Diagrammatic calculus

0:00 Okay, so I'm pleased to introduce you to Rosenberg, who could speak about what's written It's a background practice for the geeky theory, you know, and it's a non-protective Riemann-Raw. Okay, well, thank you very much for the invitation. I was told, I was asked whether I could speak in French or in English, and I was told to speak in English, but if people have questions in French, they can ask questions in French. My English is obviously better than my French, but... Okay, so that's the title. This is actually not as complicated as it might sound, but this diagram calculus for K is something that might be of independent interest aside from the other applications which I'm talking about. kind of a short name which I just developed for doing calculations and I find that it makes things a little easier. What I'm going to talk about is joint work with Jacek Brodsky from Southampton and Mathai Varghese from Adelaide and I'm never sure which reporter would write his name in. Mathai is really a renault and Varghese is such a family name but he always writes his name backwards, so there's also a... And I'm Richard Zappo from Ahead in Brookfield. So this... Since there's a program going on in the group where it's applied to geometry and physics, there's a little bit of geometry and a little bit of physics here. So there are several motivations for what we were working on, part of which came from physics, not that I'm a physicist, but part of which has to do with geometry, the growth

2:30 So basically, the objective is to take some things which are fairly well understood in a sort of standard commutative setting and try to figure out how they should work in the non-commutative setting. So let's start with something classical. So if you have a compact spin C-manifold, then you have two kinds of pointoreguality on it. There's pointoreguality in ordinary, let's say, rational cohomology, and there's pointoreguality in K-theory, which is realized analytically in terms of the index of the Dirac operator. So in ordinary cohomology, you have non-degenerative cohomology that's just given by a product. I'm assuming the metaphore was conceived in particular as orientable or ordinary cohomology. So there's a fundamental class, so you have quite very dualities that can do with a non-generated pairing like that. And on k-theory, there's a similar kind of pairing. If you have two classes in k-theory, let's say represented by vector bundles, then the pairing is given by tensoring the two bundles together and taking the index of Dirac with coefficients in those bundles. and that is a kind of pairing that's related to Planck-raduality and K-theory. And the Churn character gives an algebra isomorphism from K-theory, once you rationalize it, to irrational homology. And part of the content of the Grotman-Dietrich-Mann-Roch theorem has to do with the fact that when you take this isomorphism, these two pairings don't exactly match up with each other. So it's close, but not exactly the same thing.

5:00 Because by the index theorem, this index over here could be computed in ordinary homology. Okay, now the churn character of T tensor F is just the cut product of the churn character of E with the churn character of F. So that looks like this, and this is the same, but you have a correction term, and that's the top class. So these two pairings, if you rationalize this and apply a term character, they're almost the same, but not exactly the same, so they differ by this. Now, if you want an isometry from one pairing to the other, then that means that you have to correct the churn character by the square root of the top class. I mean, we want the pairings to match up, so that means if you put in the square root of ta with this one and the square root of ta with that one, then you get exactly something like that. So then if you make that correction, then you get an isometry of carabels. And this has been known to mathematicians for a very long time, but it's also this business by the square root of the Todd class is also shown up in the physics literature and it's known to physicists Manassian-Moore formula for the deep brain charge and I'm not a physicist but I'm one of the co-authors of a mathematical physicist and I guess he got interested in this from that point of view whereas I've always been interested in index point of view, so that, um, anyway, so that's, this is, this is a review of something classical, which is, uh, presumably well-known. There we go. Um, all right, so part of what, the motivation here is to try to figure out how to, uh, generalize this to a, uh, non-commutative setting.

7:30 Okay, now, I'll come back to that later, but first, as I said, I wanted to make a digression and talk about this diagram calculus for KK, which is actually kind of very simple-minded sort of thing, but it's something that we found very useful, and we'll use it later on when it actually comes to making calculations. So, I assume this thing operated out in the seminar that everybody's familiar with the the Kasparra product in KK theory, and in the standard, the very simplest form, it just gives a, you know, a product which is like composition, like this. So if you actually come to use it in index theory, very often you need a more complicated form of the product in which you have several tensor factors here, several tensor factors here, several tensor factors here, several tensor factors here, and you just sort of partial cancellation over some of the tensor factors, but not all of them. So that's the more general form of the Cosparov product, which also shows up in Cosparov's original paper. And when you have a lot of these to compose with each other, it's sometimes very difficult to keep track of that. And so the diagram calculus is sort of the simple-minded way of keeping track of the associativity formula for the more general form of the Cosparov product. So, let me explain how this works. So, for example, suppose you have an element in KK of B tensor A comma C tensor D. So, we represent that sort of schematically by this. Here's B tensor A and C tensor D, and B and A are the inputs and C and D are the outputs. So, the diagrams are read from left to right, and you have one input node for each tensor factor on the left and one output node for each tensor factor on the right. Now when you want to do Kaspara product, the basic rule about the sort of most general form of the Kaspara product is that any kind of product is allowed

10:00 as long as you can match up these diagrams so that the output of one feeds into the input of the next one. And you're also allowed to permute the factors here that sort of corresponds to interchanging the order in intensive products. And as long as you're in the zero graded part of KK, that's harmless in a, you know, if If you're in an ungraded part, since things are graded commutative, you sometimes have a sign factor that you see. But the basic thing about this is that the associativity of the Kusparic product shows that you can concatenate diagrams in any order. So the associativity of the product corresponds to the fact that you can concatenate diagrams any way you like. And so here's an example of the associativity rule. Now, so you'll see in a moment why the diagram calculus is useful. This is the associativity formula for the Kaskara product. This is actually written correctly, but when you first look at it, it looks as if some of the factors are out of order, right? Because here we have Z, Y, X, and here we have Y, Z, X. So it looks as if somebody made a mistake. not a mistake, and the way to get these, you know, the order of the factors right is by looking at the diagrams. So the fact that these are the same, you prove as follows. So y is an element of kk of dA, so that's represented by a little arrow like that. Z is in kk of E, B, so that's written by a corresponds to a little diagram like that, arrow over there. And X is in K, K of B tensor A, C. So that's here's a B, the A, the C. So this is X. And now to X, you see, we can attach a Z over here or a Y over here, and you can do that in either order. And the associativity formula says that the products are the same regardless and which order you do it in. So here we did the y first and then the z. Here we did the z first and then the y. And the answers are the same, except if you're in the odd-rated part of kk,

12:30 then because, as I said, you're the graded community to be able to retract the sign. So this is a sort of... I mean, maybe diagram calculus is too fancy a term. I mean, it's just a kind of a shorthand for just keeping track of the order of the factors when you multiply. So the graphical proof of this formula is that, which is certainly more of a hat. Whereas to prove this analytically, of course you can do that. Okay. So to give a rigorous proof, you need to use the exterior product in Kasparic theory. So the exterior product will denote by a time sign. So really what this thing means is, if you look at the definition of the fancy form of the Kasparic product, what it really means is this. In other words, the z going back. The z had another d in it, right? The z just involved e and d. So you can dilate z to an element of kk of e tensor d, e tensor d, by taking the exterior product of the identity on d. So that's the definition there. So that's this. Then you take the product of your B tensor D with that. But then here again, to really interpret this, you needed to inflate the Y by putting in identity on B. And then you end up with this. And if you do it the other way, you get this. and then you need to see that this and this are the same per phone and pausing. Oh no, that's all right. Yeah, because the Here we have the 1, so in the first factor, really, nothing's going on, you just get z. In the second factor, you get y, and similarly, here you get z, and here you get y, so it really does the same thing.

15:00 So that's how you check these sorts of formulas, and this is just a sort of simple shorthand for you to crack at them. And you can also do the same thing with products in bivariant cyclic homology, because it has all the same formal properties as KK. So once again, this is our independent part of the talk, and if you now want to go to sleep, you can do that. maybe you learned one thing that's useful, so this is a sort of quick way of doing some of these associative functions alright, now that was a digression, but you'll see, we'll use it later on for doing calculations so now I'm going to go back to the original subject which had to do with index theory and I want to talk about non-commutative micro-radiality We don't claim any great originality for this. I mean, lots of people have. I've studied this already. But we've just tried to summarize one by thinking of that. So two separate C-star authors, say, A and B, are said to be Poincare dual in a strong sense if there is a KK element delta in here and a KK element delta check in here, which together implement Poincare duality. In other words, you take the product this way over B, you get the identity of A, and if you take the product this way over A, except for a sign, again, because of the graded commutativity of D is odd. Okay, so now you can see why we needed a diagram calculus. It's just that it's very hard keeping track of these formulas in your head, so it's easier to draw a picture. So the picture that goes with that is this. So... So...

17:30 delta check looks like that. It's in KK of C, common A tensor B, and delta looks like that. And, for example, you can concatenate here along A, and then you get something is an input terminal and B is an output terminal. Notice that the inputs always go in this direction and the outputs always go out in this direction. So actually this one's the input and that one's the output. So now when you take a product that way you get something in KK to BB and that's required to be identity and symmetry the other way around. Okay, so So this little picture is just to show you what the diagram calculus is for, to be able to do little calculations on that. So DELT is called the fundamental k homology class, and DELT to check is its inverse, or it's the fundamental cohomology class, and they're required to satisfy these identities. So A is said to be strong pipe-raduality algebra if you can choose the B to be equal to the A opposite. So that's the same underlying vector space of the order of the chronicle version. and was I believe Kahn's idea to write it this way with the a-off because it makes it from other formulas a little nicer. I mean, for some applications, it won't matter whether you have an a or an a-off over here because, for example, obviously, if a is commutative, it doesn't make any difference. and for many algebras A and A off are actually KK equivalent to each other so again I won't make any difference better to do it this way so then you sort of have a roll of A and B separated out and see which one is which okay so product

20:00 On the right, by delta, gives you an isomorphism from the k-theory of A to the dual k-theory of B. And that can be seen in terms of the diagram calculus this way. I mean, an element of the k-theory of A is represented by a diagram that looks like this. This is C over here. So this is, something like this lies in the k-theory of A, because it's in k-k of c from A, it's the same thing. So this is an element of the k-theory of A, and now if you multiply it with delta, which looks like that, that the A's cancel, and now you have something that has C as the output and B as the input, which means it lies in the dual K-theory of B. So dual K-theory, remember, B as commutative corresponds to K-homology of the space. People sometimes call it K-homology of B, but that seems like a bad terminology also because it's contravariant and D, so I'll just call it dual-chain theories. So you get an isomorphism like that, and you have an inverse that's given by product of delta check. Okay, so that's easy to check. And more generally, and again, you can check this with the diagram calculus, you also get quite a regularity with arbitrary coefficients. So you can put in a C here and a D here, And then the point-graduality isomorphism says you can take an A off over here and put it in as a D on the other side. And you get the same thing. So again, that's by product with delta in one direction, product with delta check in the other direction. So that's what we mean by non-commutative point-graduality. Okay, so here's an example of what you can do with the, again, with these little diagram calculations. So, so I suppose you have a fiber-aual pair, a and b, you have the delta check, and now

22:30 you take an invertible element in kk of a, a, then you can, you can twist the delta by L, and as long as you twist the delta check by L inverse, you get another pair consisting of a fundamental class in my school. Now, in one direction, that's completely obvious, you see, because it's sort of obvious that delta check tensor L inverse tensor L tensor delta. So if you write it this way, it's sort of obvious that these things cancel out. So it's clear that the product in this direction is equal to the identity, or maybe with a sign after it. Because these things cancel out. But the more confusing thing is to get the product in the other direction, which is illustrated here. So this is the harder direction of checking that these things are still inverses to each other, because it looks as if the L and the L inverse are on the wrong side when you move the order around. But actually, the way you can see that is by looking at the diagrams. So here's L tensor delta, that's represented by this composite, and here's delta chap tensor L inverse, that's represented by this composite, and now you're taking a product over D, so you compose over here, and now you have A as an input and A as an output, you get, well, this thing together with that, the associativity formula is equal to 1, so that means that all of this part of the diagram cancels out, and all you have left is an L and an L inverse, and then they cancel out. So it's also equal to 1 going the other way around. So the diagrams are basically just a way of keeping track of the order of the Okay, this is all very simple, but it's just very convenient. So, there's also converse. So, we saw that if you had a pair consisting of delta and the delta check would give you point radiality,

25:00 then you can go from one to the other. You can create a new pair by tensoring with an L, which was invertible in KK of AA. Well, what about going back? So suppose you have two different fundamental classes. Then if you multiply delta 1 check with delta 2, then you get something invertible in KK of AA, numbers to the previous statement. So in other words, all the hyper-legualities, this is, again, very easy to check. You just use the diagram topics the same way. So you've got the corollary that the set of all fundamental classes is isomorphic to the group of invertible elements of So now you can actually figure out how big that is. So for example, if a is commutative, the group of units in k, k, c of x, c of x, is by the universal coefficient theorem, it's an extension of the automorphisms in the k-theary of x by this x-group, so you can know exactly precisely how big that works. So five-reguralities, you see, are not unique. There's a great deal of non-uniqueness. You can always move around by this group of invertible KK elements, but that's fairly harmless because we see exactly what that does to the formulas. What is the biggest number of x? Because what is up in C and C? Well, then that's zero. That only shows up if there's torsion. Or if they're not fundamentally generated. I mean, in general... There, if you have potentiality in these portions. Well, yes, in a point-raduality, yes, if you have point-raduality, this is a general statement, with or without point-raduality. It's true, if you have point-raduality, that's the time to be generated, and then the X group part only shows up if there's portion-prank. But there could be portion, there's no reason you can't have portion. Okay, so now, alright, how many cases are there of this kind of point of radiality?

27:30 So there's an involution on a category of C-star algebers that just involves sending each algebra to its opposite. And that involution passes to KK, so that means if you have a KK equivalence between A and B, then you also have a KK equivalence between A-op and B-op. So if A is KK equivalent to a commutative algebra, then A-op is equivalent to the opposite of this, which is the same thing, so therefore, in that case, is KK covalent to its opposite algebra. I don't know if that's true in general. It's possible that all C star algears are equal to KK covalent to their opposite, but I don't know of any way of proving that. So just throw that out as a sort of side question of a certain independent interest. I have no idea how you would prove that, but at least for many algebras, it's obvious from this fact that we're here. Okay, so now suppose you have a separable C-STAR algebra that satisfies the universal coefficient theorem for KK, which means that it's KK equivalent to a commutative algebra. So the class of algebras that have that property is reasonably large, and it includes all, for example inductive limits of type 1 algebras. So there's plenty of stuff there. And also suppose there's finally generated k-theory. obviously want that if you want this to be a point-graduality algebra. So the question is when is this A point-graduality algebra? Well, it's always part of a strong PD pair and it's actually a PD algebra itself. In other words, you can take if and only if the sort of obvious condition is satisfied. So you see, if A is a point-graduality algebra, that means that the k-theory of A is up. And it's a k-k-equivalent to a commutative algebra,

30:00 so that means A and A-op are also k-k-equivalent. So then the k-theory of A is isomorphic to the dual k-theory of A. So now there are two possibilities. You can either have this isomorphism without a dimension shift or with a dimension shift. So that gives you some natural necessary conditions. So if that's true with a dimension shift, well, as far as torsion is concerned, the torsion here and the torsion here always differ by a dimension shift. It's just by the universal coefficient theorem. because that comes from the x term in the universal coefficient, which has a dimension shift. So the torsion here always matches up with the torsion here with the dimension shift. So for this to be true for all of K-theary, then it has to be true for the torsion-free part. So in that case, you need the ranks of these two groups to be the same. On the other hand, if you want this to be true without a dimension shift, then it's automatically true for the torsion-free part, and now you need the torsion to match up, so that means the torsion in K0 So those are obvious necessary conditions to have a point-graduality algebra. So the content of the theorem is that those are also sufficient, so that once you have that, you can always construct a delta and so on. So if you have that sort of obvious necessary condition, you can always construct a point-graduality element delta at delta check. Of course, I'm not claiming this is a very interesting theory, because this point-raduality element is not anything very natural, I mean, it's rather artificial, but at least it tells you how many of these point-raduality algorithms there are. So here I'll just sort of review what I said. So, since A is KK covalent to an abelian answer, you might as well see it as abelian, and then by the universal coefficient theorem you get that, so we have these necessary conditions. And then you need to show, well you can show it for A to be commutative, but if you have this isomorphism then you can implement it by a delta. So, there's sort of two ways to do this. One is you can build delta and delta check directly from the universal coefficient theorem. that computes KK of AB in terms of the K-theory independently,

32:30 and that will sort of show you that there has to be a delta that implements any given isomorphism. So that's one way to do it. Or alternatively, you can do it topologically. You can realize A as it functions on some, well, maybe vanishing infinity on some manifold, and then, so the way to do that is, for example, if A is a finitely generated K-theory, and it's commutative, that means that as far as changing A up to K-K equivalence, for example, you can always assume that A comes from a a finite CW complex, and then you can embed that in Euclidean space and take a regular neighborhood and so you get a regular manifold, which is not a tiny, of course, but that realizes this thing. And then take V to be the cotangent bundle, and then you can construct the delta So that's what gives you geometrically construction. So this is one way to get these deltids geometrically, but it's not the only way you could also do it. So there are actually lots of these strong, frankly, Routman pairs. Although they may be sort of artificial. So now I want to talk about... Remember, the original motivation was to try to generalize this fact that, so you remember what we started with in the very first slide, was that we had this fact that if you multiply by the square root of the Todd class, it gives an isometry between the two pairings on rational cohomology, one coming from cohomology point-radiality, and the other coming from the index of the RAC operator.

35:00 So since we wanted to generalize this to the non-commutative setting, I mean, we already saw what we mean by point-radiality, so now we need to know what we mean by this. So the general context will be explained in a moment. First, this is a sort of preliminary. So I just want to point out, first of all, that if you have two point-graduality pairs where the a's are the same and the b's are different, then the b's have to be kk equivalents. That's sort of obvious because you take the delta for one and the delta check for the other and put them together to get the s-point. So now let's consider the class of all separable c-star algebras for which there existed such that AB is a strong PD pair. So as we just saw, this is a huge class that contains all separable C-star algebers that are KK equivalent to a commutative C-star algebers finally generated K-theory. So if you have such a thing, we choose a representative for B and the B will now be called A tilde so this is just to point out that the B's are not unique but they're sort of unique up to KK equivalents so you just make a choice if A is itself a point-radiality but then we have a canonical choice so we take it to the opposite so now we have a point-radiality pair and the B will be called A tilde basic things are we have this delta, and we have the delta check. They're supposed to implement point-raduality. Alright, so this is sort of the setup. So these implement point-radiality in k-theory. But you remember that this thing, the square root of the top class, involves comparing the point-radiality pairing in k-theory with the point-radiality pairing in cohomology. So we need sort of an analog of that also. So where is that going to come from? So that means we need some kind of bivariant cyclic cohomology theory

37:30 with a good multiplicative churn character from k-k. Now there are various choices you can make. I mean, if you want to use usual cyclic homology, then you have to replace all the original C-star algebras by dense sub-algebras, and then you need to know that the inclusion of the dense sub-algebra induces isomorphism on K-theory and so on, and then you also, but not just on K-theory, but also on dual K-theory, that's more complicated and stuff. So, you know, while you can do that, in many cases, it gets kind of complicated. So, for our purposes here, we're going to use Puschnig's local bivariant cyclic chronology, which doesn't require you to cast against sub-algorithm. So, that seems to be technically more convenient. And so, there's no standard notation for it, but we'll just call it H-local. So, this is the local cyclic theory. I mean, he calls it, I think, H-E-sub-local or something like that. What does that give you from Apple? What does that give you to plug in the contact panel? Oh, it gives you the usual thing. So, it has the property that if you have a C-star algebra that has the metric approximation property, then this theory and most of the other theories all agree with each other. So, for kind of nice nuclear c-star algorithms, what we're mostly interested in here, it doesn't make any difference which of these theories you take. But this one just sort of technically works. And there's also a universal coefficient theorem for this theory also. So if A and D are in the class of algebra for which you have the universal coefficient theorem for KK,

40:00 then there's also a universal coefficient theorem for this bivariant cyclic theory, and it looks like that. So if the K-theory is phonetically generated, this then is the same as that. Although in general they're different. But, so that means that this thing is, I mean, for the cases we're interested in, since we're talking about point-radiality altruists, these will be finally generated, so this thing will just be the same thing as that. So that means that this will play the role of rational cohomology, you know, for our purposes. And we have term character, which is a sort of good isomorphism from rationalized K-theory to HL. So you have sort of good punctorial diagrams and everything. which are KK equivalent to each other, then they're also HL equivalent, but non-conversal. Right, because we've washed away the torsion, so you can't go back. And so if you have a strong pipe radiality pair for KK, then it's also a strong pipe radiality pair for HL. But the whole point here, and this is sort of the crucial thing, so that's why I put it in color, is that while you could always take the original point-raduality classes in K-theory and just sort of push them down by the turn character and use those in cyclic theory, most of the time you don't want to do that. So that would sort of not be the right choice. I mean, if you do that, you've got something that's totally uninteresting. So you'll see why in a moment. First, I want to make a definition. So, all right. So suppose we have A, which is one of these point-radiality algebras. And so you have the fundamental K homology class. So it's also point-radiality algebra for HL. So you can make a choice here also. And then the top class of A is defined to be this product in binary and secret theory.

42:30 So you take the change character of delta and multiply by C check. And this lives in that ring. And to me it's obvious that this thing is invertible and the inverse looks like that. And again, this sign here just has to do with the graded commutativity in the case where the dimension of the point-grade 1 is odd. Okay, so this is the definition of the top class. It's just the definition. But in order for the definition to be useful, we want it to match up with the usual definition of the community text, right? So, otherwise this would be pointless. So, let's look at basic examples. So now you can see why I'm doing all this. Suppose we take a compact, complex manifold, and we turn it into a commutative c-tero algebra, so that's in a continuous function. So then you have a strong-plank radioology algebra, and now in this case you have a sort of canonical choice for delta, which is given by the Dalbo operator on the product of x-wood itself. Now, the HL is identified with the usual periodic cyclic homology after it passed as C-infinity, and so this thing is identified with the endomorphism ring of the rational homology. And so we need to choose Xc as well as the delta. So why don't you choose for Xc? Well, the obvious thing is just the usual point-radiality in rational homology. Now, if you make that choice, then what I call the Todd class is just cut-product of the usual Todd class. So the usual Todd class is an inhomogeneous cohomology class. You know, it's a certain power series in the churn classes. And that thing is invertible, so multiplication by that, cut-product by that class, is an endomorphism and a rational cohomology, and that's exactly what we call the Todd class. actually do match up. So that now explains how we propose to generalize that formula that we started with, by doing all of this. So as I said at the beginning, my goal was to get to Brotundeeck-Riemann-Rock in a non-commutative setting, so now you see we have all the formalisms set up to do that. I don't really understand, because you see that like a lot of different choices you can make.

45:00 like a lot of different post-processes. Right, exactly. Well, it depends on certain choices, but you can keep track of how it behaves with respect to the choices, and there's no canonicity, and there's nothing you can do about that. Just as... But if you think about it for a second, you'll see it has to be that way, and why is that the case? Let me give you a simple example. Suppose you take complex projective space. I'm going to take the simplest possible compact manifold, complex projective space. All right, now, in the same homotopy equivalence class as complex projective space, you have lots of other manifolds with different characteristic classes. So, you know, if you're just sort of looking from the point of view of K-theory, if you didn't know about the smooth structure and the complex structure and everything, you know, the knowledge of the sort of underlying homotopy type of the manifold doesn't really determine what the taught class is. So even there, you have to make some choices. Maybe that wasn't the best example because I'm not sure you can make all of those things into complex manifolds, but there are more complicated examples where you can have complex manifolds which are on the top equivalent to each other and have different churn classes, different taught classes. So, you know, the taught class is not determined just by knowledge of the radical homology algorithm. So there's some extra stuff that has to be fixed, so we just make the choice and just study what happens. I mean, you're right that here it looks like we're making more choices than you do in a usual case, but on the other hand, the choices are of a simpler nature. because they're basically just choosing the delta and the xi, and then studying how these vary as you move them around.

47:30 We saw the delta isn't unique, but it's unique up to an invertible element of kk. There's a similar sort of non-canonicity of the xi, and then as you change the delta to a different choice for delta, and you change the xi to a different choice for xi, class gets multiplied on the one side by the churn character of that k-k, invertible k-k element that was used to move the delta around, and it gets multiplied on the other side by the invertible cyclic homology element that's used to move the c around. So there's a formula for how the top class of k's would be a particular choice. Okay, so now I want to talk about Grotendieck-Riemann-Roch, so I want to introduce the Giesen map, and in general the idea of Keg-oriented maps. and the formalism here is just sort of basically copied from the paper of common standard on the index theory of pollinations. So if you have an amorphism of C-star algebras in a suitable category, then a K-orientation on that amorphism is going to be a functorial way Now, I mean, there's, we want the right one, so there's one obvious thing you could do. I mean, any homomorphism, of course, is a KK element, so we could just take the KK element associated with the homomorphism you started with. But for purposes of the Riemann-Walk theorem, that's not really the right one. So what we want is really this one, which has... So this is really the thing that corresponds to the... ...the Geistin map in k-theory. Oh, sorry. Oh, I know what's wrong. The map was supposed to... Can you have the opposite, the opposite?

50:00 Yeah, I want it going the opposite way. Sorry. This should be, oh, I know this one, yes, this should be a B comma A, sorry, okay. Yeah, we want the KK element going in the opposite direction, right? Right, so there's an obvious misturing here, this is supposed to be B comma A, right? Yeah. So we want a KK element that starts at B, inputs are A from the left, you know, so it starts at B and comes out at A. So you want something in KK of B comma A out of the F, which is going in the opposite direction. So you want this one-way map. And the way you do it is by applying hybrid reality installs. So, you can take f-op from a-op to b-op, and then compose with delta k for a on this side, and with delta b on that side, and then that gives you this. So if you think about that, this is very much like what's done in the original, you know, Sayre's original paper on the Riemann-Roch theorem. So you want the wrong way map on k-theory. So you have to apply point-raduality on both sides together with the map. non-computed cases, just we change A to A off over here, but we can get that off. Um, so this thing is actually functorial. Um, in other words, if, if you have two morphisms of C-steroalgebras, And you take this and that, the two of them, and you multiply them together, then you get the thing for the composite. And that comes from the associativity of the Kaskara product and the fact that these things will cancel out. So, once again, the whole idea of this was that to f and on the ab, we're associating

52:30 this f3, which lies in k, k going in the opposite direction. So that, and this thing is, I'm for it. Okay, so now we're set up for the Riemann Roth theorem. So, as I said, we're not claiming any great originality for this. This is sort of taking the standard thing and just translating it into this language. So, the churn character of this F-shrie, which was defined by composing with point variability on both sides. So, the f-streak was defined by this composite, so if you take the churn character of that, you get a formula for it, which is just, sorry, this L should be up in the superstructure, Sorry, that's probably a brace missing in the tech box. All right, so this is the F-street computed in cyclic theory. So that's what you would like this to be, but it's not exactly right. You have a correction factor because of the Todd class. So there's the Todd class in A and the Todd class in B, and you put those in. The F-street in cyclic theory and the F-street in K-theory are related to those two type classes like that. So if you think about that in a classical case where A and B are, you know, corresponding to compact-complex metaphors, then this is just the usual Riemann-Rothler. So, I mean, this proof is just totally straightforward. There's nothing very profound about this. So you just take the right-hand side and expand everything out and multiply it out and increase right out. And at some point, you have to use the associativity of the intersection product.

55:00 So you have a lot of complicated associativity formulas like this. and once again the reason for mentioning a diagram calculus for KK at the very beginning is that these formulas make you dizzy if you don't have a way of keeping track of the order of the factors and that's what the diagram is for. if you draw the diagram on a piece of scratch paper then it's very easy to do the calculations. You see, so now we wanted to compare what you had in k-theory, which was this. And then when you do that, you see that the left-hand side of the equation was this, but the turn character was, uh, was punctorial. So you just have to show that this is equal to that, and this is equal to that, and you do all this and sort of won't count all the products out there. So, um... As I said, I'm not claiming any great originality either because of here or more to prove there's really not much content to it, but it's just sort of an interesting exercise in the diagram calculus. All right, now, to come back to something else, we also wanted an analog of this fact that multiplying by the square root of the Todd class gives you an isometry of pairings. And so, we need to know what we mean by this. So the tot class was an element in binariant cyclic theory, so it's sort of hard to see what you mean by taking a square root. But on the other hand, you have to take a square root if you're going to get a formula like that because you have to multiply by this thing on both sides and then they have to combine together to get a tot class. So for that reason, we were led to looking at this condition here. So this puts a little more restriction on the fundamental class.

57:30 So once again, this is a restriction which is satisfied in the standard case and makes it possible to generalize that to the non-commuter setting. So we'll call the fundamental class symmetric if it's basically unchanged under flipping the order of the two factors here. So not every fundamental class has that property, but remember in the classical case that we started with, you had a complex manifold X, and then you took X cross X, and you basically took the, so this delta basically came from the characteristic classes of x cross x, and those obviously are symmetric because of, you know, the term classes of x cross x just come from the term classes of the x on the first factor and the term classes of the x on the second factor, and you can just switch them around. So, there's a natural condition of symmetry you could introduce. And symmetry basically just means that these two kinds of products are the same, in all cases. See, it sort of depends on where you put the fact of it, where you put the pin question. So that's sort of the natural condition. And now, finally, we get the thing which is the analog of what we started with. So suppose A satisfies an universal coefficient theorem for local cyclic homology, and for KK, and the cyclic homology is a finite dimensional vector space. So, of course, when the k-theory value is finally generated, then that's going to be automatic. So that's a finite-dimensional vector space over Q. Or C, I guess.

1:00:00 Thank you.