Pt. 3 – Notions of equivalence & complexity for knotted surfaces | Dave Auckly, Kansas St U | PCMI

Next Wednesday, we'll do a hike. And some people might want to know about the hike. And so this is actually a map of the trail for the hike along with a map of our um talk. I think when I get overly caffeinated, I can go a little fast and that may have happened last time. Um so, you know, yeah, this is the start of one of my talks. Uh, we've seen that joke before, but we ought to slow it down. Do I have notes? I concentrated on making the slides. I really take your request seriously. I had some family issues this year and I will get the notes done. I have an hour after this talk to get the problems done before the problem session, but no notes for now. Um, so let's see. Last time there were a few points that I wanted to get across and I think I went a little fast and obscured some by getting too many details. So the first point is just tubes rock. If you internal the fact that internalize the fact that tubes rock, that can help you with lots of geometric constructions and seeing that things are equal in four manifolds. Another thing you'll see is that I kind of like overviews. It's kind of the way my brain works as little icons and pictures. And I'm getting uh our middle volunteer pointing to tubes rock. Let's see. Jesse rocks we can get and Sophie. And then another that we had and I could have probably spent the entire second lecture just talking about these standard tools without doing an application but standard tools in for manifold topology include finger moves which is a way of moving intersections around. They often produce new intersections but okay. And then Whitney moves also move intersections around but can also remove intersections. And then there's sort of a weird weird one the boundary roll. If you have a surface in a four manifold with boundary you can roll the surface around the boundary changing the um framing. And these are things you should play with. There's problems in the notes. There'll be more problems in the notes. Um, to keep me from being a caffeinated crazy, you guys need to ask questions and make sure we're on track. So, cool. There's a little bit more that was implicitly in there last time. So, um, one of the things that we saw is that, um, I actually, uh, presented two theorems. One is one from 2015 um that Inanch and Nathan proved showing that when you internally stabilize um that they become equivalent after enough. I gave an alternate well I did this for the examples in sort of the mark one paper. The second thing I did is I did an updated version of the scram. One is enough. Of course, I'm taking the initials of my colleagues and myself when I collaborate to try to make things sound like things. In particular, there were really two theorems. There was a 2017 one for submanifolds and there was a new one for um embeddings where you have the maps that's uh getting published in these notes the first for the first time. And so you have copies, but it's not on archive. Um, then the other thing that I just used without talking about it is a whole slew of different ways of representing things in fourdimensional space. There's little icons underneath the words that actually represent it, which probably only my pointers can help see these things. I know that my poor eyes don't work at this distance, but if I walk back, I can go read the monitor. And so, um, what we see is we've got finger moves, curvy diagrams, oh, I guess movies, I said then, whoops, and then curvy diagrams. You can draw a real analog is a very useful thing. And so in complex geometry they often look at things you know parameterized curves which when you complex parameterizes a surface and you put it in two dimensions. Two complex dimensions is four real [sighs] dimensions but just drawing the real picture is very very helpful. It shows you intersection patterns. Um you can push stuff into three dimensions like with the mark surfaces. You can um do three sections we haven't talked about. You can also use dimensional reduction and in a different ah yeah the first so with dimensional reduction a very typical one is I might draw a picture of a um say circles inside a surface or zero spheres inside a surface to rec to represent something like two spheres inside a four manifold. So one of the pictures I had that Yeah. And so for instance, we had um the picture of a surface of genus 2 next to a Taurus and we put a couple dots, one on each component. Um so the picture you're pointing out here, so um Mea is pointing out some of my blue pictures. All of my blue pictures were on the lines of the real pictures. I was drawing a bunch of curves in the plane that had intersection patterns and the intersection patterns of those curves matched the intersection patterns of surfaces in a four manifold. The picture when I'm talking about dimensional reduction is a fancy it looks like glass has a surface of genus 2 a surface of genus one and a dot on each piece from a prior lecture and in the lecture notes and you can't connect them but when you add a handle you can and that's analogous to uh four manifold where you attach a two handle to save some uh fundamental group that allows surfaces to go together and And then so thanks for the question. And then um the uh next thing you can look at is um ways of talking about four manifolds that don't need use pictures. And so you could use jets tailaylor series polomials. You can uh use algebraic geometry is a great guiding principle in four manifolds. I mean, there's a time when people conjectured that essentially all you had was algebraic varieties connected some together. That's certainly in my lifetime. Um, and branch covers Maggie mentioned is another technique. So, that's kind of a review of how I've been a little bit more leisurely and a little less caffeinated. What the second lecture was about. I hope that's helpful. questions about it before we jump to get um well actually more things to remember. It's good to remember maybe take a backpack, maybe a hat, some sunscreen when you're hiking. And so when you're giving a talk, you also need to remember things. And so when you're presenting new results, you should uh mention who your collaborators are. And so in this case um market is the uh name I make out of Hakudo on above Danny Masaki and myself and so we'll see a number of new results by this collaboration in this particular talk. And so now like many of my talks we start with a highlevel overview. And in fact, um, we get a high level overview if we're looking at, um, going up to, uh, Clayton Peak. And so the view from Clayton Peak, if I had a little bit more time, I would have stuck one here, but this particular image in the thing is on top of Clayton Peak. Um, and what it's showing you is in the lower left it shows a Taurus with a zero handle, so a disc, couple neighborhoods of a couple arcs. It's just the standard CW picture of a Taurus thickened up, which is a handle decomposition of it. And this is an example of a dimensional reduction. And so if you do the same thing instead of in two dimensions in four dimensions, you would start with the four disk and you would attach on [snorts] instead of couple one handles, a couple two handles. And just like when you go around the boundary of the Taurus, you see a dot from the first handle, a dot from the second, a dot from the first, a dot from the second. So those one handles link. When you do the thing up a dimension, the two handles link and you actually see a pair of linking two handles. You see two of them. You see the um aqua and blue for A and B on the left that's kind of deformed, but that really is a standard hop link. And so that is a copy of S2 times S2 that maybe is easier to see if you look at the Cerelia and daisy. So the pink and orange uh for the C and D colors that really is a hoff link. And so what I've got there is a picture of S2, two copies of S2* S2. And so before showing you the next picture, let me tell you some things that are easy and things that are hard to give you a sense of what's new about some of the results in today's talk. Um, when people first started constructing exotic for manifolds, exotic for manifolds were rather big things. You had to have a lot of stuff going on in order to wiggle around and get the stuff that was crazy. And it was really part of a project to get things that were exotic that were much smaller. Okay. So one of the things we're going to do is we're going to construct exotic pairs of surfaces and really subtly exotic surfaces and just ways you know well things that if you wouldn't have thought of you say yeah it's kind of same but as soon as you think of the definition for manifolds if it can be weird it's weird. Um and so if you think about um the way many exotic surfaces are constructed, one standard way would be take a manifold that has non-trivial cyberwit invariance and do a surgery operation on it. Oh wait, you've got a sum of two copies of S2 times S2. That's not going to work because there's a vanishing theorem. All gauge theory and variance die under that puppy. Another thing you might do is take a pair of exotic manifolds and stabilize them. Add an extra S2* S2. I promise you if I had an exotic S2* S2, that's what I would be speaking about and the audience would probably be a little bit larger. But um this particular surfaces are going to be in here. So this is really very a very very small manifold to consider when you're getting exotic surfaces and there's going to be lots of them that have lots of properties in it that are kind of surprising. And another very very cool is we have very explicit pictures. We can draw banded unlink pictures. The four manifold is lumpy the way it is because you see the four manifold um in the picture in the upper left and in the picture in the lower right. Okay. And these are banded unlink diagrams just like Maggie's been talking about of surfaces sitting in two copies of S2* S2. And so you can do the exercise of remove all the four manifold handles and all of the bands and see that that's an unlink and then you can serger all of the bands and see that all of the components will be parallel to the component the two handles in the for manifold so that they can cap off to be discs and so that this really is a pair of closed surfaces. And if you were just changing things by a diffomorphism, there is the surreal daisy, you know, the all right, pink and orange uh hop link up in the upper right. I could just turn that thing over and that gives me a diffomorphism. But when I'm talking about surfaces being isotopic, um I can't change the manifold by a diffomorphism. And so you see how you get from the surface on the upper left to the surface on the o right is I'm just you know flipping part of that for manifold. What I'm doing is I'm applying a diffomorphism in the for manifold and you could compute the action of the diffomorphism we're doing. The diffomorphism I'm doing is not a teriomorphism. It's acting in a non-trivial way on the homology because it's flipping the pink and the orange components over. So, it's changing their sign. That should hopefully be pretty visible and pretty u visual in this picture. Um, however, if you follow those curves with your fingers, you'll see that each time a curve wraps around the orange curve or another copy of it d goes around in the other direction. And same with the pink, which means homologically the surface doesn't see that. And so these two surfaces are in the same homology class. And in fact, they have a dual and or primitive ordinary. So by scram one is enough. they become equivalent after one external stabilization. You can prove that they're topologically isotopic, which actually takes work, but it's true. Um, and so they're smoothly equivalent. Uh, they're topologically isotopic. We're going to prove that they are not trilli equivalent, which takes a brand new tool and argument. And even more than this, when you start talking about stabilization, we had a whole list of modification methods for surfaces. Um, all of the internal methods, the finger moves, the ambient one-handle surgeries. Um, you can get examples of these surfaces that are separated by large. You tell me a number like 10,000 and I'll give you two of these surfaces that are separated by 10,000 internal or ambient one handle stabilizations. Another comment is we can make these surfaces with um almost any self intersection any genus any type any divisibility. the type, the divisibility and the self-intersection exactly classify the uh homology classes of surfaces. And so this is happening for almost every surface you pick in this manifold. It's not just this one unique one. Um this is the first one we constructed and there's a reason why it looks so complicated and where it came from. It'll turn out at the end that there's going to be many simple ones, but this is Clayton Peak, which is where you hopefully will be going next Wednesday. And it's also the overview that you would see from here of our talk. Any questions about the result we're going to do today? Yes. Alice. Ah so terrelli equivalent means there's a diffomorphism taking one to the other and the diffomorphism acts as the identity on all the homodopy groups of the for manifold. Yeah. And so you see there's a diffomorphism turn the uh pink orange hop link 180 degrees but that difforphism switches the homology of both the pink and the orange. So now you can ask can you do this by any devomorphism that leaves the homology alone which is all there is in the homotopy here because there's no fundamental group. Yeah other questions. Thank you. Thank you. Thank you. I owe you what what ah so if they if you're isotopic then you're teri equivalent because the isotopy by ambient uh by the isotope extension theorem would give you an ambient isotopy which would give you the diffomorphism at the end but since that's an ambient it's a one parameter family so it's in the homotopy class of the identity and if you're homotopic to the identity you act as the identity on all homotopy groups and all homotopy functors. Yeah. Thanks others. Yeah. >> Yes. you you have to wa wait for the end of the movie. We're going to get there. Yeah. Yeah. Yeah. That's that's exactly what I'm doing today. And it takes a story. And so yeah, I I hope I have I hope the trailer has you hooked and wanting to ah thanks. So the question was cyberwiten invariant vanishes for these things. So how do you detect them? So we're not using just the ordinary cyberwitten. And by the way, we're not using the stabilize and serger argument either because there's no exotic structures on S2 times S2, right? And so we have to be doing something different. And indeed we are. [sighs] So there's different types of exotica that you can have in four dimensions and they are all related and there's a you know I call it a bicycle because the cycles go in all directions. You could take an exotic pair of manifolds and get an exotic pair of surfaces. You can get an exotic pair of surfaces and get exotic diffomorphisms and exotic diffomorphisms back to exotic manifolds. That's kind of a funky one. you don't see very often, but it is there. And you can also go backwards from exotic manifolds to difficults. These go in all directions. And so I'm going to look at a couple of these arrows. Yeah. So we're winding down the trail. I would have shown you on the overview here on the map. Um, maybe we'll go back. There's different places where you could choose to make a fork. This is a fork in the trail above Bloods Lake where this slides is where sometimes people choose to go one way or the other. Um so one way people have made exotic surfaces and certainly this was the first way and what was done most of the time started with exotic manifolds and then built exotic surfaces by some surgery and inverse surgery operations um or man fiber sums or submanifold sums. Um, but by the very way that you're doing it, since you're starting with exotic manifolds, you do something to get some exotic surfaces. The way that you know your surfaces are exotic is there's an arrow that goes backwards and takes the exotic surfaces to the exotic manifolds. And so if these surfaces were related by a diffomorphism, the manifolds would be related by the diffomorphism. And so this means that these surfaces are not smoothly equivalent. So if you want to get exotic surfaces that are smoothly equivalent related by a diffomorphism, then maybe the right thing to do is to look for exotic diffomorphisms and you take a surface, you hit it with an exotic diffomorphism and you get another surface. And this has certainly been a very productive way. I had tons of examples in a paper posted in 2023. So let's dig into this a little bit. You know, stop and look at the things along the trail. And so we need to see some fun exotic difforphisms which we call reflections. And in fact here once again this dimensional reduction works just beautifully. You can start with lines in the plane and write down just linear maps that take these structures to themselves. And then when you bump up the dimension, you get some really cool diffomorphisms that can be used in four manifolds. And so I've drawn on the left um you know in XY coordinates which I'm really thinking of those lines as this is the complex algebraic geometer's picture of P1* P1. It's just a line times a line um projectivized. And you know there's the two coordinate axes which is exactly s_ub_2 * s2 which um has one axis and another. It's the a sphere s2 times a point in the bphere the point times s2 the x and y axis. You could roll this in a taurus if you wanted another more schematic picture. And then it has the diagonal like the one one curve and the line y equals x if you will. and it has the anti diagonal the line y=x if you're getting there. Um the diagonal the homology of the diagonal is a + b. The homology of the antiagal is a minus b. And we can do this thinking of s2 * s2 sitting inside r3 * r3. If I use whatever coordinates I use on R2 to write down reflections in these lines, the same exact equations work on R3. And so what I call the reflection in A +B is a map that's going to change the sign of the A plus B homology class, but it will fix the A minus B homology class. And you can check if I plug in the line y =x there. So if I plug in a point like 2 -2 into ra + b, it comes out still as 2 -2 because I switched the order of the two and switched the signs. And so it fixes that thing on the nose and it's changing the homology. And similarly there's an r of a minus b. I'm saying r these things have r primes. You can check that these things are actually orientation preserving. And in fact, if you look at a little neighborhood on top of your S2* S2, this thing really looks like a copy of R4, which looks like an R2 um time R2. One of those R2s is just the map that you're multiplying by -1. The other one's the identity. And so in one parameter family of rotations, you can unrotate so that it's doing the identity at the the point where you're touching and then it does that rotation when you go out and making the difforphism the identity in the little neighborhood means you can connect sum it to the identity difforphism on another copy of S2* S2. Yes, Brandon. Ah, yeah. There's a really close uh so an old school way to say how do you do a Kirby picture of difforphisms would be to take the Kirby diagram and do a sequence of moves that slides the Kirby diagram through a collection of Kirby diagrams till it comes back to the original Kirby diagram. You have to be careful. There's a little bit more structure. Part of this is in its typical Kirby diagram, you're not drawing the four cell and so it's sort of modular. What happens to the four cell? It seems like David and David might be telling us that there are exotic diffumorphisms in the four cell ro boundary and so that that method doesn't work unless you also specify really the flow of that family as you go on. But that's a very good question and an interesting description of what's happening. And by the way, if you don't like the schematic picture on the left, there's the honest to goodness picture on the right where I've got a Kirby diagram that has a hop link with again the A and the B, the aqua and the blue. And that hot flink has a Z2 squared symmetry which is just three involutions and the identity the involutions being 180 degree rotations about those axes. And I've labeled them in the picture. You know there's one that changes the sign of a + b and fixes a minus b. There's one that changes the sign of a minus b and fixes a plus b. And then there's the composition of them that changes the sign of both which I call RR r. So do these diffos make sense? And these are the diffios that we're going to attach. In fact, you can really sort of see the idea. Now when I wanted to get things that were um teri equivalent but not isotopic, I used an exotic diffomorphism. So a diffomorphism that was homotopic to the identity which I could construct in a big manifold. Um and then it didn't really matter what the surface was because it went to its same homology and because it was teriisotopic it would in fact be um or teri equivalent. That's pretty close to showing it's topologically isotopic, which make them topologically isotopic surfaces and make it exotic. Um, it didn't matter what homology class you took here. What we're going to do is realize that you don't need your diffmorphism to be homotopic to the identity in order to get exotic surfaces. You just need your surface to live in the part of the homology that's fixed by the diffomorphism. So if the diffomorphism fixes some chunk and you hit it with the diffomorphism, you're going to get a surface and most likely that's going to be an exotic surface if there's proper interaction. So that's the construction idea. So, um, there's a philosophy. I've been doing work in parameterized gauge theory a while, and I like having pictures in my mind and philosophies that help guide my work when there gets to be a big mess. And so, I go back to being a grad student and Frank Raymond teaching me about obstruction theory. and you look for a section of a bundle for instance and you can ask is it there and there'll be an obstruction that lives in some coomology group this could be like asking is there an orientation and so there's something in the first coology with Z2 coefficients or is there a spin structure and it turns out that whenever you have this obstruction to say something exists the next question you might ask is if it exists is it unique how how many are there? And the way you would see if it's unique and how many there are um is you look at the so-called difference class that compares two, which is just basically kind of taking the product of your original thing with an interval. And so the interval cuts up one and so the difference class lives in uh comology down one dimension. So far so good. So if you're going to make this work, what this tells me is a good idea in gauge theory is, you know, somehow the original kmological thing where the obstructions live kind of appears in the old school gauge theory in dimension zero, which means the uniqueness should appear in dimension -1. And so to find things that are exotic, I want to look at negative onedimensional gauge theory and make sense of that. And this is the land of parameterized gauge theory. So um one thing that we have is this same thing takes over. Hopefully this has a junction inequality, but I'm worried that that doesn't. Is this the same slide? Yeah, that's the same slide. Whoops. Ah, okay. We we'll we'll survive and worry about what I maybe didn't stick in in a little bit. Um, so there's an adjunction inequality for just straight up manifold with non-trivial gauge theory and variance will tell you that if you have a certain homology class with certain conditions that the genus needs to be this big or bigger. And we can really think of this as an obstruction class to that existing. Um, some newer results happen in family gauge theory. And so here I guess I can get my volunttolds to do a demonstration with me. And heck, I can even get another volunttold. So Shannon, my sister, will stand up and hold the mic so I can talk or just hold the mic close to my mouth. And um so what we're going to do is our volunteers can hold their hands out like this. And what we're going to imagine is that there's a surface on this side and a surface on this side. And there's some diffmorphism that like turns the hands over. So I turn this hand over. And if I glue my two hands together after turning it over after diff bundle of intervals with my hands. If I do this, it's a mopus band. If I do the same trick with an interval times the four manifold, I will get what's called a mapping taurus of the diffomorphism that I apply to the four manifold. So some weird funky fivedimensional shape. Well, if I'm starting with a surface on my right, stage left, and it's isotopic to a surface on the left, when I turn my hand over and glue it like that, that isotopy will give me an embedding of the trivial surface bundle, the surface times the circle mapped into this non-trivial family of manifolds. And so exactly the situation that we've got is you would like to have a junction inequalities for families of manifolds. A first one in the case when the base is S1 was proven by Buralia and that's really all we need today. There's a higher parameter family that could get you to more and bigger separating things by using this philosophy that the market group and so my collaborators have proven but not uh posted yet but we can tell you about and there's a diagram that's showing you the way it works. Um there's an open question lingering there that I won't go on with and so now we can hit the next bit. So, uh, on the map, you know, there was supposed to be I was supposed to zoom out to an overview here when nobody has trouble going from the trail head to Bloods Lake or from Bloods Lake up past to one of the peaks. But um on the way back there's a fork which you pass in the direction where it merges together on the way out and in fact the fork is goes to something called wow trail and so there's nothing wrong with taking the wow trail but it's definitely a detour that will have you explore further just like if we devise into gauge theory which is what I put at this point of the fork. And so if you're taking this map with you, you'll see that the gauge theory fork is the one that takes you around. Um, how much time do I have left? What time is it? >> I have 13 minutes. So I I get to do gauge theory in 10 minutes. So let's see how this goes. Um I will say there have been a remarkable course by um John Morgan on the old school Donaldson gauge theory here at PCMI in one of the old PCMI volumes and there's another really cool one on the cyborg witten by Ron Stern where they spend a whole week those are going to be much better sources than what I can do in 10 minutes but I want to give you a taste of some of the things that happens and so You're going to start with a manifold. Could have different topology. You're going to start with some discrete data. You'll start with some continuum data. And then you'll have some crazy nonlinear map from one infinite dimensional space to another. um and there'll be a symmetry group which in the cyber witten case is a collection of all maps from your manifold into the circle and your crazy nonlinear map will be equivariant with respect to that. So if you first multiply by a map to the circle and take the map uh you'll get the same thing as if you first do the map and then multiply by the map to the circle. And so this is a structure which you can actually see a lot about the way things are going to work from the structure. The other things that you need to know is this crazy nonlinear map has the structure of a first order differential operator. So first derivatives just like the exterior d in dramology and um a quadratic term um so uh yeah that's the structure that we can actually say a fair bit about what's going to happen just by looking at that structure. the group actions tell us a lot. And so to understand where the group actions come in, I'll go back to a differential topology thing which says that things look like they're linearization. So if you have a nonlinear map and you want to take the inverse image of a point, you linearize it. If you can solve the uh linear problem, in other words, if the derivative is subjective, then this nonlinear solutions locally look exactly like the solutions to the linear problem. So now we do the same thing the implicit function theorem but with a group action. And so now our map has a group acting on it. And in this case um if you're at a fixed point uh or a point that gets mapped to the fixed point set then uh any group element that's sort of fixing that thing um is going to act on the inverse image and you can take the infetes ones. And so then you'll get a sequence of three spaces. the tangent space at the identity to the group, the tangent space to the domain, the tangent space to the co-domain. This is what happens when you linearize the problem. This is called the deformation complex. And um you can look at things about the coology. And so really if things were working and there were no fixed points and it was just regular um implicit function theorem. the derivative was on to then the dimension of the kernel that's really going to turn into the dimension of the first coology of this complex which is really just negative the oiler characteristic of this complex um in the case where the zerooth and second vanish that uh negative uh oiler characteristic is called the virtual dimension it predicts the dimension of the space of solutions for you and that's how you can have negative dimens dimensional spaces. Um and so the analog is that if the second coology the so-called obstruction space vanishes then um this your space is locally like you know the inverse image of an image mod out by the group is locally like the first coology mod out by the stabilizer. And this is you know you could prove this in finite or infinum dimensions. The finite dimensional proof is just a slight improvement of the proof of the implicit function theorem. You believe that this will help you go on. So once you have this structure, how do you get information out of these equations? So there's what's the bo modern best way to do it? And so it's Fuda and um Bower uh Chiprian uh Manalescu plus many more have used this idea and the idea is to take a finite dimensional approximation and then look at this map on the spheres and then apply your favorite homotopy functor to that map and take a limit as your finite dimensional approximation goes up. in particular because you have S1 stabilizers, you often have an S1 symmetry and the S1 equivariant homology or coomology work really well to use as your homotopy functor. Um, this was the idea that Miko used when he proved the 108 theorem. Um, he passed away recently and I'll dedicate this slide to him. He was such an awesome guy. So I guess I need to tell you the um old school way to do it. Instead of looking at the map, you take the inverse image of a point and you if it's a zero dimensional moduli, count how many you get and that gives you a numerical invariant. What you can do if you have families or you can do and there's various maps you would write down like just the inverse image of the map or sometimes you augment your space a little to make it a little bit bigger so that you get rid of the singularities by the circle action so that the space you really care about would be the quotient of the circle action so that you can at least see uh only a finite dimensional group by doing that trick of adding framings in. And um another point that's a really good point to make is that the um so-called reducables which are going to be the ones that are fixed by the group action are the ones where the second component is zero. And if you take [laughter] the a comma 0 and you square it that equals zero. And it's zero if and only if that second component is zero. And what that really tells you in terms of practice is these complicated nonlinear equations turn into linear differential equations. They turn into dram theory. They turn into regular algebraic topology when you have reducables. And that's not nearly as scary as wandering down the um well I shouldn't say algebraic topology isn't scary. homology and coalology are not nearly as scary as gauge theory. It gives us something that we can attack earlier on. And so the reason that why these invariants are well defined is you have collection of continuous data. You pick some point of continuous data and you might have some number of points in your solution. But then since it's continuous and connected, if you pass to another set of data, you do that in a family and the solutions will be a one manifold. And every one manifold, if you hold up your walking sticks, has two ends, which is zero ends counted with sign, which means that the number of points you see total are the same. That's a great argument to keep track of and know. It tells you a lot in differential topology. And so then if you want to compute these things, oh that uh dog bone doesn't show up very well at all. Um oh well. Um the uh some things you can think about is what would happen if you took the modulized space of an x union y glued together along Z. And the answer is you should get a solution of X, a solution of Y, as long as they agree on Z. That should be right. If we do this at the framed level where you make it so that you have to mod out by the extra S1, that's kind of what you should have and then divide out by the extra S1 action. That's certainly what makes sense. Doing the analysis to prove that is harder, but having the picture just at the formal level helps show you what to expect. And in particular, there's one really nice case, and that is when you've got the generic side on the left, the X, which is irreducible, so it has no singularities. And on the right, if you only have reducables, because if everything's fixed, the frame modulized space is the same as the non-frame modulized space, which means that when you mod out by the S1 action, it's just going to mod out um you know, nothing happens on the Y side and it'll happen on the left side. And so you end up with a theorem that says if you know the invariant on the right hand side and the left side is reducible which means just algebraic topology even for a family on the left hand side then you can compute everything. And so several examples of this particular theorem it's in a recent paper that Danny and I put up is for instance if you have the manifold with cylindrical ends and you attach on a little thimble that looks like an R4 folded up to get the closed manifold you get the same answer. And in fact anything that up to the virtual dimension and to the number of reducables looks like just a symbol. So any negative definite manifold you get the same thing. So that gives you the blowup formula and the fact that you have the same cyberwitten invariance when you take a connect sum with a negative definite. If you add in a p a higher dimensional family like a family given by a reflection that just changes uh the sign of H2+ where H2+ has only one generator that looks exactly like just a standard R4. So you have exactly the same invariant. This is how we compute these family invariants for difforphisms. And you should you have to have a starting point. The starting point is the quote complete the square argument that Whitten gave to compute for projective algebraic varieties. And if you want to see master at work using these ideas, look at Ron Stern's course and you can using a theorem like this, you can compute almost all the examples we know. Um there's a uh different technique where one side's not necessarily good. Um that in fact will only depend on the homology uh of the action. And this is the case where the co-dimension of the space of reducibles is the dimension of the second positive coology. it's a positive definite subset in the classical case like if you're looking at CP2 that has one positive mode um you'll get a structure where when you hit that co-dimension one set which in CP2 it's just a line with a point in the middle of it um let me there's uh the picture on the left is a little bit hard to see there's some lines going across but and you get to the point where there's a wall where there's a reducible one line merges into the other. And you can figure out that by linearization. And what it means is that the invariance you'd see on one side differ from the invariance you see on the other side by a plus or minus one. If you're doing a Z2 invariant, it just means that they're different. Um when you're doing this inside of um a family like we glued together to make our four manifold mois band my S2 * S2 has B2+ equal to two and the thing that you see is when you're looking at the sphere bundle there is you see a circle and a plane hitting the origin gets you into the trouble and you pick the class so that it reflects just one of the hop flinks in in the original picture. So it changes the sign of just one of those modes, which means that when we have this R2, that's the B2+, the H2+, you're changing the sign on one side but not the other. And you take a circle bundle and glue the ends together. That's a Klein bottle. And so the collection of good parameters looks like a Klein bottle. And a uh thing that corresponds to the continuous parameter space would be a curve, a section of the Klein bottle that wraps around it the long way once and around the other way some number of times. And in the Taurus, that's the framing that you see in the surgery. In the Klein bottle, there's only two things. You can sort of be you're related by a one or a not. There's not a Z's worth. And that's a fun algebraic geometry or that's a fun algebraic topology question you can do even just with pi one or with other techniques. And so the theorem states that if you've got two solutions that differ by one of those things which is exactly what our twist does then they're cyberwiten invariants are going to differ by one. So one of them has to have a non-trivial cyberwiten invariant which will give you an adjunction inequality for one of them. But if you take, you know, that means that gives you an adjunction inequality where you make your bound the worst of the two that you could take. And that adjunction inequality tells you that you can't patch the surfaces together. And you can't patch the surfaces together until you make the genus sufficiently large, which is exactly how we get the internal stabilization results. I'm coming to the end. And so this is another picture of the same of a different example built with a TF foil. Um the first example used the RR diffommorphism. Um this example um and had the surface not interact with the orange and pink. This time the surface homologically interacts with the orange and pink, but it interacts in a way that it's not changed by the R+ reflection, which is the one you do. So, this gives you even simpler pictures of surfaces. And this is a key step that can get you lots of crazy surfaces both in two copies of S2* S2. You can make it a little smaller, two copies of CP2 and a CP2 bar if you gave up the spin. You can also do tons of varants of it as I'm coming to an end. And so if you are Irving's talk um today, he had theorems about um exotic discs. Um so one of the things is any examples that we can do with a closed manifold. If the construction happens in a contractable not contractable compact piece then you can do it for manifolds with boundary. So we also get results from manifolds with boundary. My eyes are a little too poor to see this. Um but um so here's a theorem for any knot that you take whatsoever and for any genus you can find pairs of surfaces of that genus um who in the punctured S2* S2 that bound the knot that are exotic and they can be exotic even after doing lots of one ambient one handle surgery. ies or lots of finger whitney moves follows exactly by these same techniques. You can just cap off the S2* S2 punctured. Um if you want to get smaller than a twice punctured S2* S2 somehow adding a flare end in a contact structure makes it the same as making it closed. And so in this case, if you have knots that have a positive max TB, you can do this in a punctured S2* S2. Same sort of thing. So this means like the TFO bounds a pair of uh exotic discs in a punctured S2* S2. If you're motivated to think about hs slice surfaces and hs slice discs um you can do that in s2s*s2 which by the way Irving's result is fantastic looking at the singular instanton theory but the singular instanton theory these gauge theory invariants die when you stabilize so if you're just doing the classical invariant his techniques won't attack what happens when you add on an s2 times s2 it looks like you stabilized They won't attack what happens when you put on two copies of S2* S2. The invariants that don't die when you add on the extra S2* S2s or the parameterized invariance. Exactly. The reason the invariants die is when you add on that S2* S2 the invariant the dimension of the moduli space becomes negative. So if I have a family of spaces I add the dimension of the family to the dimension of the moduli space. I can get back up to zero dimensional and we're back in business again. Um and then I guess uh last theorem is talking about the H slice case which you know we certainly get examples by doing white head doubling of surfaces and of discs. Um, the reason we have the first crazy picture I did is that crazy picture didn't interact with the C and the D, which meant that if we could prove it was topologically the two were topologically isotopic, which I think we can, then it would mean that you know any knot bigger than uh T23 in the assoc correct contact sense of bigger would bound exotic surfaces in a puncture. S2 * S2 and um any not bigger than T25 would bound exotic H slice surfaces in a punctured S2* S2 is what this is getting at. I think this is probably getting us yeah to the end. Um that's the link to the prey. There is a link somewhere you will find my website and at some point KState will fix their snafu so that web pages at KA state can be um monitored. Also have this go up at BCMI