 I'm very pleased to be here and very honored to be speaking at Tom's birthday celebration. I could basically beg the talk, which you might be grateful for, and just tell stories about Tom, which I will attempt not to do. It's just that, you know, like more or less everything, many things I had to say here sort of reminded me of something to do with Tom, but maybe just one little one just beforehand, maybe an expression of pleasure for the degree to which Tom has influenced me and my work, and a lot of that has been because he's been in Boston, and one of the things I remember really clearly, although you'll probably contradict me because what I remember things clearly, they're probably not true, but when I was in Oxford, I spent the year, the cyber-gwitten year, 1994, 1995 there, which was a great experience, visiting largely with Peter, and in the spring Tom came to visit, and Peter and Tom were on the job market, if you can sort of imagine that, and finally made their decision, and we went out, they went out for lunch to celebrate, and I joined them, and we had a bottle of wine with lunch, which is not a really mathematician thing to do, and I do remember they toasted each other, because the decision to go to Harvard and MIT, and as I recall the toast, I don't know who said this first, but one of them, I think maybe Tom said, thank you for making me a lot of money, and the other one came back with the same toast, so they of course were very pleased, but I was probably more pleased than either of them because the idea that Gates theory was, well, of course Cliff was there already, and it made a big impact, but the idea that Tom and Peter were coming to Boston was great, and over the years that's meant that there's been a million informal interactions, and a million times when I've had a chance, when I just, some point I'm stuck on, or maybe a crazy idea that needed encouragement or shooting down as the case may be, and Tom was really good at that, so I hope to illustrate that point, because it actually one of the most striking instances of that, well, as you'll see, it wasn't the insight that Tom, I will explain a small point that Tom made to me that was a root around what seemed to be a very large stumbling block at the time, so I don't know whether you will remember the story, but it certainly looms large in my mind, in any case, this is a joint project with Dave Ockley, we're almost done, we had vowed to finish writing our paper by the end of the summer, and we might actually achieve that, I don't know, so we have 90% or 95% written, depends on Dave's supposed to be finishing one last little bit, so it might be long, so maybe it's more like 90%, in any case, we're almost done, so, okay, so the theme is actually not so different, although different in techniques and intentions from what Dave talked about earlier today, and it has to do with the interaction between difthomorphisms of four manifolds, and in a way, with embedded surfaces, and also embedded three manifolds, so let me start with something which I assume everybody, knows, but anyway, I'll say it anyway, just to set the scene, you know, by the early 80s, Donaldson had given us examples of homeomorphic manifolds, smooth four manifolds, I probably should say four here, manifolds which are not difthomorphic, and here's a concrete example, actually this isn't the concrete example that I want, but let me start here with, so I'll write V, or V4 for the hypersurface of degree 4 in CP3, so there's an equation, as a smooth manifold, you can think of this in various ways, the K3 surface, or an elliptic surface, which is somehow the guys in which we use it, we use the fact that it has elliptic vibration, although, again, as I said, I'm easily distracted as you'll see, so this isn't exactly the example I want to discuss, of course, the existence of more than one smooth structure on the K3 surface is one of Tom's early and truly spectacular results, and one of the amazing things about that, of course, and actually in the paper that he wrote with Bob Gumpf, is that this jointness of the planes that Larry was referring to, X1, X3, and X1, X4 planes, that plays a crucial role, because what that tells you, if you think about it right, is that the K3 surface has three different elliptic vibrations, and the fibers in those cases are all disjoint, so you can do that on the torus, and then that sort of descends down to the K3 surface, so I thought that was just a funny coincidence, that that exact, what's that, is it a coincidence? Yeah, I don't know, I mean, you tell me, but tell me afterwards, you know, so anyway, I thought that was, you know, just to see those same pairs of things written up on the board, I was rather mused by that, any case, so here's the example I actually want to think about, which is probably, I don't know, easier, but more directly taken, you don't have to do all this hard cutting and pasting, to do this, so take, instead of X itself, take X, sorry, instead of the K3 surface, take it blown up once, so adding a CP2 bar, so that I'll call that manifold X, and unfortunately there's a couple of things later on called X, but I hope those will be, it'll be clear, this one is really X, okay, so there's X, which is that's blown up K3 surface, and that's, the reason for the blow-up is to make it not spin, and that's homeomorphic to another manifold in the same, which is three copies of CP2 and 20 copies of CP2 bar, okay, so those are homeomorphic, but they're not difiumorphic, and the way you can tell them apart, by the way, through the work of Donaldson, is that you, that Donaldson defined for us these invariants, I'm just thinking of sort of the simplest Donaldson invariant, which would be a numerical invariant, so I'll just call it D, without any definition, so DX is some sort of, is an integer, so it's non-zero for X, the blown up K3 surface, but for this connected sum, Donaldson proved this connected sum theorem, which said that in fact it's zero, so those are different manifolds, okay, so, and the fact that they're, but on the other hand Friedman told us they're homeomorphic, so in fact they're an exotic pair of manifolds. Wall had proved, you know, in, you know, sort of, even before I was a student, Wall had proved in the 60s, in a series of seminal fundamental works, I should say, that homotopy equivalent manifolds are stably difiumorphic, that's actually, his proof is sort of harder than it really needs to be, it's actually a pretty straightforward handle body argument, but in any case it's a very nice argument, and in particular if you add some number of S2 cross S2's, you can turn X into Y, if you add same, they're homotopy equivalents, so it would be the same number. Actually Mandelbaum whose, actually name I misspelled, but forgot to correct, and Moises on, this I think may be originally a result of Moises on, but hard to understand and made comprehensible by Mandelbaum in the 1970s, in fact showed that in this and many, many similar examples, you could take K equals 1, it's just one stabilization that suffices, and it's a wonderful and interesting problem whether that's true in general, I don't know an example of homomorphic manifolds, simply connected manifolds where you need more than one stabilization, someone knows, let me have the microphone, you could feel free to take over and do the lecture instead. Okay, so it's also true that you could add just a CP2, and they would become difhumorphic, so that's also part of Mandelbaum and Moises on's theory. So let me call, actually I don't think this is quite the right name, but so I want to take X connects some CP2 and Y connects some CP2, and I want to think of them as the same manifold, so they really ought to be, as we did it in our paper, there ought to be sort of a choice of difhumorphism between them, and when I identify them with the same manifold, I want that identification to have the following property, I'd like it to be, to look like it's homotopically, like a homeomorphism from X to Y, and then a homeomorphism from CP2 to CP2, so put it more simply, I'd like the homology class of the generator of H2 of CP2 here, whatever difhumorphism I choose, I want it to go there, so they should be identified, but in such a way that the homologies are sort of kept track of. Okay, so those manifolds are the same, but as I say by Donald's, so there's of course an important principle that it matters that we're adding CP2, and if you added CP2 bar, they would remain distinct. Okay, so an important point about this is that, and sort of maybe the beginning of the connection that I alluded to in my title, if you keep track of various submanifolds of Z, you can actually, so Z is sort of kind of a connection, it comes from X and comes from Y, but you can remember which one it came from if you remember a little more data, so the simplest data would be to remember a particular generator of the homology of this piece. Okay, so to Y, how is that, well this is a straightforward argument, remember whenever you have a sphere, an embedded sphere in a four manifold of self intersection plus or minus one, let's, mainly we're going to use, well it doesn't matter, plus or minus one, but in this case plus one, you can blow it down, so it's not really algebraic geometry blowing down, Dave insists on calling it anti-holomorphic blow down, but he's not here, so I'm going to just call it blow down. If you're here, Dave, then online, don't interrupt. Anyway, so, and the point is if you add CP2 and then you blow it down immediately, you just get back to whatever your manifold was, so in other words, the pair Z and the sphere coming from the X picture on the sphere coming, or the sphere coming, remembers X and likewise the Z coming from the Y picture remembers Y, so if you like that pair is, so one point is that the pair is exotic because in fact there would be a homeomorphism taking one to the other. In fact, if you sort of just start with a homeomorphism from X to Y, then an extended in the obvious way would have that property. Okay, so that's sort of an exotic pair of spheres, and as I say, those spheres remember X and Y. You could also, instead of remember, right, oh, this is, I have it before, so yeah, this picture will recur a few times, so the picture is somehow, like in this little bit here, there's a blue sphere that's supposed to be the one from X and then there's a red sphere from Y. So just to point out, those have algebraic intersection one because they're homologous and the self-intersection is one, but they actually have to hit each other in a number of points. The simplest picture I know, they hit each other in five points. It's not hard to prove that they can't intersect in just one point. That's a fun exercise that they must have sort of, so-called excess intersection. Anyway, similarly, instead of thinking about the pair of the Z connect with either this X sphere or the Y, X two-sphere or the Y two-sphere, you could take Z with the three-sphere, which is the three-sphere that sort of separates it into connected sum. So that also remembers. So that's kind of a sidelight, but also sort of fun. So maybe another way of saying this is that, but again, those things are all topologically isotopic, so maybe I'll say it unnecessarily. So this is, of course, a well-known observation. I don't know who made it first, but it's sort of been around for a long time. But let me just say it in a way that perhaps provokes an obvious question, which is if you look at pi zero of the embedding space. I actually thank you, Dave, for sort of warming up the audience to be sensitive to things like homotopy groups of embedding spaces. Maybe that seems like a good thing to study, I don't know. So here's pi zero of the smooth embedding space mapping to pi zero of the topological space of topological, maybe locally flat embeddings. That's not injective or if you do the same thing for the three-sphere. Okay, so that's all about embeddings. How does that relate to difumorphism? So as it turns out, here I'm actually going back quite a ways, some 25 years to older work of mine that says that there's another way to remember the X and Y, and that's through studying a certain difumorphism. So let me, again, I assume remind everybody that on CP2 you have complex conjugation that takes CP1 to CP1, and actually I'm always going to do a little isotope. So rho is the name, that's sort of for reflection or something like that, because writing bar as a map is just problematic, I don't know. I have enough, anyway. So rho is that map, but I'm going to isotope it so it's the identity on a ball, so it's no longer an involution, but who cares? Okay. So, but it's supported as in is on a neighborhood of the CP1. So you can sort of imagine that ball as being maybe relatively big. So supported on, I think I will say a few times, meaning like the places where it's not the identity. I don't know if that's standard terminology. Okay, so you can do such a thing on any format as long as you have a sphere of self-intersection plus or minus one. Again, I think these observations, in some sense, go back to Wall's early work. So, and one easy way to say it is if you have, so here this Z and X are not the Z and X, this is like any Z that's a connected sum of, with either CP2 or CP2 bar, you can do the complex conjugation on the CP2 part extended by the identity of Z and then you get a diffeomorphism, which, so just to emphasize, it really depends on the placement of this sphere. Okay, so again, a simple idea, but crucial. So sort of informally, I'm sort of writing it as the identity on one part and then this complex conjugation on the other. Okay, so in my theorem was that in the situation above, so now back to the X is now this, I forget what it was, the blown up K3 surface. It actually remembers the Donaldson and Mary, and in lots of other circumstances. So actually I'm going to expand it all a little bit because it's the same principle and what I just said is correct, but it's actually better instead. So here I'm going to change my notation, but now I'm done. Z's and X's just mean what I'm writing here from now on. X is this blown up K3 surface and this is other manifold which I wrote as M, which is CP2 and throw in a couple more CP2 bars for good measure. So if you like, you could have just sort of always added CP2 and then instead of blowing up the K3 surface once, you could have blown it up three times. So it doesn't matter, but it's awfully handy to do this. Okay, so in that manifold, well, so what was my point? Of course, this is the same as why connect some M because there's a CP2 sum N. Or if you like this M, maybe I pointed out, right, this M is also diffeomorphic to S2 cross S2 connect some CP2 bar. So if you'd like the stabilization with S2 cross S2 is better than you can, either way you have the stabilization. Okay, so the Z has two identities and it also has lots of spheres. So here's one sort of generic, so M itself is, so let's look at M. So there's the CP1, which I'll call H in CP2, and then in each CP2 bar there's essentially a CP1, but I can't call them all CP1, so the one in the CP2 is H and the other two are E1 and E2, so I can pretend I'm an algebraic geometer for a minute and call them exceptional spheres. Okay, so on M we could do this reflection. So I could do a reflection, so if I take H plus E1 plus E2, that has self-intersection of 1 minus 1, or I could do a reflection in E1, or I could do any number of compositions of reflections. Now, there's an important reason for doing this composition, which if I weren't telling stories and getting all sentimental, I would probably explain more, but anyway, I won't. But anyways, so the important part of this F, which is the composition of these reflections, is really the first part, because that's the one where I can sort of do it in terms of X, or I can do it in terms of Y, and I have, in principle, maybe different looking diffeomorphisms. So I have, I think something called, well no, sorry, that's not Alpha, excuse me. All right, so I could take the, I could, I could transplant all this onto, onto, onto Z, either thinking of it as X connects some CP2 and a couple of CP2 bars, or I can transplant it onto Z using the decomposition into SY. Okay, so, so now I'm going to, so this all provoked, so this all provokes a certain, again, a memory. So, so I'm going to drag you through this memory, and I don't know whether you will remember this, Tom, as, as, as distinctly as I do. So, okay, so what's the, what's the diffeomorphism I want to look at? So if you do this with sort of the X picture, and you do it with the Y picture, and you compose the one with the inverse of the other, I'm going to call that Alpha. Okay, so that's by construction, that's provably that, well that just acts as the identity on homology, because the X sphere and the Y sphere were homologous, and by work of various people that's homotopic, and therefore topologically even isotopic to the identity. So in, so in, you know, whenever, and years ago in the, in the 90s, I had defined, actually I think I did it, worked on this when I was during that spring in Oxford, and I defined this invariant, an invariant of diffeomorphism. So I will say something very brief about it if I have a minute at the end, but it's defined using one parameter gauge theory. So this is the Donaldson invariant version, there's a cyber-gwitten invariant version, I don't know if there's a Higgart-Fleur version, but probably there is, but I think it has some technical issues. Okay, and the question was okay, well it's fun to define, it's fun to prove it has various basic properties, but I wanted to construct examples. Well I constructed it, and I had an example where, you know, it clearly should be non-zero, and I wanted to calculate it. And so here was my original idea. So this is, this thing in, in the box here, this sort of messy picture, that's supposed to be an example, that's supposed to be like the Sx, which colors, which X is blue, so let's do blue. So this blue thing, this is Sx plus E1 plus E2. The E1 and E2 are the black lines that are cutting it across. Okay, so what should I, so my idea was, well I have this thing that involves an X sphere and a Y sphere, so I should take a regular, I should take the union of those spheres, and more or less nothing, you know, take the union of those spheres, and kind of take a regular neighborhood of that, and pull apart along that. So that would isolate all the interesting stuff on this one manifold, which is the regular neighborhood of a couple of spheres. Well, you know, that's kind of a complicated manifold. The bound, I mean these days it's sort of more tractable with various other kinds of gauge theory. It's some kind of graph manifold, and probably lots of things to be computed about. But you've got to remember, this is like 1996 or so, five when I'm thinking about this, and that was a pretty hopeless manifold to think about. So I was stuck, but I had some ideas where you sort of pulled apart along one, and then somebody pulled apart on the other, and anyway, well I was just not making progress, so maybe I'll skip the argument and tell the story and come back to the argument. So I'm presenting it in the form of a very short play. I always wanted to write a play, and you know, well, so this is my best, this is going to be it. So if I don't win the, whatever, Tony award for this, I'm out of luck. Okay, so there are two, it's a play in one act with one scene in Tom's office. You could guess who the players are. So here's my line, first line is, well, hey Tom, I'm kind of stuck on this, the boundary of this neighborhood is really complicated, and I don't know what to do with it. And so Tom says, well, this is kind of a fantasy recollection of this. I have no idea what he said, but he said, well, do you know how to calculate the two separate, this invariant for the sort of the X piece and the one for the Y piece? And I said, yeah, sure, I know how to do that. And he said, well, do you have a composition law? And you know, I had already done that. I said, yeah, yeah, that's fine. He said, well, why aren't you done? Well, so, okay, so that's the end of the play. And you know, so, I mean, so what do you learn from this story? Well, you learn that sometimes you could be really stupid and you also learn that sometimes like a simple observation is really, really crucial. So this was a very simple observation, but it was really crucial because, and here's what, you know, like here's what it happens. You notice that this E1 thing, which is actually crucial to the calculation. And in fact, my original version of this had a different composition. So it's not quite this. This is, again, revisionist history. But who cares? So the point is that with this row E1 stuck in there, which was in there in the first version of all this, there is a Donaldson invariant for this piece. It's a little tricky because it doesn't act trivially on homology, but anyway, there is one there. And likewise, the Donaldson invariant for this piece is defined. And there's a composition law, which says that the invariant for a composition is the sum. This one, you notice there's an inverse, so that actually turns it into a minus sign. So I get the invariant for the one defined using x and the one defined using y. And the key calculation, which is in my paper, well, with a factor of four, but it somehow simplified it, so it's actually, it's a slightly different de-fumorphism, so it's actually a factor of two. For y, it's kind of clearly zero. Well, the point is it's what's the actual calculation is that the invariant for f of this form is simply the Donaldson invariant of the manifold, the underlying manifold, or the manifold without the CP2 on it. And likewise, the Donaldson invariant for y is the Donaldson, for the de-fumorphism defined in terms of y is the Donaldson invariant of the underlying manifold y itself. So in that sense, the de-fumorphism remembers that Donaldson invariant. It's actually an interesting question to me as to whether it, like, the sphere has actually remembered the manifolds. So I don't know whether the de-fumorphism remembers the manifold in the same sense. I mean, but as far as the invariant that tells them apart, these two, this pair apart, it remembers them. Any case, so if you choose the right x, then you get one part zero, the other part's not zero, and then you're done. So this works for blown-up CP2, sorry, blown-up E2. It works for lots of other examples. Okay, so there are things stood. Oh, we've seen the play. Let's see, in the movie? Anyway, okay, so somewhere around, so I presented this, I think probably the first time I presented it, well, the first time I presented it correctly in public. I had presented it incorrectly before, but the first time I presented this correctly in public was at Rob Kirby's birthday conference in 1998, and Dave Ockley was there, and Dave was talking about a whole gang of results. He was quite interested in the stabilization results, you know, proving that things are stably de-fumorphic. So we got to talking, and we agreed on the principle that one ought to be able to do this for higher homotopies, and how should you do it? Well, basically, the idea was if you had, so remember, all this depends on these two-two spheres, so over here is the X sphere, over here is the Y sphere, so suppose you added maybe another S2 cross S2, and suppose you could prove that the X sphere and the Y sphere became isotopic, well, that would give you an isotope between the X diffumorphism and the Y diffumorphism, so that would give you probably an interesting path from one thing to another. So that was an idea. So we thought, well, of course that's nice to have paths, but if you wanted to show Pi 1 of something, you need not a path, but a loop. So we had two problems. One was showing that spheres became stably isotopic, and the other was showing, and then coming up with something maybe a little more clever than just taking these paths, taking the path in such a way, cooking up a path that closed up so we would actually get a loop. So anyway, so let me state the theorem, so at least I've stated a theorem by the end of the talk. So this is sort of a theorem that contains, I think, it's not everything in the paper, but it's a lot. So remember, there was this manifold Z before. Z is going to be the first, it's got two indices, and so what are the two indices? There's a K which is going to tell you something about, the ZKs have something to do, I'm going to tell you something about Pi K of their diffeomorphism groups, and then there's an R, and that tells you how big Pi K is. So let's just, we can just read it. So for any K and R greater than or equal to one, we find manifold ZKR and interesting embeddings of two spheres in those manifolds and also three spheres, and all of these groups have Z to the R subgroups and actually many of them have Z to the R summands. So what are those? So the one, sort of the basic example is so Pi K of the diffezero, diffezero meaning the identity component of the, not listening, the identity component of the diffeomorphism group. Or, so that's sort of the basic, the construction is there, we prove that they survive into the homology, so they're non-trivial, so if you think about the homology, the Haravits map from Homotopy to Homology, so they survive as homology elements, and if you think about whatever the universal bundle that would give you elements in Pi K plus one of B diff of the B of diffezero, but those are actually homology elements in B of diffezero. So all those groups, all those things are non-zero and so that's sort of the basic, so the construction is sort of here and the detection is in some sense here. So the point is that you're getting interesting elements, but they're spherical families. If you like, we're building families of four manifold and they're families over a sphere. So that's sort of the constructive part and all those things are distinct in the groups that I mentioned. And while we're at it, we get non-trivial elements in Pi K of the embedding spaces for either the two spheres, sort of like the two spheres I mentioned, or Pi K of the three spheres, again, as I mentioned, and those also survive into homology groups. So the fact that you're getting something in Pi K is also a bit related to Dave's talk because there's all these vibrations relating homotopic groups and so on. It's actually not coming directly out of that but it's sort of morally in that same way. And one of Dave's PhD students, Josh Drouin, actually modified our construction so it works with spheres of self-intersection bigger than one or not equal to plus or minus one. And then the same... So the three manifold then would be the boundary of a tubular neighborhood of that, so some lens spaces. So he gets sort of similar-looking results out of ours. Okay, so I say... Oh yeah, so what are some addenda to this theorem? So as I said, at least for diff zero, you're getting a sum n, a z to the r sum n. And so that sort of follows just by the construction because these are... It's kind of the usual thing. They're coming... They're detected by a homomorphism to something which is a free abelian group and then so any case, it's just some algebra. More, maybe more interesting is that, or more interesting to me, is that all of these things are tri... If you passed... I didn't want to write it all down, but if you passed to the topological carat, in other words, if instead of writing diffeomorphisms, you wrote homeomorphisms, instead of writing embeddings, you wrote topological embeddings, all these things become trivial. And all of them... Sorry, not in the sum n, because the sum n just doesn't have to expand a subgroup, but all the things in the subgroups that we detect become trivial after a single stabilization. So in other words, the families of embeddings become trivial families of embeddings after you add a single s2 cross s2 to those diffeomorphisms, become trivial families of diffeomorphisms after you add a single s2 cross s2. And in fact, that property, sort of that's, as I said, as I tried to suggest earlier, that property is kind of the key to the whole thing. Maybe I'll say one other point about the topological category, you know, for pi zero, in other words, for isotope, there's a very clear criterion for when two homeomorphisms of simply connected manifolds are isotopic, which is that they're so-called pseudoisotopic, which follows from them being homotopic, which follows from just them, you know, inducing the same map in homology. So that's really, oh, it's a very hard theorem, by the way. I don't know, I know very few people who've read the proof of that one, but including me, I have to say, or remember it anyway, but there's no such principle for these higher homotopy groups. So somehow the topological triviality of all these things has to get boosted up along with the interesting constructions. And all of that goes via this stabilization. So let me say a few words about that. So this is just the same picture I had before. Instead of having K and R, I have Z. Well, I used to have Z, but it's the first of all of those. So it would be the Z01. Okay, so again, we have this difumorphism preserving things sort of homologically. And actually, a small point, we could preserve the CP2 bars on the nodes, because we could have done the... I was going to hold my hands over the screen here. That didn't really work. We could find those difumorphisms even without the CP2 bars there. Okay, so in the service of this goal of finding these higher homotopy groups, Dave and I were talking about this problem-stable isotopy of submanifolds, and we worked with Hee Jung Kim and Paul Melvin, and we proved that those spheres... So this is a more general theorem, but in this particular instance, we sort of worked on this. We worked out this one example, because we were interested in it, that the sphere Sx and Sy are smoothly isotopic, excuse me, isotopic if you add an S2 across S2, or if you add a CP2, or just because you threw in some more stuff if you added a copy of M. So that's by a very explicit isotopic. And it follows from what we did is that the difumorphisms you get are therefore isotopic. So let's think about just adding M. So we're going to add all the things. We're going to add a CP2, and I'm going to add S2 cross S2, so everybody's happy. So the construction... So let me at least indicate how you go from Pi 0 to Pi 1. I think I could... Yeah, I think we're okay with that. So one of the consequence of the explicit stable isotopy from the paper with David and Hee Jung and Paul is that in fact, the isotopy between those spheres all takes place in the complement of a torus of self-intersection 0. So remember this manifold started out as being an elliptic surface blown up. So there's lots of torus. You can sort of take a torus and sort of send it off into the corner and it just will stay there the whole time. So that's extremely handy for any number of reasons, but in particular it's handy for this. So what we're going to do is just take two of those. Do I have a picture? Oh, here's a picture. Take two copies of this Z, or the Z01, and do a five or so. So remove a neighborhood of T2 cross D2 and glue them in some obvious way. So this is artist's conception. Anyway, it's not really getting flipped over, but that's the best I could do here. Okay, so you glue a couple of them together and so let's just look at this sort of final picture. So notice that that's got two copies of this M in it. There's sort of the left one in this picture. Again, left and right don't really make much sense here. And then there's the right copy of M and the right copy of M has in it some two spheres. It has some copies of C, it has some sum ends of Cp2 or S2 cross S2. So I can use the right copy of M to kind of clean up the spheres in the left copy of that. Okay, so in other words, what does that mean? That means in this picture, there's an isotope between those spheres and therefore if I go down here, this diffeomorphism that's sort of supported on the left part is after I add on this other part, the second copy of Z, I've made the whole thing a lot bigger by the way, of course, that's isotopic to the identity on this manifold C11. Okay, so that was sort of the scheme I was saying where I said, okay, let's add an S2 cross S2, but we added more. And that's, again, it's sort of technical thing that comes out of the computation. So let me call F the isotope that you get in this fashion. Okay, so again, it's, we sort of want to use the explicit one that we used. Okay, so here's the, so this is the gen, I'm suggesting a general sort of inductive thing. So imagine, you know, think of a sphere in this way. So, you know, you take a sphere across I, it's basically I'm making the reduced suspension. You know, you add a cone point, cone point, and then you collapse and arc. And if you think about that, just sort of, you know, so again, we have sort of two variables. We have an isotope F sub t, and then you just sort of, in each t, you make this commutator. Okay, so literally commutator in terms of, you know, as commutators of diffeomorphisms. Anyway, what you see, that's actually a loop, just because, you know, sort of up at the end, various things commute in one end, you have an identity at the other end, the things you have commute. You're getting, if you like, you're getting a path from the identity, an interesting path from the identity to itself. So it's the same thing I said earlier. It's just that with this kind of tweak to make it into a loop. And then, well, so then you could kind of imagine the scheme. So, okay, we have to discuss in a second, a very short second, you know, how you're going to compute the invariant, what kind of invariant might detect this, but you can imagine a similar sort of scheme. Okay, you add another copy of this z, which among other things adds a copy of this sum ends of Cp2 and S2 cross S2. That, using a similar device, which I didn't tell you, would then create a contraction of this loop. And by a very similar commutator construction, that would build you an element in Pi2. And then, well, that one would be stably trivial. That would build you a contraction. The contraction using this commutator construction builds you something in Pi3 and so on. And so that's the construction. That's sort of making the Ks bigger, making the Rs bigger. It's just kind of like, you know, we're supposed to be knowledge about four manifolds. You just add some more junk on and make more basic classes, so to speak. So I won't do all that. So let me say in the last few seconds, so how do you detect all these things? Well, so here's just sort of a step-by-step. So, you know, as I said, Donaldson taught us how to compute invariants of manifold. So that was in terms of a generic metric. You count points in some modularized space. You organize it so that it has virtual dimension zero for turbid, so it really is a zero manifold oriented. Count them. Okay, what about one parameter invariance? So to detect Pi0, so this again is old work of mine. You have a diffeomorphism. You choose a generic metric. You pull it back, connect them by a path, and you've organized things. It's not clear necessarily until you do the arithmetic that you can form a one-parameter modularized space, which would be a zero-dimensional object. You can count points in that, and that's an invariant of the isotopy class of your diffeomorphism. So that's what I cooked up years ago, and it was again clear at the time, but I didn't have no idea how to calculate it. You could do something similar if you have a case sphere's worth of diffeomorphisms of some manifold. This is back, X is back to being any old manifold with the right homological invariant. So that, so again, what do you do? You have a case sphere's worth of diffeomorphisms. You pull back a case sphere's worth of, you get a case sphere's worth of metrics, but the space of metrics is contractible. So you fill that in with a k plus one ball, and you count points in a k plus one parameter modularized space, and that's an invariant. And then the final, let me just say the last theorem, well the theorem is again an inductive calculation using this particular alpha and that particular commutator construction is that this one parameter, well, so this was kind of my original calculation said in slightly different language, is that the Donaldson invariant for the original diffeomorphism is the difference between the Donaldson invariant of X and Y, and in general, if you take, at the case stage, you take sort of the case family built in this way, the Donaldson invariant for that is twice the Donaldson invariant for the one at the previous stage. So I don't know, yeah, let me just skip all that and go to the last thing. So again, this is straight out of the paper I wrote and I guess published in 98. So it was the sole acknowledgement and really this is sort of, I'd like to thank myself for writing a good acknowledgement because it really hits the nail on the head. So Tom listened to this complicated version with me battling and drawing all over the board and just asked the right question. It's quite a gift and thank you. So the question was referred to work of Donaldson and Sullivan who showed that actually the original Donaldson theory would work just fine to show that these non-diffumorphic manifold detected by Donaldson theory, so to speak, are not quasi-conformally, which is weaker than smooth. So I guess in quasi, so they're not quasi-conformally difumorphic. So the question was, does our work show that these difumorphisms are quasi-conformally isotopic? And I guess so, but I don't know. I would assume so, but yeah, I didn't think about it. Yeah, it seems, yeah, yeah, yeah. And I suppose so there's, I don't know, is there a category of quasi-conformal embeddings and all that, so yeah. I would suppose so, but I don't know. It's an interesting question, I don't know. Oh, so yeah, well, so Tom makes a comment that that paper, which is an intimidating paper, sophisticated paper to say the least, is not necessarily known to work for like the full glory of Donaldson invariance, which involves you know, it gives you polynomials and so on. So there's a whole bunch of other technical stuff you'd have to get to work. That I should say that the Pi zero version of this works to give polynomial invariance, the higher homotopy version, I don't know how to do the polynomial. I don't know how to do the version with cutting down modulate spaces and stuff like that, just because you'd sort of, anyway, you'd have to study families of cutting things down. And actually that's, it's problematic, I don't want to try to explain it, but it's, it's conceptually problematic as well as I have no idea how to do it, so, yeah. So the question is what's the behavior of these things under adding CP2 bar? So not the blow-up which I said or the anti-hologmorphic blow-up kills things. It's presumably the case that the actual kind of blow-up preserves all these invariants. Yeah, I'm pretty sure that's true. We're just trying to get the damn thing done, so we haven't done things like that. I should say, by the way, it's a perfectly reasonable thing to ask and related to the point about these polynomial invariants. For instance, if we could get all the polynomial things to work, we could probably prove that you could take the K to be, sorry, the R to be infinity. So it's a good question. All these things should have Z and stuff like that. I don't know how to prove it. Yeah, so this is very much, so for instance, one of the reasons I was keen to prove results about the homology of B-diff or B-diff zero, so our results are about B-diff zero, homological stability, so there's a whole theme about, it's not just one of these weirdo four-manifold things to study. In some dimensions, they study diffeomorphism groups and classifying spaces and families of manifolds and there's very strong patterns of how things behave under connect sum with, in that case it would be Sn cross Sn in even dimensions. Sorry, this is, so I wanted to sort of say, well, you know, the four-manifold thing is really different. We don't have homological stability or we have the opposite of homological stability because things die for the group dif-zero, but that doesn't quite match what the high-dimensional people are doing. Yeah, but the invariance, you really need the dif-zero, or I don't know. Anyway, it's a little bit, I'm not quite sure how this fits with the homological stability of high-dimension. Well, so the question is, so the question is, thank you for making that observation because I was supposed to say it, which is that all these manifolds are about as trivial as can get, so there's sort of, maybe not the manifolds one wants to study. By the time you're done with these constructions, you have, first off, they have trivial, Donaldson, Cyberg, any kind of invariance. They're connected sums of CP2s and CP2 bars, so they're not exciting manifolds from any point of view. So there are lots of, there's lots of work by the assortment of people, Baralia, Kono, Smirnoff, I can't name everybody off the top of my head, although we have a lot of references in the papers that construct not so much Pi-0, but elements in say Pi-1 and one of the diffumorphism groups are various interesting manifolds, like algebraic surfaces. As far as I know, like even Pi-1 of diff of S2 cross S2 in the paper of Smirnoff, Gleb Smirnoff, very nice paper, but those diffum, those families are actually non-trivial, sort of homotopically. In other words, those are, the bundles, if you like, are not even, are not trivial. They're not topologically trivial, for instance. So the problem of finding, so there are interesting examples and of course, as Dave alluded to this morning, there was the work of Watanabe who does this on S4, which is sort of, you know, in some sense like really what you want to be doing. None of this says anything about all that, but yeah, I don't know how to make our method work for manifolds with non-trivial invariants. And there is work of other people that does this, well, there's the work of Tom and Peter which, you know, does K3 connect some K3 and the work of Jean-Pierre and Ling, which proves that that element is actually stable under connect some with S2 cross S2. Sort of conjecture that one stabilization is enough is not true for diffeomorphism which caught me entirely by surprise. So another great problem not really connected to anything I said today is, you know, how many so it's known that some number of S2 cross S2 would do that, that's the theorem of Quinn. So yeah, there's lots of interesting questions about all this.