 here. So I'm going to talk about, mainly about instanton fluorohomology. So going back to somehow, you know, there are all kinds of different fluorohomologies. And Peter Cronheimer and I sort of started off working on instanton fluorohomology. And the way life seems to go sometimes you find that it's a good idea to look at what you did a long time ago and think about it again. There's some interesting fresh things that seem to happen. So instanton fluorohomology. So the setting is that we're going to look at a three-manifold. So usually it'll be closed and oriented. We're going to pick a principal bundle over the three-manifold. Usually we'll just take G to be either SU2 or SO3. And in fact, for the moment, let's think about the case of SU2. So there's an interesting function, functional on the space of connections. Recall, so for an SU2 bundle, P is necessarily trivial. And so we can think of a connection as the same thing as one form with values in the algebra of SU2. So maybe we'll call that one form little a. And the function that we're going to look at is the Chern-Simons function. So the Chern-Simons of little a. I'm going to write it in two different ways. It's the integral from 0 to 1 of minus the trace of FATDT, where AT is just this kind of silly path that goes from the trivial connection, the one coming from our trivialization to the connection a. Writing it like this, it connects it to the original definition of the Chern-Simons function. The Chern-Simons function was originally introduced by Chern and Simons. And as you hopefully all know, you know Chern-Vay theory. So if you have a connection, you get a connection and a homogeneous polynomial on the Lie algebra of the corresponding structure group, then you can get a characteristic class. Of course the homology class doesn't depend on the particular choice of connection. And Chern and Simons looked for a canonical expression for the primitive of the difference. And in this form, that's how they wrote it down originally. That's how you see it easily as the primitive of, so it comes from the trace of FA, yjFA. That gives you this expression. But another way of writing it, sorry, then we integrate this over y. So we want to get a number. And another way of writing it is the trace of a times da plus one-third a, racket yj. So here my notation is a tiny bit sloppy. Here I'm going to think of these, the connection forms in this case as matrix valued one forms. And then here I mean matrix multiplication and wedge product. Here I mean using the bracket and the wedge product. So that's how the formulas work. And yeah, you should never trust my conventions. Probably. And in the notes there's a half. But for the purposes of the lecture, there's no half. Now a nice reason for writing it this way is that you see what the first variation is with respect to the connection. So if I look at d by ds of churned simons of a plus sb at s equals 0, using the first formula you can easily check that's got this expression. And here you see something rather remarkable. The critical points, so stationary points for the first variation, that's if and only if the curvature of a is 0, i.e. a is flat. So critical points are flat connections. Now of course the churned simons function has a large symmetry group, the gauge group, which in this case we can think of as maps from my three manifold into su2. And g acts on a by this formula. And it nearly preserves the churned simons functional, except that it picks up some factor of the degree of the gauge transformation. In this case it's a map from a three manifold to su2, which is the three sphere. There's a degree. And this difference picks up a multiple of the degree. So alpha is probably something like 4 pi squared or 8 pi squared, depending on whether I put that half there or not. Anyway, all right. So in particular churned simons descends to a function which I'll still give the same name to from space of connections mod gauge to the circle r times alpha z. Now, so this is what we would like to do Morse theory for. So Floor's amazing contribution was explained how to do Morse theory for churned simons. Now, anyway, so for this lecture we're not going to do too much with that except I just want to explore this a little bit and get some feeling for what this is supposed to tell us about three manifolds. All right. So let's look at some examples to begin with. So, well, it's always good to write down the trivial example if y is s3, then there's only one flat connection up to the G action. Oh yeah, sorry, I forgot to say something important. Need some little bit of notation here. So we're looking at this critical set. So critical set of churned simons, that's flat connections mod gauge and it's important to understand that that can be identified with representations row of the fundamental group into the structure group mod conjugation. Now, you have to be a tiny bit careful with this equivalence. So in the case of SU2 this is fine. So because there's only one principal SU2 bundle up to isomorphism. There may be more than one principal bundle up to isomorphism and then you'd have to, if you pick up your principal bundle, then you have to check that your representation gives rise to the same principal bundle. So if you have, so you note that if you have a, if you have a row, then you can look at the universal cover of your 3-manifold times G divided by pi1 via row. So that's by deck transformations and by row. That's a G bundle over y, which is, has a natural flat connection coming from this product connection. And this bundle doesn't have to be trivial in general. So anyway, that's a story. And I want to observe that there's already something a little bit interesting going on in the simplest possible case. It's our next simplest possible case. Suppose y is LPQ, a lens space. So pi1 of y, of course, is Z mod PZ. And you can use this construction, well, so, you know, representations, so representation from Z mod PZ to SU2, that's just given by picking a generator here and sending it to some pth root of unity. But of course, this lens space, I mean, there's another parameter Q in, in its definition. And you can check that while the space of representations, sorry, I need a notation here. So this is the representations of y into G. And if G is obvious, I'll just call it R of y. So here the representation space is easy to figure out. It's just these, here's up to conjugacy or representations of this form, but you get to switch these guys. So if P is, if P is odd, they're P minus 1 over 2 such representations. The interesting thing is that, so the representation space itself doesn't notice Q, but we have this churned simons function, which you can compute for these representations. And even the image of churned simons on the flat connections notices, notices the lens space. So, so, so actually, let's say omega is 2 pi, let's say omega n is this guy, then you can check that churned simons mod pi squared on the corresponding rho n is n squared R over P mod 1. I've gotten rid of the, made alpha essentially be 1 here, where R is an inverse of Q mod P. So, so to say, you know, although the representation spaces are the same as you vary Q, but fix P, the churned simons behavior of the churned simons function is already interesting. Some other examples that will be important as we try and develop the story a bit. So let's think for a second about tori. So if I look at, say, representation of fundamental group of n torus into SU2, that's just, so rho is specified by choosing choice of n commuting elements in SU2. And so, so since when SU2, there's simultaneously diagonalizable, so, so I, up to conjugacy, I get a bunch of guys like this. Up to conjugate, so I can simultaneously diagonalize, diagonalize them, but that's still not given me a unique representative. I can simultaneously, I can switch these factors. So the representation space of the n torus into SU2, I can identify with S1 to the n mod Z2. And there's a very, you know, here we're sort of exploiting the fact that SU2 is simply connected for non-simply connected groups. It's a, there's a very rich and interesting story which Greg could tell you everything about. It's, it was only one, one case of that that's going to be important for us, which is the case again of SU, SO3. Let's look first, well actually, let me say one, one, let me digress one second here. So, so if I have a representation or a connection, but let's think about it from the point of view of representations, a representation into G, then I can ask what is its stabilizer under conjugation. So, and it's an easy exercise to see that what that is, it's the kind of, essentially by definition, the commutant of the image of rho. Now the groups that can be commutants of other groups in a lead group that's an interesting story there for SU2, so for SU2 it's, there's a rather simple list, it's either the trivial element, a circle subgroup, or the whole group. For SO3 there's an interest, it's a richer story, you can have the identity, you can have Z2, this is the non-trivial generator, you can have Z2 times Z2, which we'll call the Klein four group, you can have SO2, you can have O2, or you can have SO3. So, this for example is matrices of the form 2 by 2 matrix A and it's determinant. And you notice this picture exhibits a nice feature, this kind of graph is self-dual, if you have a look at a representation, look at what its image is, then its commutant is the guy over here. So the stuff that commutes with this is a Z2, which is this Z2. These guys are their own commutants, and these guys are commutants of each other. So, that tells you that there's another, you know, so if we're looking at representations of the fundamental group of say the two torus for the moment into SO3, one possibility is that they're already the ones we saw coming from SU2 where their image is in SO2 subgroup. Another possibility, however, is that they lie in here. So, there's a representation K4 from pi1 of the two torus into K4 sitting inside SO3 and its stabilizer is K4. Now, to explain some of the other things that we're going to do, let's stare at this example a tiny bit more, take another point of view on it, so SU2 of this example. So, what do I want to say? So, of course, what we're going to do is we're going to look at the punctured torus, make a point P, of course pi1 of T2 minus P is a free group, but this, if I pick generators, pick my base point, pick a generator X and another generator Y, then loop that just goes around P is the commutator. Let's call this loop gamma. Gamma is equal to, depending on how I chose things, is the commutator. Now, what I can look for instead, instead of SO3 representations, I can look for representations rho of pi1 of the punctured torus to SU2, but with the property that rho of gamma is minus the identity. If you think about it, SU2 mod plus or minus the identity is isomorphic to SO3. So, this is picking up some kind of SO3 representations, but if I set, so let me call this R0 of T2 minus R0 of T2 and P SU2, that's this set of representations, SU2 acts by conjugacy still and now acts by consciously, but it turns out it acts freely. In fact, here an example of, so we had this rho K4, there's a, I'm going to call it rho H from pi1 of T2 minus P to SU2, which sends X to the quaternion I, Y to the quaternion J, and then you can check that part of the commutators of these guys is minus one, gives you an element here and exercises, see that this is, this is the unique rep up to conjugacy. So, just to say, you know, if we were looking at this, at these representations into SO3, there's a very nice interesting representation with image decline four group, that's great, but it has non-trivial stabilizer, which is a bit annoying, but if I look at it from this point of view, kind of look at it, try to look at it as an SU2 gadget, but with a bit of a singularity, it gets much better, yes. Rho sub H for quaternions, Y goes to J, so these are the quaternions I and J. So remember that SU2 is the same as SP1, it's the three-sphere, I can think it was the three-sphere in the unit quaternions, and you know, so anyway, just, so let me introduce another bit of notation then. So, let's set R of T2P mod SU2, that's, sorry, R0, that's just R of T2P, and the exercise is that this is a point, is a unique guy like that with trivial stabilizer. Okay, and another exercise for you, so we can, well, it'll be useful to generalize this, so a little higher level view of what we did, we took an SU2 bundle, sorry, we took it really an SO3 bundle, and we tried to lift it to an SU2 bundle, we failed at one point in this case. And, but in doing so, we can still study representations successfully, and what that point represents is the second Stiefel-Whitney class of that SO3 bundle. So, we can generalize this picture in a way that turns out to be useful for technical reasons, I mean, for trying to make some of the gadgets that we're going to construct into functors. It turns out to be useful to generalize this picture to three manifold, so we're going to look at SO3 bundles, but we're always going to look at them, try to think of them as SU2 bundles, and the way we do it is we pick a particular representative of W2, a curve, so we choose gamma in Y3, a closed curve, it might be null homologous, and consider, well, first R0 of Y and gamma, that's representations rho of pi1 of Y to SU2, so that rho of, sorry, rho of mu is minus the identity, and here I need a picture to explain what I mean. So, the generalization of the picture we had of the torus is, well, let's imagine our three manifold, something like the three torus, we take some curve gamma, maybe it's interesting, maybe not, whatever, this is gamma, and we pick our base point, and what mu is, is some kind of meridional curve, any meridional curve, so these are, okay, and then I can also set this to be the quotient space. Anyway, so it need not be free in general, yeah, but it makes it a little freer. That's close, consider, okay, all right, and yeah, lovely, good, yeah, right, then the next exercise is that R of T2 times S1, point times S1, so the three torus, and I just take the product of the previous picture with the circle, this is now two points, I mean, the sort of elementary exercise to make sure you understand the definitions. So, I'm going to give you some, a little bit richer family of examples, which are beautiful computation due to Fintichel and Stern, so these are ciphered fibrous spaces, so remember, so we're going to just look at the kind of basic example, examples of rescorne examples, so on the one hand that's the link of the singularity z to the p, z1 to the pq, z2 to the q, z3 to the r equals zero, intersected with the five sphere, so it is, there's S5 v6, that's this zero set here, and the sigma pqr is here. So, it turns out that if pq and r are pairwise and co-prime, then,