 This part is not here, and the fact that this is trivialized means that this guy is trivialized, but this is the tangent space to Fw minus one of y. Right, so we have trivialization of the topic serial power of the tangent bundle, and this exactly gives us an orientation. Now, if we have any two w zero and w one, again, you consider m gamma, right? This is a set of xt, such that F of x, gamma t is y. Again, you show that this is compact oriented manifold, this boundary as before, and this is actually an oriented borders between mw one and mw two, and that's the end of the proof. Here is one particular corollary of that, so if assume d is zero, right then, right then, Fw inverse of y is actually a finite number of points, so say x one, x k, and this is oriented, right? So in this particular case, this means that each point comes equipped with a sign, say epsilon one, and so on, epsilon k, right where each epsilon j is just plus minus one, and so we can take the sum of all those signs, and the statement is this does not depend on w, and this is, you could call the degree of w, of Fw, right then, even more generally, so if you have d is bigger than zero, then assume we also have a cohomology class of degree d in x, say with integral coefficients, but this is not really that important. Now we could integrate this eta over Mw, so what I mean is it takes a pairing of the fundamental class of w with eta, so if you think of eta as a Duramko homology class, this is just the integral over Mw of eta, and this is also independent of w. And this sounds now pretty much similar to what I started with in the first lecture, this will be exactly the form in which our invariance coming from gauge theory will arise. Okay, are there any questions to that? If this is not the case, let me describe one more generalization, and this is an equivalent setup. So what I mean is this, assume we have, again, a manifold as we had before, but it will be convenient to assume that x is in fact a Hilbert manifold, this is not actually important, but we'll simplify at least one step, and we have a Hilbert leak group, G, and this acts on our manifold x, so I assume that the action is on the right, but again, this is of course not really important, but what is a little bit more important is that I will assume, so assume Gx freely on x. Now, again, this assumption is not really essential, we could do this without that, but the arguments would be a little bit more messy. But what we have is, we have an infinitesimal action of G on x, that is a linear map from the Lie algebra of G into Gxx, so this is an infinitesimal action. I will say that Hilbert sub-manifold S in x is a slice for the G action if, so what we have, we have a natural, well, first of all, if x is a point in S, and natural map, so let me denote G times S to be the set of points S times G, so I require this to be an open subset, so this is a subset in x, this is required to be open, and the natural map from G cross S into Gs is a diffeuomorphism. Yeah, Gs and this is a subset of x, so it doesn't need to be a diffeuomorphism onto the whole x, but on a smaller subset. So here, as an example, to make it perhaps a little bit more visual, so assume we have a two-sphere, so this is not quite infinite dimensional manifold, but still, and let us take here an action of u1, so of the circle, which rotates this sphere around the z-axis, right? So what I'm looking for is a subset or a sub-manifold here in the two-sphere, such that each orbit intersects this subset once. Well, this action is of course not free, right here are fixed points in the north and the south pole, so let me delete those, but if I do this, what I can take, I can take one of the great circles, so half of the great circle, so for instance, this guy, and it's easy to see that, so this is now s, that this has the required property, so this is a slice. The proposition is, so assume, so let G act freely on x, so I assume this sort of from the very beginning, but let me state this again, and assume there is a slice each point in x, then the quotient x mod G is again a manifold. And it's easy to understand why, right? So locally near each point, you can identify the set of orbits with a slice, but since I have required that s itself is a sub-manifold, so the quotient is a sub-manifold. So now let us assume the previous setup, that is what we have is, we have a family of maps, Kali f from x cross w into y, so let me assume that this is a family of G equivariant for the whole maps. So this means in particular that I have an action here on x, I have an action on y, and f respects this action, but I don't need to have an action on w. And whenever we have an equivariant map, or here we have a family, we have the so-called deformation complex. So what we have is first a map from the Lie algebra of G into the tangent space to x, so this is our map rx. Now we can take the differential of fw and we end in tyy and then in zero. So this is called deformation. And the name complex here is not a random one, so what you can check easily is that if fw is G equivariant, that actually implies that what we have here, so that b star is a complex. Now why is this really an interesting thing to study? So let us consider the cohomology groups of this complex. So what you have is h0, h1, and h2. Now since I have assumed that my action is free, so in particular it's locally free, this means that rx is injective, so in this particular setup, the first cohomology, I have noticed as zero's cohomology group is trivial, so let me say this is just r, this is zero. Now if y is a regular value for this map, this means that this map is surjective, and this means that they don't have any cohomology group on this place. All right, so this is again trivial, provided y is a regular value. All right, and so what is h1? h1 is the kernel of the differential mod the infinitesimal action, and this is precisely the tangent space, so this is the tangent space to fw inverse of y mod the g action. Okay, so this deformation complex describes us the tangent bundle to the modulate space, so to this mw. So let me denote here gx to be rxql gxfw, so this is now a map from gxx into the le algebra of g plus gyy. All right, so what I did is I just took the joint of this map, and this is precisely where I sort of needed that my spaces are here, but spaces to have the same origin here. Okay, so let us, yeah, so here is main theorem, so assume the following holds, so we have gx freely on x, y is a regular value for fw, set condition is there exists a local slice through each x in x, and the first condition is set dx as a linear for the whole map, say of index, then mw is a smooth manifold of dimension d. Again, this is all really very similar to what we had before, so I won't prove the theorem. The only thing what you need to realize here that if you have an action of a league group and you have a slice, you can restrict to the slice, and what we have done before goes through essentially unchanged. All right, so let me skip the proof. Now, a corollary of that is, sure. No, I mean, you're right, let us say, on the G-equivariant maps, and what I meant for the whole is that this complex is either elliptic or you say that this operator is for the whole. Well, sort of literally that's false. Now, a corollary of that is if in addition to the terminal bundle of dx is trivialized, then, well, and we also need to require that this trivialization is G-equivariant and G-invariant, say, then Mw is more oriented for this class Mw does not depend on W. Okay, that is more or less the main theorem that I wanted to present today. So, the strategy for our particular examples that we have in mind will be the following. So, we will have exactly, you know, we will have one map from x to y, which will be G-equivariant. It will turn out that y is not necessarily a regular value, so we will be looking for a family so that in family we have a regular value and we can apply the machineries that we have developed today. Okay, that's all I wanted to tell you today. Are there any questions left? That's the same as slice. So, what I meant is, so here, for example, you have a slice which works for any point, so then it's called a global slice, but if you have the slices through each point which are not necessarily global, sometimes you would say it's a local slice. So, it's just a slice, the same as slice. Okay, then, see you in the afternoon.