 Thank you very much for the invitation. So we just heard the title and we'll talk about the action of the diffeomorphism group of a manifold in the space of metrics of positive scalar curvature. So let me begin by fixing some notation. So we talk about compact manifolds with boundary often. If we have on the boundary a fixed metric of positive scalar curvature called a curly R plus Mg, this is space of Riemannian of PSC metrics, which near the boundary have like a color form, the form g plus dt squared. And if M is closed, I just omit the subscript g. So that's my notation for the space of metrics of positive scalar curvature. And then we have the diffeomorphism group of M, the group of all diffeomorphisms denoted dif M and denoted dif delta N. By that, I mean those diffeomorphisms, which are the identity at the boundary and a little bit over the boundary. So you have an action that's very obvious that if your morphism group acts on the space of positive scalar curvature, fixed by pullback. And from this action, you get two things. You could say, okay, look at the space of, so each diffeomorphism F gives you a pullback map F star. And let me denote by H odd of R plus Mg. Johanna, we don't see your slides. You don't see the slides? Yeah. I see the slide. Oh, you see the slide. Okay, so then it's my problem. That's it. Maybe, is it a problem? Okay, sorry. I see it. Maybe you need to pin it. So this, can you see that? Yes. Some of these are not for me, but well, let me figure out. Yeah. Okay, so H odd of a space X, that's the space of homotopy equivalence from X to X. That's not a topological group. It's something a little weaker. It's a topological monoid. So you have this map sigma coming from diffeomorphisms going to the homotopy automorphisms. And also you could look at a metric H in R plus M and look at the orbit map from the diffeomorphism move to the space of metrics. So that's a matter of notation. And so that's of course, we have heard it in Bernard's talk. One idea, so people asked, does this space R plus M have non-trivial topology? And one idea of course could be you take an element in the homotopy group of dif M and let it act. So look at the action map and you get a different and get an element in the homotopy groups of the space of PSE metrics. And then your house is non-trivial. So there we know we have, if M is spin, we have all these tools from index theory, this index difference map. And you could ask, do you get a non-trivial element here? So that was an idea by Nigel Hitchin, the long go and he proved some results for spheres and then later other people proved results with that. But that has been difficult. So you learned in Bernard's talk, together with Schick and Stimler, that's exactly, they did exactly that. They constructed for certain manifolds, constructed such elements in the homotopy of dif M and prove that by index theoretic means that the resulting homotopy class in R plus M is non-trivial. And I want to talk about the phenomenon that's slightly strange. The phenomenon is that this action map is often fairly close to a trivial map. So this explains why it was so, why it was so difficult to implement this strategy. It's because very often, and sometimes it's even a theorem, but that sometimes this action map is actually the trivial map. So that's an insight that we had. And this is what I call a rigidity phenomenon. And I want to start with some examples of this rigidity phenomenon. And they all come from the surgery theorem. And the first example is, was also already mentioned by Bernard. So you are, okay, let's try to implement the strategy. What's the first manifold you can think about? You think about the disk. And not dimensional disk, there we have a classical theorem that calculates the homotopic groups of the diffeomorphism group of a disk in low degrees. And the answer is slightly surprising. You get in every fourth degree, you get a copy of Q, if you're tensile with Q. And for an even dimensional disk, you get actually zero. So that's a very classical result where Farrell Schoen, so it's almost 50 years old. And then Bernard together with Boris and Thomas Schick and Mark Walsh, they asked, do you get non-trivial elements in the homotopy of the space of PSC metrics on the disk? We're here, I mean, I mean those PSC metrics which near the boundary are just the wrong metric. And what they proved is, well, this map is rationally zero. That's bad news in a sense, yeah? So you don't get for a disk, for a disk you can't see that the space R plus is not contracted by. So that was the, maybe that's the first instance of this phenomenon. So that's the second thing. So let me talk about spin manifolds and it turns out, it turns out that the theorems are not about the group, the action of the diffeomorphism group, but about the action of what is known as a spin diffeomorphism group. So the spin diffeomorphism group, it's not only those diffeomorphisms which preserve the spin structure, there will be a finite index subgroup of the diffeomorphism group. Instead, you have an extension of this by Z mod two. It's because a spin structure has an automorphism of order two, so-called spin flip. So it's not quite a subgroup, but it's very close to it. And this is a group we will be talking about. So what we discovered some years ago is a following phenomenon. If you take a spin manifold of dimension at least six, which is simply connected and spin mild-bordant, and then you have the following result. You take two spin diffeomorphism manifolds, you look at the pullback maps on R plus M, and we found out that these pullback maps commute to homotopy. So why is this remarkable? It's remarkable. By slight reformulation, we see why it's remarkable. So this action map, it starts from pi zero of the spin diffeomorphism group, that's so-called spin mapping class group, and goes to pi zero of the space of homotopy automorphism of R plus M. And the result says it's a group homomorphism, it's homotopy automorphism, that's the pi zero of it is a group. And it factors through the abelianization of the mapping class group. If you take a commutator of two diffeomorphisms, then it acts trivially. And this is remarkable because this group, this is a highly non-abelian group. We know a lot about it. We know that it's an arithmetic group by a theorem of Dennis Sullivan, but it's complicated, it's big. And this one is completely mysterious. So we don't know the homotopy type of R plus M. And even if we knew the homotopy type, we don't know what the group of homotopy classes of homotopy self-equipment should be. It's a completely mysterious object. But this homomorphism, it factors through an abelian group. And this abelian group is often small. So one example that played a role in our work was you take S3 cross S3. And then you take the connected sum of many copies of it. And if you take enough connected, enough copies, then it turns out that this abelianization of the spin mapping class group is actually the trivially group. So it tells you for this manifold M, whatever diffeomorphism you take, whatever spin diffeomorphism you take, the pullback map on R plus M is a homotopic to the identity. So that's the phenomenon, that's something we discovered. It was on the way of the proof of the theorem that this index difference map is subjective. This index difference map to KO theory is subjective. So now if you know it's an abelian group, it factors through an abelian group, you ask what abelian group could it be? And they have the following construction you take. If you have a diffeomorphism of a closed manifold, you look at a cylinder and you can glue the two ends of the cylinder using this diffeomorphism F. And then you get a manifold of one dimension higher, the mapping tools. And if you take a spin diffeomorphism, this mapping tools has a spin structure. And you can look at the spin cobaltism class of this mapping tools. So we get a map from the spin mapping class group to the spin boredism group in one dimension higher. And it turns out to be a group of morphism. And of course the spin cobaltism group, that's an abelian group. So that's a candidate, that's a candidate. And it indeed it turned out to be two. It was a PhD thesis by Georg Frenk. The main result is so in this under these hypothesis, you have a spin manifold simply connected and often of dimension at least six, then there's action map factors through the spin cobaltism group, the other mapping tools. So this says you take a simply connected spin manifold, you take a spin diffeomorphism. And if the mapping tools of the spin diffeomorphism is null boredom, then the pullback map on R plus M is homotopic to the identity. So that was Georg Frenk. And he has more general versions which work for arbitrary manifolds, but then you have to closely look at tangential structures. And I don't want to talk about it right now. Yeah, so that's just a special case. So you have a more general theorem which works for all closed manifolds. So let me say in a couple of words how he did it. So you have seen in Bernard's talk that if you have a spin cobaltism between two simply connected spin manifolds, then you have to make that spin cobaltism simply connected and then you pick this suitable hand and body decomposition. And this tells you how to obtain the M1 from M0 by a sequence of surgery in high co-dimension. And then each step you get homotopic equivalence from one manifold to the R plus and one manifold to R plus on the next one and the next one and the next one. And what Georg did was to look very closely at this and to find out that the homotopic class of this map does not really depend on the choice of the antibody decomposition. That was previously done by Mark Walsh. But then he also went one step further and it was that this map really depends on the cobaltism class on the cobaltism. It really only depends on the cobaltism class of the cobaltism. And if you know that, then it's not so far to get to the theorem. So it has an interesting special case. The interesting special case are manifolds where you have all rational Pontriagin classes zero. For example, you take a stably parallelizable manifold. And then you look in the old literature on cobaltism groups and you find a theorem that for such manifolds, the mapping tools of any spin diffeomorphism has finite order in the spin bothersome group. Not too hard to prove, it has to do with the multiplicativity of the signature. And, but as a conclusion, you get for such manifolds, for a manifold with trivial rational Pontriagin classes, for any spin diffeomorphism, the action map on R plus M does have finite order. Some power is extravially. And so you see, you see for those manifolds, very difficult to get from diffeomorphisms maps on R plus M, which are non-travelable. So then we have, and this is maybe the main new result I want to talk about is, we have a generalization of georg theorem to higher homotopy groups. So this was about pi zero of the spin diffeomorphism group and pi zero of the space of homotopy self-acquivalence as we could ask, what about higher homotopy groups? And then we have the following result. We need some technical adjustments. So we couldn't get it done for simple connected manifolds. We had to require that the manifolds are two connected. So pi two is also zero. We have the same dimension bound, D at least six. And we could not talk about the whole spin diffeomorphism group. We have to talk about those which fix a certain disc. So you fix it, you take a disc inside M and you look at those diffeomorphism which are the identity on that disc. So you still have a map like this. And what we found out, it factors through a certain abelian group, which I call, or which is pi K plus one of empty spin D. And I want to explain what this is. What is pi K plus one empty spin D. So that's, if you know a little bit about cobaltism theory, you know that Tom identified the cobaltism groups with the homotopy groups of a certain spectrum. And the type of spectrum, which is nowadays known as a Tom spectrum and empty spin D is a certain Tom spectrum. Which was introduced in this specific version of spectra was introduced by Martin Tillmann. And the homotopy groups of a Tom spectrum have always, always have an interpretation in terms of cobaltism groups. And then what the right interpretation is depends on the spectrum and here it's as follows. So it is, you take manifolds of dimension D plus K plus one, which are closed, you take a vector bundle of rank D on the manifold. And you fix an isomorphism of this vector bundle V, or a stable isomorphism of this vector bundle V with the tangent bundle, the N. But it's not an isomorphism of vector bundle. It's only if you stabilize, if you add sufficiently many N and I wrote I infinity to be a bit more weight. Yeah, so that's the type of cobaltism group. And then what is the cobaltism? Yeah, sure, it's a cobaltism of manifolds K plus D plus two manifold. And on this cobaltism, you have a vector bundle also of rank D, which restricts to the given vector bundles on the two ends. And then you have an isomorphism of the same sort of what I said. So that's an abelian group. And we claim, and I didn't write our names, here to the theorem, that's the theorem by Oscar Render Williams myself, proven in 2019. So that's a new theorem I want to talk about. So how do we, let me explain the map A. So you take, we take an element in the homotopy in the homotopy of the diffeomorphism group. And what can we do with that? An element in pi K of the diffeomorphism group that gives me a fiber bundle over a sphere of one dimension higher. Yeah, the dimension of the sphere is one higher than the degree of the homotopy group with fiber M. So what you do is you take two copies of DK plus one times M and you glue it together along these maps from B and X. And this manifold N has a spin structure and it has a vertical tangent bundle. Yeah, you have, that's the kernel of the differential of pi, that's a vector bundle of rank D, dimension of N. And then you'll see, you exactly get an element in this group pi K plus one of MG spin D, exactly in the description here. So this map A generalizes the mapping towards construction to higher homotopy groups of the diffeomorphism group. And that explains the, that explains the theorem. Yeah, that explains the map A. And this map A is not at all an isomorphism. Yeah, it can be often be quite prevalent. So we can, there's one thing. If you look at this homotopy group, you see the manifolds in this co-border in this picture, you have K plus D plus one dimensional spin manifolds. And you just forget that the tangent bundle has a specific form. And by this, you get a map to the spin co-border group. And if K is zero, then the composition, you first apply IA and then this forgetful map, you get the mapping towards construction. So this says that for K equals zero, we get a result which is quite close to Georg's theorem, but not quite. This obvious map is not an isomorphism. If it were an isomorphism, it would just recover Georg's result, but this is a little bit weaker. And yeah, then I already mentioned, we had these strange conditions that the manifold is too connected. We would like to get rid of it, but we don't know how to do it. And also we don't know how to do it for the whole different morphism group and not just those fixing a disk. But this is, I believe, can be done, but not with our proof. It has a consequence. Like homotopy groups of spectra or homotopy groups are difficult to compute, but what is always fairly easy to compute are rationalized homotopy groups. And in this case, you can compute them in terms of characteristic classes like Contralian classes and Euler classes. And they are finally generated a building groups. So it's not so hard to calculate. And one can prove that if the manifold is, doesn't have Pontriagin classes, then this map A is rationally zero. So that's, again, that's very similar to the theorem that the mapping torus of any diffeomorphism of a stably parallelizable manifold has finite order in the group. That's very similar. And one manifold where all rational Pontriagin classes are zero are the spheres. And in this way, you get, we get back the result by Botvinnik and Anke and Walsh and Schick. So that's the corollary. If the manifold doesn't have, doesn't have any non-trivial rational Pontriagin class, then this orbit map has actually finite edge. Even through the pi k of this diffeomorphism group, that's just unknown. I think the only structural thing we know here is a theorem by Alexander Kupas that these are finitely generated groups, that's a billion groups, if k is at least one. So there's almost nothing is known and about pi k of r plus m. We also don't know much, but we know this orbit map has finite image, which I find quite striking and strange. So when I planned this talk, I thought I could explain the proof of the theorem, but I couldn't, somehow it was too long. But I want to give you an impression how such results, so all these results are like of a sort, the action of the diffeomorphism group on the space r plus m is fairly trivial. I just want to say you have 30 more minutes. Yes. Okay. Yes, yes. Okay, just to reassure you, yes, okay. Yes, I still have seven slides. But I want to go into a proof now, but I don't want to go into the proof of the new result. I want to go into the proof of the Abelianness theorem, because there you see, it will be fairly explicit. And there you see exactly how the grammar flaws and surgery theorem comes into play and works behind the proofs of both theorems. This is why I picked this Abelianness theorem as an example. And so let's go through it. So what's the setup? Is we have to simply connect with spin manifold and it's spin null-bordant and high-dimensional and we have two spin diffeomorphisms. And I want to show that the pullback maps of those two commute up to homotopy. So what we do? Okay, let's, so in this M we find two disks. So you remove the two disks and we get a co-borderism from the sphere to the sphere. Well, let's call it W and if you take the union with the two disks, you get back M. So now there's an elementary theorem in differential topology. So if you have a diffeomorphism orientation present, so if you have, so look at it diffeomorphism of M and look what it does with one of those embedded disks and it gives you another embedded disk. And if there's, they have two embedded disks in a connected manifold and a classical theorem, an elementary theorem tells you these two disks are isotopic. Let's see, so we've got disk lemma. And you can use this isotopy of disks, you can use it, you can extend it to an isotopy of the diffeomorphisms. And that tells you after isotopy, after applying an isotopy, you can assume that F zero or that spin diffeomorphism fixes this disk, point wise. You can do it with one disk, but then you can do it with another disk as well because it doesn't change it. So we can assume that our two diffeomorphisms both fix two disks and then you can remove the two disks and you get a diffeomorphism of W, which is the identity on both ends. Now, okay, we have this surgery theorem, now it makes its first appearance. The first appearance is in this, in R plus M, there's a homotopy equivalent to a subspace and it's homotopy equivalent to the subspace of matrix which on both of those two disks are equal to the so-called torpedo metric. So that's torpedo, it's a rotationally symmetric metric. There's the type of metric which shows up in the surgery theorem. By the way, I avoided that word and therefore I introduced the half spheres. Yes, but then you come into trouble with the boundary condition. So I- Not using my theorem with Christiane, I think that's okay. Yes, yes, yes, yes, yes. I mean, the theorem is so surely out of what you formulated, you can deduce what I use here. Yes, I mean, it's the same theorem in slightly different variants, yeah? It's a variant of that. So it tells you the space of matrix which on each of those disks are torpedo metrics that's homotopy equivalent to the whole of R plus M. And then of course, if the metrics are of a standard form on these two disks, you can cut out these two disks. And you see that the space of metrics on the cobaltism W, which on the two ends are G zero, where G zero is the one that's here. So by these two steps, so I firstly achieved that the spindle film often fix the disks and the metrics are of a standard form on these two disks. This means I can restrict two manifolds with boundaries. Yeah, and then let's look at W instead of M. So that was the first step. So I introduced two holes and you see, now how I use these two holes. So let's shorten the notation C means the cylinder on the D minus one sphere and M is the round metric plus the metric, the usual metric in the interval coordinate. Let me write it M. Now, one assumption was that this manifold M is bordered to, is null bordered, spin null bordered. And then it's also a spin bordered to a D sphere. And then you'll see, okay, then you can play a bit around and what you see is you get this. W, which is M minus two disks, you get from a cylinder by surgeries in the interior of the cylinder because that M is co-ordined to SD means you can obtain M from surgeries on starting from SD and you can obtain it by surgeries from that and a little refinement of that would be you obtain W from a cylinder by surgeries. Now I use the surgery theorem for a second time. And it tells me, okay, the space of matrix on the cylinder subject to the boundary condition that you have the round metric on both ends. That's the same as the space of matrix on W with the same boundary condition. And the theorem says there is a homotopy equivalence. You should not try to, you should not try to think there's a map of spaces which takes one point in the space as an argument and spits out the point here. So it's just a map, it's only defined up to homotopy but what you can always do is it gives you a well-defined bijection on pi zero components. And this is what I do. Yeah, I take a metric H zero on this R plus W which corresponds to the cylinder metric on R plus C under this homotopy equivalence, just this plate. So that's a metric H naught that I constructed on R plus. Now this has an important property. Yeah, what property does it have? It's what we call right stable and left stable. So let me explain what these words mean. So you could, so if you have a metric on a cobaltism, this makes sense in full generality but let me, I only wrote it from the situation here. So if you have a metric on a cobaltism W, you can look at any other, any cobaltism V which goes from one end of W to some other manifold. And then I can talk about a PSC metric on the other end call it G1, an arbitrary PSC metric on this other end of V. And I get a gluing map from metrics on V to metrics on W union V and it just takes a union of H with H naught. And the property that this map has is that it's a homotopy equivalence. This just follows from the surgery theorem in a couple of lines. So it's a homotopy equivalence. This is a property that we call right stable. Yeah, so why is it right stable? So we had to get to distinguish these two sides. So the metric on W is so that whenever you glue a cobaltism V on the right end of the cobaltism you get a homotopy equivalence of this sort. So in this situation, you can also play the game with the other end and it has the same property that's called left stable. So if you glue it the other end, it's left stable. So another metric which is both left and right stable is that the metric and the cylinder metric. And by the way, this notion is we develop this notion and this paper and we proof later to versions of the surgery theorem, which I just tell you it's, if you have a cobaltism from one end M0 to M1 and if the inclusion from M1 into the cobaltism is too connected, too connected means you have an I, you have bijections on pi zero and pi one and you have a surjective map with pi two. And then you can prescribe a PSC metric on one end of the cobaltism. Then you get from the theorem, you get a PSC metric on the other end and a right stable metric in between. That's one theorem. And the other theorem is if both, if the inclusions of both boundaries are too connected, then right stable is equivalent to being left stable. So, but that was only an aside. So we have, so I constructed this metric H0 and I found out it has this property. It's right stable and it's also left stable. So let's go on. And this, now I can look at a spindiff heomorphism, F0 and I can pull back the metric H0 with F0. And I take the union with the round metric and because this H0 is left stable, I can, I find out that this metric, is in the same path component of the space of PSC metrics as a metric MF0, union H0 and MF0 is a certain metric that I just constructed. So that's what we get from H0 being left stable. And now I do what I need to apologize like, I make a diagonal trace. I have W and I glue cylinders to both ends. So here I have, here I glue the C to the right end and here as well and here as well and I use the metric M, the cylinder metric to do it. Here I have the pullback with F0. Here's also the pullback with F0 but I extend F0 over the cylinder. This is the square as commutative and here I glue this M0 from the left side. This is also a commutative diagram and here I glue the H0 and all maps are homotopy equivalents. So that's clearly a commutative diagram and I want to find out something about this map F0 stuff. But now you look at this diagram and you compute the right hand column. What does the right hand column do? And the right hand composition, what it does is I take a metric on a cylinder, I glue on H0, I threw it over the manifold with F0 and I glue on this M. But now that's going to be the same as taking the union with M and the union with F0 star H0 because this metric here is in the same path component as this metric. I see that the two maps are homotopy. So that means the right hand column is the same as taking the union with H0 and then taking the union with MF0. But now these maps are all homotopy equivalents and then you see this map F0 star is essentially the same as taking the union with MF0 on the right hand side. That's what I get from this argument. So here's the conclusion. So if I first apply F0 star and then I glue on this cylindrical metric M, this is homotopic to the map that I obtained by gluing on this metric MF0 to one side of the cobalt. But now I want to talk about two different morphisms and the other one is F1 and I played exactly the same game but I used the other end. So it says F0 star is given by taking the union with MF0 from the left and F1 star is given by taking the union with MF1 from the right. But if you take the union from the left or from the right, it really doesn't matter in which order you do it. And this shows that the two maps are commute up to homotopy. So that's the proof of this ability in that theorem. So you see, I presented it because it really shows you how the surgery theorem can be used to manipulate the action of the diffeomorphism group and to obtain interesting results. Johannes may ask something. So my intuition is that, of course, if you apply these two diffeomorphisms, one after the other, each of those effects, basically the whole manifold, more or less. And so if you pull back the metrics, it's completely unclear why these two actions should commute up to homotopy, but why is it possible? So is there a kind of a rule of thumb explanation or giving an intuition why it's possible to still separate the kind of the regions where the positive scalar curvature is affected by the... I mean, that's a surgery theorem. In a sense, it's a set. I mean, the real intuition is, what really happens is here, this step. This H naught is right stable. So you see all of the topology of R plus W, I mean, W union C, it's essentially W, yep, it's diffeomorph to W, but it tells you all of the topology of R plus W is somehow concentrated on this cylinder around the boundary which is not affected by F at all. That's how, this is really the intuition. The intuition is it's all the complicated topology of R plus M, it can all be compressed into little disks. That's the reason. But we don't get, I mean, we don't get that the action is trivial. You don't, this doesn't follow from that. And it's also, it's not true, yeah. The action is non-trivial. One can concoct examples of manifolds where this is non-trivial. But it's somehow, this is the best you, the most you can trivialize. So it commutes up to homotopy. So it factors for the abelianization. And maybe I should say, maybe I should say that you should say this explicitly. This map A, which goes from pi naught of the spin diffeomorphism group to pi one of empty spin D. This map is some, for some, for some even dimensional manifolds. But it is actually the same as the map as the abelianization. So that's really something one can show. Galaxius Rendeweim really proved this. It's part of their big theorem. For some even dimensional manifolds, this map is actually the abelianization. But that's somehow, you can't make the action completely trivial, but the fact that you can compress, that you can compress all the non-trivial topology into disks makes it possible to prove such a thing. You can't prove more, but this is exactly what you get. So we can also, then later, we also showed it for manifolds with arbitrary fundamental groups. And then the role of the disk is then played by somehow the unions of all the two handles in handle body decomposition and things like that. It's a very similar result there. Yeah, this is a basic argument. The basic argument is such an Eckman-Hilton trick. It's very similar to the classical Eckman-Hilton argument which proves that higher homotopy groups of spaces are commutative. And that's not very much, not different. I could also, no, maybe I shouldn't say this. No, no, that's not true. Okay, let's go on. So you see that the surgery theorem really allows us to prove things about the differential morphism action. Let me also say that this is a key lemma in this work with botvinic and random variance because it allowed us to use knowledge about differential morphism groups to conclude finally things about R plus of the sphere. So it also tells you, this is also a theorem which tells you that this action of the differential morphism group, it's not just a random action of some topological group on some topological space. There's very distinctive, very interesting and rigid features. So that's, I found this very interesting when we, and I still find it mysterious. So I promised to talk about a little bit about the higher dimensional version for higher homotopy groups. And then we don't do it in a way that we look at homotopic groups. We look instead at a certain vibration. So there's a standard construction in algebraic topology. If you have some topological group which acts on a space X, you can form a fiber sequence with the base space. It's a classifying space and the fiber is X and the total space, that's the so-called homotopic quotient. Yeah, X, yeah, it's EG cross over G with X. And for such a fiber sequence, you get a long except homotopy sequence and it has a connecting homomorphism. The homotopic groups of BG are the homotopic groups of G shifted in degree by one. And then you apply the homomorphism and the map is exactly, I mean, things are exactly done in such a way that the connecting homomorphism is exactly this orbit map. Yeah, that's exactly true. When you want to say something about the orbit map, it might be better to say something about this vibration and to see, for example, why is the connecting homomorphism this vibration? Yeah, so the theorem that we actually prove makes a statement about this vibration. So here we need to talk about the group Diffen that's a whole of a big group from a group theoretic perspective, but it has a nice geometric model. So it's like, you know, what is BOD for the orthogonal group? It's this, you know, it's a Grasman manifold. The Grasman manifold is the space of all linear subspace of r infinity of say, dimension D. And here, Bdiffen that can really be modded as a space of all submanifolds of r infinity, which happen to differ morphic to M with a, of course, that needs to, you need topology to make sense out of it. But that's a nice model. So you can really think about points of Bdiffen as some manifolds, which are differ morphic to M, but not equal to M. And similarly, this homotopic quotient has a nice description like you take such a submanifold which is differ morphic to M, and you take a PSC metric on it. Yeah, that's the, there's homotopic quotient. And that's also what one knows as a modular space of PSC metrics. It's very close to what Bernard calls the observer modular space. It's the only difference is I apply this usual trick. So the usual trick here is the action of Diffen on R plus M doesn't need to be free, but you make it free by introducing this extra EG, this contractable free G space. And then you get, then you get this. So that's a modular space very close to this observer modular space. Now let's, let's return to the situation where you have a general group acting on a general space. And you look at this vibration that I just introduced from the homotopic quotient to the class frame space. And now I suppose that I can induce this vibration from another vibration over some space set. Yeah, suppose I have a vibration over some space set and I want a commutative diagram which is homotopic Cartesian. And that's, what does it mean? Homotopic Cartesian means the following. It means the following if you turn this maps Q and P into vibrations, you can always, there's a standard way of doing this. Then you get an induced map on the fibers from Q, fibers of Q to the fibers of P. And you want that this map is always a V-comotopic equivalence. So that's the meaning of being homotopic Cartesian. It's the same as saying that the, that the vibration Q is induced from a fluctuation P over some space set. So assume that this, and then you, then you make a little diagram trace and what you see the action map, the action map is from the homotopy of G. It's the same as the homotopy of BG, but drifted in degrees. You take connecting homomorphism and you make a little diagram trace. The fibers are the same. And you see that this connecting homomorphism factors over F. And that's exactly the theorem of the sort we want to prove. Yeah, that's exactly how we do it. So instead of looking at individual homomorphisms or individual homotopy classes of diff M, you look out for such a vibration over some space set. And that's what we want to do in this. And you can play a similar game. So we have two versions, one for the orbit map, the other one goes to this, that looks more fancy. So the homotopy automorphism, but you can get the same deck. Yeah, so they both of these maps factor through the homotopy groups of set. And then what we do here, okay, we look at the action of B diff M on R plus M and we do such a diagram. So what are these spaces there? Explain the left-hand column and the right-hand column is looks very scary. The infinite loop space of the spectrum. That's a very big thing. Its features are the homotopy groups of this looks infinity empty spin B are exactly the homotopy groups of the spectrum that I just talked about half an hour ago or 20 minutes ago. And then there's a map alpha. And this really, so I should say what are the hypothesis? The hypothesis are that W is one connected and spin and the boundary is also simply connected. And those hypothesis you get such a map alpha. And that's a very, that's a very important map has been, this was introduced by Eben Watson and Michael Weiss 20 years ago and Ulrike Tillmann, they did it. Yeah, these three people introduced those maps. And there are many firms saying when this map is some sort of equivalence, it can't be a homotopy equivalence because the right-hand side is a loop space and it has a billion fundamental group whereas the left-hand side, the fundamental group that's pi naught of B this delta W and that's, as I said, a big non-nabirian group. But sometimes for some manifolds, it's known to be a homological equivalence in small degrees. And it's, I gave you one example where the statement is literally true. It's if you take connected some of many copies of S3 cross S3. You said something is wrong with your microphone. I don't know. Is it microphone? Yeah, I mean, we can hear you but it's sometimes distorted. If he stays a little back, it's not distorted. Okay, yeah. Okay, so this map, these are really well studied and have been well studied in the last 20 years. So it's not a random construction, not something we came up with. But it's some standard thing. It's like a, if you know the comparison of cobaltism groups with homotropic groups of the term spectrum, it comes via this so-called Pontriagin-Turm construction and it's exactly the same in a parametrized way. And that's how these maps alpha come into life. So that, an important thing here is that this infinite loop space is the classifying space of the spin cobaltism group category. So what is the cobaltism category? The topological category, objects are close D minus one manifolds, morphisms are D dimensional cobaltisms, both have spin structures and then you take the classifying space. It's a general construction homotopy theory and then Galatius-Machin-Turman-Weis proved that this classifying space is homotopy equivalent to loops infinity, I should say minus one to be correct. Loops infinity minus one of empty spin D. So that's it. So this space here is a classifying space of the cobaltism category of spin manifolds. And then the idea was, what is this mass cal X? It is a classifying space of the cobaltism category of manifolds except the metrics of positive scalar parameter. It's not exactly that, it's a variant and then we get a diagram and then it's quite a bit of work to show that this is homotopy cartesian. There we need at several places, we need the surgery theorem in a very essential way to do it. So that's how it goes there. It's really the use of the surgery theorem here is really hidden in the construction of X or rather in the proof that this is homotopy cartesian. But I think it's a good time to stop here. I hope I convinced you that the surgery theorem can be used to prove powerful results about the different morphism action. Okay, I think I stop here.