be an axiom which says it's a nice morphism, onto some open set, which is u empty set alpha inside here. So this is exactly what we have over here. I realize now I forgot to explain some notation. What is this? I only defined s alpha. So as before, si will be the direct sum of s alpha for alpha in pi. Now there's a little bit of ambiguity in this notation because there are many functions called s alpha because there are various possibilities for i here. And so that's why I have to specify the domain here. I take the direct sum of all the s alphas for alpha in j and not in i, and look at the action of that on x j. So this is the picture I get on one alpha. It's just a single element, which is more or less what I wrote over here. The difference is only here I required x alpha to be a manifold. Here that requirement, we haven't seen that requirement yet, but it'll come later. And finally, just the other thing to say is that in this picture, I assumed gamma alpha is trivial, but of course, gamma alpha is non-trivial. There's a similar picture. We have actions here. This map is aquavariant. And now this is not an isomorphism, but just a quotient map by gamma alpha. This is a gamma alpha cover. So what are the axioms for this data? So there are lots of these axioms, but most of them, as you'll see, look rather tautological, and they are. So in any particular, so first of all, there's just some universal construction of an implicit atlas on any reasonable modulized space of holomorphic curves. And when you do that construction, almost all of these axioms are trivial consequences of the definition. There's only a few that you really have to carefully. OK, so up here, 6 was just some map, but now let me sort of be more precise. We actually demand that this is a homeomorphism after quotienting the domain. So this is one of the axioms, which is somehow requires a little bit of argument in practice. Not so serious, but there's something there. j x i rag is contained in x j rag 7 gamma j minus i acts freely on here. Now finally, some axioms, which are very important. So this should be open. As the locus where this modulized space is transfer should be an open subset. Second, s j minus i from x j to e j minus i should be locally modeled on this projection over the inverse image of the regular locus. So in particular ways this mean, if I take j equals i, then the target is just a 0 vector space. If I take j equals i, this tells me that the regular locus is a manifold of dimension d plus d me i. So let me just write this. x implies, as a special case of axiom 9, x i rag is a manifold of dimension d plus dimension e i. And moreover, whenever I have some, say, x i and x j, and I have some point in x i, which I declare to be transverse, when you view that point as being cut out inside x j, the corresponding section by this section, this section is transverse over that locus, locally modeled on this. So here I'm only demanding topological condition. This is locally modeled topologically here. This is a topological manifold of dimension, d plus dimension e i. Yeah, of course, there's an analogous definition, analogous notion of an implicit Alice with a smooth structure. You could require things to be smooth. The point here, though, is that for this second arrow, we don't need a smooth structure. I'm going to give a definition of the VFC via some algebraic topology, and it won't actually use any smooth structure on these thick and transverse modulite spaces. And this first arrow is easier if we don't have to construct a smooth structure on these spaces. So these axioms require some argument in practice. Finally, last axiom, which is also not important, is that this space should be the union of the transverse loci. So if I look at the transverse x i rag, this gives me, this gives me, or via, is some subset of x empty set via this footprint map. And these subsets better cover all of x empty set. This is basically just saying, somehow, we have enough charts to cover all of x with transverse thickenings. Obviously, there's no hope of recovering the virtual fundamental cycle if these thick and modulite spaces, the locus of a transverse dough doesn't cover all of x. So that's what we're requiring here in axiom time. So this looks like a lot of axioms, but I should just remark that in practice, we have this canonical procedure for constructing one of these. And in practice, there really only two things which require some sort of argument. The first is this. You have to show that this procedure gives you enough charts. In the case of contact tomology, and usually in general, what this amounts to is just saying that any curve in your modulite space that you carve out, you can stabilize it by intersecting with a transverse divisor. That's somehow the ingredient which goes into this. Five is actually, maybe I should erase this. Five is not actually all that hard. It's something about comparing Gromov's topologies. For eight and nine, this is some amounts to a gluing theorem. So if you want to prove eight and nine near a point where the domain of your holomorphic curve has no nodes, then it's easy. This is something that just follows in the implicit function theorem for Banach-Menthefels. Neighborhood of a nodal curve, you have to work harder. This is where the analysis comes in, verifying eight and nine. So just briefly, the example you should keep in mind is x is just some modulite space of maps u, del bar u equals 0. Then thickening xi is some space of maps u, along with choices of e alpha in e alpha for alpha and i, such that, so first, u satisfies some open condition depending on alpha for all alpha in i, and del bar u plus e alpha equals 0. So this is just sort of a cartoon version of the definition I gave in the previous lecture. But even given just this cartoon version, you can check that most of these axioms, things like this, are just sort of tautological. xi reg here is a locus where xi is cut out transversally and has trivial isotropy. That is the automorphism group of the particular curve with whatever marking we've chosen in addition here is trivial. OK, so some remarks about this definition. The set A is, in practice, the set of all thickening, I call the elements of a thickening datums, is some universal set of all possible choices you might make of a thickening datum. So it's a very infinite set, uncountably infinite set. And one thing to point out is it's somehow, it's not completely a triviality to formulate a definition which allows you to make thickened modulized spaces for arbitrary subsets of thickening datums. Remember, if A is a set of all thickening datums, somehow I could, I'm demanding here that I can construct these thickened modulized spaces even when I pick two thickening datums which are very badly non-transverse to each other intersecting simultaneously with two divisors which are not transverse to each other. And the fact that we can formulate a definition which allows this sort of universal construction taking into account everything at once is not entirely trivial. And it's somehow not, it's not obvious in other constructions that this is possible with other notions of atlases. Another sort of difference between this definition and others is that XI, OK, so one is that XI reg, you might want to require that this XI is transverse everywhere for non-trivial thickenings. So as long as we thicken at least some, then we get a transverse modulized space. So this is tempting to include this as an axiom because of this picture right here. This X alpha I drew as sort of transverse everywhere. And this is the case which is most intuitive. On the other hand, this requirement makes it much harder to construct an implicit atlas. And it doesn't let us do lots of other things we want to do. So I'll give an example. So let's just say we don't want to include this axiom. So more generally, these XIs are treated on equal footing independently of the cardinality of I. So in other definitions of similar to this one, you have one sort of formalism for how X empty set is cut out inside X alpha. And you have another formalism for how XI is cut out inside XJ when XI is non-empty. And this somehow leads to sort of duplication of the axioms that you have to check. So this is, I think, a good simplification. Another reason why both of these points are very important is that there's this construction of the product implicit atlas. So if I have space X with an implicit atlas A, space Y with an implicit atlas B, I can construct an implicit atlas on the product with index at A disjoint union B. Namely, the thickened moduli space X cross Y thickened by I and I prime is just the product of thickened moduli spaces. And the regular locus, well, there's sort of only one reasonable possibility. It's XI reg cross Y I prime reg. OK, now we see immediately that this construction, even if you started with implicit atlas as satisfying this property, the product doesn't necessarily satisfy this property. Because I disjoint union I prime could be non-empty, but one of I prime could be empty. It's also clear that this somehow completely fails if we don't treat XI as an equal footing for X empty and non-empty. So some people find this notation quite strange. They use the same notation for the moduli space and the thickened moduli spaces. But I think this is a good justification for why I think that's natural notation in this particular setting. Right. OK, so finally, this is the definition for somehow moduli spaces without boundary. I require the thickenings to be manifolds. Of course, there's a natural generalization of this implicit atlas with boundary. If you have a space X with closed subset known as its boundary, or just called its boundary, there's a notion of an implicit atlas on this space, which just consists of thickenings inside each thickening re-we specify closed subset. And then all of the axioms are the same, except the local model we now allow is a manifold with boundary as a local model. And in the local model, somehow the boundary of XI reg as a manifold with boundary should coincide with this subset that we declared to be its boundary. And you can do the same thing with more general stratifications. Just a matter of requiring that the stratification be compatible with the footprint maps and specifying the right local model that you want. So last time I constructed implicit atlases like this on the moduli spaces for contact technology. So now I've given you the definition. Last time I gave you the construction. Now I want to give you the construction of the virtual fundamental cycle. So as I said earlier, what I'm going to do is just simply write down a chain map which represents this virtual fundamental cycle on the chain level. And this construction will be canonical. In particular, it's functorial with respect to inclusions of boundary, inclusions of other strata in implicit atlases. And as a result, it'll work well for homological theories where we really need a theory of somehow coherent virtual fundamental cycles over a collection of many open sets, of many moduli spaces. So VFC machinery. So this will be a compact halter space. And A will be implicit atlases on X. I'm going to assume for convenience that A is finite right now. This is mostly for notational convenience. Everything I'm saying works for infinite A simply by taking a direct limit over finite subsets. So this isn't a big assumption. Plus, choosing a finite subset of a given atlas isn't very much of a choice. So the first thing we have to get out of the way is orientations. So if we have an atlas A on X, we get an orientation sheaf. OX on X. It depends, of course, crucially on A. I won't write that dependence here. It's stock at a given point P in X is given as follows. So we choose a lift, P tilde in XI reg. And then the stock at a particular point is the orientation sheaf of XI reg. Remember, that's a manifold. So it has a well-defined orientation sheaf. Stock at P tilde tensored with the orientation module of EI dual. Finally, we take invariance under the stabilizer of P tilde under the action. Again, I. So we say this atlas is locally orientable, even only if this invariance is always non-zero. So our orientation modules are always isomorphic to Z. The orientation module of a vector space is a free Z module of rank 1. And choosing a generator of that copy of Z is equivalent to choosing orientation on a vector space. So that's what orientation modules are. This action, crucially, is just by a sign. So if gamma I is trivial, or if I'll have odd order, you always have local orientability. But if not, we need to impose this condition in practice. It's always satisfied where the modular spaces we're interested in. So henceforth, we'll just have this as an assumption that X is locally orientable. So this is just a local system on X. And of course, the virtual fundamental cycle will be in homology twisted by the dual of the orientation sheaf. So what's the goal of this VFC machinery? Well, it will be to define the following structure. Actually, maybe I'll write it here. So the VFC machinery will consist of the following. So we will define, first, a chain complex, which I'll call virtual co-chains on X with respect to A. This is just notation. I'll say what this is. I mean, the main thing I'll say is, what is this complex? It'll come with a natural push forward map to chains on a tom space of E. It will be canonically Clausi-isomorphic to check co-chains on X. So this is what we'll construct. After making this definition on orientations, now I'm going to leave orientations out of the notation. Really, I should twist this by the dual of the orientation sheaf of E. And this should have local coefficients in the orientation sheaf. But for simplicity of notation, let me leave this off. So this is what we define. Why is this the virtual fundamental cycle? So this chain complex is very easy to understand. It's the tom space of EA. This is just EA is a finite dimensional vector space. This is a sphere. And so this is just concentrated in degree D. So Z concentrated in degree D. And what we have, or up to Clausi-isomorphism, is a map here. So i.e. what we have is, so you get a cycle, denoted to x, v, in the dual of check co-chains. So let me write it over here. Z in degree D. This is check co-chains. X, we have this map. What this is is simply an element in the dual of check co-chains in degree D. So as an aside, this is check co-chains. And recall what is this. This is a direct limit over open covers of co-chains, the nerve. Now, what is the chain level dual of this? Denote the chain level dual by C upper bar. And it's called Stienrod chains. This is used much less than check co-chains. And I'll spend most of the construction just in the check world. But it's somehow the final answer can be understood this way. Stienrod chains is not the dual of check co-chains, which is the same as a derived inverse limit over open covers of the homology of the nerve. So this is somehow the right way of doing homology in the check sense. I mean, you can also take the homology of the nerve and take just the usual inverse limit of the homology of the nerve. You get something called check homology, but it's not a good object to consider. This is better. So for probably wondering why, we have to use all these exotic homology and co-homology theories. Well, so the exotic is not meant in the technical sense. I just mean all these strange co-homology theories. So the reason is the following. So the virtual fundamental cycle, unfortunately, does not lie in general. It does not lie in singular homology for the following very simple example. So let's take this space, w, as the Warsaw circle. Let's cut out inside r2 by a function, which is 0 on the inside, negative on the inside, positive on the outside, s from r2 to r. Now, clearly, the virtual fundamental cycle of the space should be non-trivial. It goes around the annulus, goes around the loop. If you push it forward to a regular neighborhood, which is homomorphic to an annulus, then you get something non-zero. Unfortunately, w singular homology is 0. Virtual fundamental cycle will never lie in usual homology. But Stienrad H1 is indeed z, as you think, generated by going around the loop. So now our goal now is to define this object and define this diagram that will give for us this virtual fundamental cycle on the chain level. So let me first just say, how do we do this for a single chart? That's already not obvious. And then we'll generalize from a single chart to multiple charts, a single chart. So we want to consider the following diagram. So let's look at chains. So recall that the dimension of x alpha reg is d plus the dimension of e alpha, where d is the virtual dimension. So now we have chains x alpha reg, x alpha reg minus x. So just consider the chain group. So let's first let's recall that x alpha reg, that's a manifold, we have a map to e alpha, and x is simply the zero set of this map. This is a global viantimensional reduction. So I just want to say how this works in the global case first. We'll assume that x alpha is cut out transversely everywhere. So now there's a push forward map, s alpha push forward, going to chains on e alpha, delta to zero, the time space of e alpha, which, as we said before, is quasi-isomorphic to z in degree d. And now by punctual duality, there's a quasi-isomorphism between this and check co-chains on x. So what we get is a map like this. And I claim this is the virtual fundamental cycle. So let's just discuss the smooth case first, where s is the smooth section, maybe even transverse to zero. This is not entirely obvious, even in that case. It is true. It's the assertion that this is the virtual fundamental cycle is very similar, in fact, basically equivalent to the assertion that the two standard ways of defining the Euler class of a vector bundle are the same. Namely, you can either take a generic section, take the fundamental class of its zero set, or rather the Poincare dual of the fundamental class of the zero set of a generic section, or you can pull back the Tom class from the fibers. This definition of the VFC is similar to the pulling back the Tom class. Basically, it is pulling back the Tom class. And this is a relative Euler class. And the usual perturbation picture where you perturb s alpha, take the fundamental class of the zero set is analogous to the corresponding definition of the Euler class. So in this case, oh, we get the definition of the VFC and the degrees with the definition that you're used to. So in this case, this complex right here, this is this virtual co-chains complex. So this diagram I wrote down is exactly what we wanted. So now let me say how to generalize this to multiple charts. Again, let me just do this very explicitly in a single case of two charts and an overlap chart. Although what we do for this will somehow, say, exhibit all of the complexities necessary to deal with the general case. So first, let me try to draw one of these things. So here's our space x. And I have one chart, say the alpha chart, which looks like this. So we have a manifold. So again, I'm going to assume that all the non-trivial thickenings are just regular everywhere. So manifold x alpha, which is the same as x alpha reg, maybe I'll just write x alpha reg. And there's a function from x alpha reg s alpha to e alpha, the zero set is identified with some open subset. So that's one chart. It's the same as what happened over here, except it just takes place over some open piece of x. Now I have a similar thing happening on another piece of x. So this is x beta reg. And then we have exactly the same situation. There's a map, s beta to e beta. And s beta inverse of 0 is u beta inside x. And x is covered by u alpha and u beta. Now, our atlas consists of another chart, namely the alpha beta chart. Indeed, as it should, because somehow the alpha beta chart is exactly what we need to glue together the local virtual fundamental cycle from the alpha chart to the local virtual fundamental cycle from the beta chart. So what does the x alpha beta chart look like? Well, it's some higher dimensional manifold looking something like this, x alpha beta. It has two functions, s alpha to e alpha and s beta to e beta. If I look at just the zero set of s alpha, that gives me an open subset of the x beta chart. I look at the zero set of s beta, gives me an open subset of the x alpha chart. These are both cut out transversally. So on x alpha beta, I have two sections. Both of them are transversed to 0. S alpha cuts out x beta, or rather open subset thereof. S beta cuts out x alpha, an open subset thereof. Now if I look at s alpha direct sum, s beta, this is now an open subset of x, namely the footprint of this alpha beta chart is exactly the intersection of the footprints of the alpha chart and the beta chart. And that's not necessarily transverse. So s alpha and s beta cut out transverse submanifolds, but these submanifolds don't necessarily intersect transversally. Their intersection is exactly x. Or, again, rather of an open subset of x. So this is the picture. So now let me say how do we want to glue together the virtual fundamental cycles. So the first try is the following. So let's take, again, chains d plus convention e alpha on x alpha reg as before, direct sum chains on x beta reg. So why are we doing this? Secretly, this is Clausiasomorphic 2. This gives me check co-chains with compact support on the footprint of the first chart. And this gives me check co-chains with compact support on the footprint of the beta chart. Now we have to glue these together using information from the alpha beta chart to do this. Oh, we take a mapping cone. So we look at chains d plus convention e alpha beta x alpha beta reg x alpha beta reg minus x. So we take some map. So what I've drawn here is a map of chain complexes. What I mean is you take the cone. So now this is Clausiasomorphic 2. It's Clausiasomorphic 2. Check compact squared, check co-chains on the intersection. This mapping cone, which is Clausiasomorphic 2, check co-chains on x. Again, compactly supported, but OK, x is compact. So it's the same. So this looks very good as a replacement for this complex. Unfortunately, defining this map on the chain level is rather cumbersome. So sort of intuitively, what's this map? It should be the cat product with the savings map as cat product with S beta pullback at Tom class. But somehow making this on the chain level is possible, but it leads to a mess. So we don't want to do this. We want to do something a little bit different. So the problem with this construction basically is that the dimensions differ between these charts. So to fix that, let's just take the alpha chart and cross it with, there's not enough room here. Let's do it somewhere else. So to fix this, we'll just take the alpha chart, cross it with e beta, and we'll take the beta chart and cross it with the alpha. Now everything's the same dimension, and things should go much better. So we're continuing this example. So virtual co-chains will be defined as follows. Let's start sufficiently far to the right. Chains d plus dimension e alpha beta on. So we take x alpha cross the beta chart, and then we take the complement of x cross 0. We write complement like this. And we also want chains e alpha cross x beta relative to the complement of 0 cross. I'm leaving room for something in the middle of which we'll use to glue them together. So how do we glue them together? We're going to do a deformation to the normal cone. So let y alpha beta be the following space. We look at x in x alpha beta, e alpha in e alpha e beta in e beta, and t in 0, 1, such that s alpha x is t e alpha, s beta of x is 1 minus t e beta. So this space has a natural map to 0, 1. I think there's a family of spaces over 0, 1. So to understand what it is, let me just tell you what the fibers are. So if t is strictly between 0 and 1, then you can divide by t. And you can divide by 1 minus t. Include that e alpha and e beta are determined uniquely by x. So for t between 0 and 1, the fiber is just x alpha beta. For t equals 0, you can still divide here. But now this equation says that s alpha must be 0, and e alpha can be anything. So I get e alpha cross the 0 set of s alpha, which is the beta chart, or rather an open subset of the beta chart. Similarly for t equals 1, I get u alpha alpha beta cross e beta. So this is called a deformation to the normal cone. And so why is that? Well, so you think of it the following way. As t goes to 0, we're blowing up along the sub manifold u beta alpha beta. So you have a manifold x alpha beta and a sub manifold, given by the 0 set of s alpha. And as t goes to 0, we blow up along in the normal direction so that when we actually get to t equals 0, all that's left is the total space for the normal bundle. So the motivation for this construction is that we'd somehow like to pick a tubular neighborhood, but there's no canonical choice. However, there is a canonical deformation. There's a canonical deformation from x alpha beta to the total space of the normal bundle. And this didn't require us to choose a tubular neighborhood. And so instead of choosing a map from the tubular neighborhood, we defined a family containing both. So how do we use this space? So now we'll get y alpha beta relative to the complement of x cross 0 cross 0 cross 0 1 complement. And now we'll glue together again using a mapping code instruction. So I use, say, u alpha alpha beta cross u beta complement of x cross 0. So this maps in here just by inclusion. It maps to here by the t equals 1. And similarly, here with the alpha and beta swapped. So this whole complex now is, by definition, virtual co-chains on x with respect to its Alice. And as before, you can write down this isomorphism. This is check co-chains on u alpha. This is check co-chains on u beta. This is check co-chains on u alpha intersect u beta, all with complex support. And you get this. This is quasi-isomorphic to check co-chains. So I'll say a little bit more about how to make this quasi-isomorphism precise very soon. So I defined now the virtual co-chain complex. There's a canonical push-forward map from this to chains on the tom space of e alpha beta. Namely, we just take the obvious push-forward map on all these spaces. And then the degrees and the inclusions of these. So you just take that push-forward map here and 0 here. That gives us this map. So this is the definition for two charts. Just for illustration purposes, let me write down the definition in general. So for general x and a, you define virtual co-chains on x with respect to a is the following. So direct sum over p, direct sum over chains of finite subsets of a of chains of dimension minus p plus d plus dimension set e in ea t in. So this is some complicated definition, which is a natural generalization of what I wrote up before. Just the only thing you have to do is figure out how to organize everything when you have many charts relative to the complement of locus where. So here's the definition. So this is some rather complicated chain complex. What you gain here, though, is that this definition of the virtual fundamental cycle, one is functorial. It didn't require any choices. And second, it works without having a smooth structure on a thick and marginalized basis. And since we are just using algebraic topology to define things. The other remark is that this is somehow sort of homotopy co-limit of chain complexes. So what's the deal with homotopy co-limits? Well, so a lot of times when you want to try to define the virtual fundamental cycle, you take some big, this joint union of the thick and marginalized basis, maybe the regular loci, and divide by some equivalence relation. Now this is a very, sort of modding out big equivalence relation here. It's a very drastic thing to do, because there are simple questions about this topological space, very basic properties which require a certain amount of work to verify and to ensure. I mean, in some cases, they just aren't true if you do this for certain atlases that might fail to be housed or formatizable. So this is an example of a co-limit. Just glue things together naively. If you instead do sort of homotopy gluing by taking a homotopy co-limit, sort of automatically has good properties. And you don't have to worry about this. And this is sort of an example of a homotopy co-limit. So you could take this construction and just forget about chains. Just do this at the level of spaces. Glue together these spaces according to this simplicial diagram. And you just view this as a simplicial space. And this automatically has good properties. It has the properties you want to use it to define the universal fundamental cycle. You don't have to mess around with the atlas at all. OK, so just aside, let me just give a basic example of this difference. If you have some, say, two spaces, x and y, and you have a common subspace. Between them, you can take the dysteroenia of x and y, modulate this equivalence relation which identifies them. They say open subspace. Of course, when you do this, you can get a horrible topological spaces, like line with doubled origin, things like this. This is the naive columnit, homotopic columnit, which is better to take x union, y union, u cross in interval and say that if u in x is equivalent to u cross 0 and u in y is equivalent to u cross 1. So that is I take the u cross 0, glue that to x, take u cross 1, and glue that to y. And now this is a much better space to work with. And the point is that this is what you really wanted anyway. So I defined this virtual fundamental cycle on the chain level, just some canonical construction. One thing which I didn't explain, which I want to explain now, is how do we show the isomorphism between this, the quasi-isomorphism between this complex and check co-chains on x? This is, I gave a sketch of why it should be true morally using point of duality. But now let me say in more detail how this actually goes. So everybody knows what a sheaf is on open sets. For when you're talking about sheaves on compact spaces or locally compact householder spaces, it's often convenient to also have an alternative definition of a sheaf which uses open sets instead, which uses closed sets instead. So we say a k sheaf is a functor f from compact subsets back to abelian groups, such that following axioms are satisfied, the f of empty set is 0, f of union is left exact, and 3, this is an isomorphism. So now you get this category of k sheaves on x, and it's equivalent to the usual category of sheaves on x. Namely, if you just have a k sheaf, you send the corresponding sheaf is the inverse limit over u. And if you have a sheaf, to get sheaf on compact sets, you take the direct limit over open neighborhoods. So that's the correspondence. Now there's something called a homotopy sheaf, which I'll only define in the k setting. So a homotopy k sheaf is a functor f from compact subsets of x to chain complexes with the following properties. So first, f of empty set is acyclic, f of k1 union k2, f of k1, f of k2 to f of k1. Magnetite k2 is exact, meaning the total complex is acyclic, and this direct limit is a quasi-isomorphism. So this is a homotopy k sheaf should be thought of as a complex of sheaves, which calculates its own cohomology. So for example, any complex of soft sheaves is a homotopy sheaf. So a sheaf is called soft if and only if all of the restriction maps on compact sets are surjective. So another example of a homotopy sheaf is k maps to singular chains on x relative to the complement of k for any topological space x. This axiom is obvious. This axiom amounts to Myroviatoris, and this is some compactness of simplices. So this proposition, so I said in words, homotopy k sheaf is something which calculates its own cohomology. Let me write it precisely. Then there's this natural map from f to check cohomology of f, and this is an isomorphism. There's a natural map here for any complex of pre-sheaves here, and if the complex of pre-sheaves is a homotopy sheaf, then this is myomorphism. So why is this machinery useful? Let me give a non-trivial application here. So here's what we're really interested in. So as opposed to f, this homotopy k sheaf is pure, i.e., the homology of the stocks is concentrated in degree 0, and the cohomology on any compact set vanishes for i sufficiently negative. So one is somehow the key condition, two is technical and might not be even needed. I don't know. So then 1h0f is a sheaf, and 2, the homology of f, is naturally kinetically isomorphic to the check cohomology of that sheaf. So given this proposition, what this is really just saying is that if f is pure, then it's a quasi-asomorphic to a sheaf. So f should be thought of as an injected resolution of some underlying sheaf, h0. This all looks very abstract, but here's an example. It's really the main example we care about. So let m be a topological manifold of dimension n. Then you look at k maps to chains on m relative to the complementary k. And this is a homotopy k sheaf, as I remarked before. And now using the fact that m is a manifold, we can show that it's pure. This is just the standard calculation of the homology of r, in retrospect, the complement of a point. And now, thus, by this lemma, we conclude that the homology of this homotopy k sheaf is isomorphic, canonically, isomorphic to the check cohomology of k with coefficients in the associated sheaf, which is simply the orientation sheaf of m. So this is point creativity for arbitrary closed compact subsets of a topological manifold. This is clearly a non-trivial interesting state. So this is how this machinery is useful. Now, how is this relevant for this? Well, there's a natural extension of this virtual co-chains complex to be the global sections of a homotopy k sheaf on x. Namely, you simply require here, basically, to do the analog here instead of taking the complement. This is sort of the complement of all x. If you want to say what are sections over a closed subset k, you take the complement of where the x lies in k. And now, this is a homotopy k sheaf. And you can check that it's pure. And this gives you the identification you want of this with check co-chains on x. So this completes the construction of the VFC machinery in the case x has no boundary. And as you saw, we just wrote down some big complex and the map of complexes. And that was the virtual fundamental cycle. So now, at the end, quickly, let me just say how this works in the case with boundary. Because somehow, that's what we're really interested in. Case of floor homology is much more interesting. So first, so with boundary. So let's say x has an implicit Alice with boundary. I already said what that is earlier. In particular, you get an implicit Alice on without boundary on the boundary of x. So now, our earlier construction, a construction definition I just erased, it does not give something which deserves to be called virtual co-chains on x. It gives something which deserves to be called virtual co-chains on x, rel the boundary. So this is somehow, the reason we get rel boundary is for the same reason that in the unusual Planck-Riduali what happens, you have the homology of the boundary of a manifold going to homology of the manifold going to homology of m, rel the boundary. This is isomorphic by Planck-Riduali to co-homology of m. This is isomorphic to co-homology of m, rel the boundary. And then there's this restriction map, hn minus star of the boundary. And this corresponds to the connecting homomorphism here. Connecting homomorphism corresponds to this just restriction on co-homology. And the push forward map here corresponds to the co-boundary map on co-homology. Just somehow, trading boundaries is what happens. This is why we get something, rel boundary. So now there's a map, right? So there's a map from virtual co-chains of the boundary to virtual co-chains of x, rel the boundary. And the fact that it increases degree by 1. We have isomorphisms between this and check co-chains on boundary x via the co-boundary map to check co-chains on x, rel the boundary. So this is just some natural push forward map on singular chains. And what it gives us under these isomorphisms is a co-boundary map. Now we have maps from this to z in degree d. This diagram commutes just simply, basically, by construction. This diagram commutes by construction. This is some non-trivial statement which you prove using the Schief theoretic machinery, which is used to define these vertical isomorphisms. So what does this tell you? Well, this map here is the virtual fundamental cycle of x. This map here is the virtual fundamental cycle of the boundary of x. So what this tells you is that we have this boundary map on chains. And this sends the virtual fundamental cycle of x to the virtual fundamental cycle of the boundary. So this is important, for example, if you want to, even in the most basic settings where, say, you want to prove Gromov-Witton invariance are independent of the almost complex structure, you take a path of almost complex structure, and you have a modular space with boundary. One boundary being the modular space for complex structure number 0. And the other boundary being the modular space stable mass for complex number 1. Complex structure number 1. And this fact that the total space can be given the virtual fundamental cycle, whose boundary is the difference of the two sides, it gives you this independence of j. A bit more sophisticated is that you can, this construction somehow, we did everything at the chain level. So this diagram allows us to have a notion of a coherent collection of virtual fundamental cycles over a bunch of different moduli spaces, which is what we need to do for our homology. And this also extends for the case with manifolds with corners. Just you have to draw bigger diagrams, that's all. So finally, let me just say there's also a product map. You have two spaces you can look at. So we really wanted this nice definition of the product of two implicit atlases so we can make a construction like this. This is isomorphic to check-co-chains on x. The whole boundary, and there's cunith-causiasomorphism here to check-co-chains on x cross y. Real boundary, and this commutes, again, by some sheaf theoretic considerations, which are used to define these maps. So in this compatibility of virtual fundamental cycles with products, probably what this gives you is, x times y, for is x for y, for this compatibility on the chain level with products is also crucial for floor homology. Because you want to be able to say that the virtual fundamental cycle of the boundary of a moduli space is expressed as a product of other moduli spaces. You want to say the virtual fundamental cycle of that boundary is the product of those virtual fundamental cycles on the chain level. OK, so this completes the really definition of the VFC machinery. Next time I will show how to apply this to construct the virtual fundamental cycles for contact homology given the atlases that I constructed last time. I'll also review a little bit the definition of the atlases.