Transcription
[Music]
Um, it's my pleasure to introduce the second speaker for today, Liisa Pillo from MIT and UT. Liisa Pillo got her PhD from UT after an undergrad at Boston College. After that, she has continued to alternate between Boston and Texas. She is most famous for solving the Conway knot problem, and she has been awarded the Mariam Meakan New Frontiers Prize, the Clay Research Fellowship, a Sloan Fellowship, among other awards. So it's my pleasure to introduce her and her talk on exotic phenomena in dimension four. Thank you!
Hi! Yeah, thanks so much for the introduction and for coming. I was very pleased to be invited to speak here. I was also a little bit surprised because every time I've spoken at Harvard before, I've really just given an awful talk. So hopefully today, we're going to break the curse, but also you've been warned.
So, I'm a low-dimensional topologist, so I'm primarily interested in studying manifolds. And if you want to study manifolds, maybe a particularly obtuse goal that you might have is to classify them. Okay? But if you're going to do that, you probably first need to specify a little bit more carefully what you mean by manifolds.
So you probably should declare what dimension you're going to be working in. But you also probably should declare what kind of manifold you want to study. Historically, there's kind of three types of manifolds that people are interested in. The one you're thinking of, which is just a nice topological space, local atlas to RN, that's what we're going to call a topological manifold today. So that's kind of no extras.
But we also study smooth manifolds, where we're going to just require that, in addition, the transition functions are C∞. And also, historically, people are interested in something called a PL manifold, but I'm not going to tell you what that is today. The reason being is that in dimension four, it's the same as smooth, so it won't really come up.
Okay, let me also just make a couple of convenience assumptions. Today, all of my manifolds are going to be oriented. Unless I explicitly say otherwise, they are compact and without boundary. So these are the kind of manifolds maybe that I'll talk about what we know about the classification of. Today, I'm going to be primarily talking about dimension four, but let me just very quickly tell you what we know in other dimensions.
So in the sort of classical low dimensions, dimensions 1, 2, and 3, we have classification theorems. In dimension three, that was not easy, but that's a story you've heard before and not the one I'm going to tell today. Let me also just tell you that there's a theorem in dimension three of Moise that there's no difference between smooth and topological manifolds in low dimensions.
Okay, the classical high dimensions, greater than or equal to five, in high dimensions, there's something called the surgery program or surgery theory, and this is a theory that was developed in the 60s by a whole bunch of folks. Let me call out, Rouche, Novikov, Sullivan, and Wall in particular.
In slogan form, what you can think is that surgery theory is going to reduce the classification of manifolds to problems in algebraic topology. I want to comment here that I think there is a misconception. I had this misconception for a long time that because surgery theory exists, the classification of high-dimensional manifolds is kind of done. That's not true; it's kind of not my problem anymore, but it's not known in general.
For example, Simply Connected smooth five-manifolds are classified, but there are lots of things we don't know. For example, the smooth Poincaré conjecture is open in dimension 126.
In high dimensions, as perhaps you know, we can have a difference between the smooth and topological categories; that's originally the work of Milnor.
Okay, so that brings us to dimension four. Let me start off by talking about the topological category where groundbreaking work of Freedman from AD2 tells us—one way to phrase it is that surgery theory works topologically. It works when the fundamental group is what he calls good.
Sometimes this machinery will let you get classifications in dimension four for topological manifolds, and he carries this out. For example, he gives a classification of simply connected topological four-manifolds.
So the set of X4 top with no π1, Freedman tells us that this is injection with some other set of algebraic doodads that are more well understood.
So it's in B with this set of pairs. There's a form from Zn to Z, which is whatever unimodular, bilinear, symmetric, and there's an integer in Z.
So, right—the theorem, and I wanted to kind of state this to see what classifications in high dimensions have a tendency to look like. There's usually a big algebraic set that you maybe do or don't think is any nicer than this, but anyway, there's a big algebraic set, and you say that all of these correspond to a unique manifold.
Let me interject here with just a couple definitions for people who don't do topology or geometry all the time. This π1 thing I keep referencing, this is the fundamental group. It's not important that you know what that is today; just know that it's a group that you can associate to a manifold. It's kind of a primary invariant for them.
And this thing right here, I'll mention a lot as well—this is called the intersection form. Again, it doesn't matter that you know what that is, but it's a form which is an invariant of a manifold. And Freedman tells us that for simply connected topological manifolds, it's basically the invariant. If you know what this means, the intersection form is the cup product on H2.
Okay, any questions so far? This is kind of a board on everything I’m not going to talk about. Let me also say that I really like to get a lot of questions.
So what I really want to try to tell you about today is what we know about smooth four-manifolds, which is nothing. We have no classification theorems; we have no conjectured classification theorems. So far, the field is in a sense really still in its infancy. One of the primary things we study is just the extent to which whatever is true topologically fails to hold smoothly.
Let me make a definition to make that a little more precise. We're going to say that a smooth four-manifold X is exotic if there exists another smooth four-manifold X prime which is homeomorphic but not diffeomorphic.
The smooth four-dimensional Poincaré conjecture says that S4 is not exotic, but in fact, we know from Donaldson's work in the early '80s that there do exist exotic four-manifolds. And these two statements together serve to motivate the question that will sort of guide the talk today, which is: which smooth four-manifolds are exotic?
And since we're maybe particularly interested in one day eventually understanding the Poincaré conjecture, maybe we're particularly interested in understanding which smooth four-manifolds are exotic, where "small" will maybe be taken to mean today that the fundamental group is trivial. Donaldson's examples have that, and maybe I also am going to ask that B2 be fairly small.
So this is this integer, and I want it to be... So B2 is the final invariant that I'm going to sort of impose upon you today. This is the second Betti number; it doesn't again matter that you know what it is—it's an integer. It's an invariant of a manifold; it's really crudely measuring how complicated it is. B2 of the four-sphere is zero.
So this is the question that I'm going to use to guide the talk today. The plan is to tell you classically what are the techniques and what do we know. Then I'm supposed to, I guess, tell you about current developments—so what do we know and how do we know it? But then what I really want to talk about, and what I'll spend a lot of the second talk on, is starting to try to say something about the structure of where exotic comes from and maybe a little bit about quantifying it—so asking how you could measure how different a pair of smooth structures are.
Okay, so before I move on, let me just first give a little bit of a disclaimer. This is a question that gets a lot of air time. I'm going to give it more air time today, but it's really not the only question in smooth four-manifold topology. There are a lot of things we want to know about four manifolds, and we are also interested in things other than the manifolds themselves.
For example, submanifolds, particularly in codimension 2, that's kind of higher knot theory. We're also interested in diffeomorphisms of manifolds, and for these two types of objects—submanifolds and diffeomorphisms—the story I'm going to tell today, by and large, has a parallel story.
In fact, both of those stories are very active right now as well. And maybe the final comment is that, from now on, unless I explicitly say otherwise, all manifolds are smooth and simply connected.
Okay, so classically, how do you build exotic? So this is really kind of a three-step process. The first step is to give yourself some candidates—so build a pair of smooth four-manifolds X and X prime, which you know you think reasonably might be homeomorphic but not diffeomorphic.
Step two: show they're homeomorphic. This is really riveting stuff. Step three—you know where this is going.
Okay, but I wanted to point this out because I’m going to do it a lot. It'll give me a little structure, and I also wanted to sort of point out that this is basically under control by Freedman.
So really what you need to do is build yourself some manifolds, and you just get the intersection form right, and then Freedman says you’re good to go here—now you just need to distinguish them.
So then the techniques I need to tell you about are really how do you build four-manifolds and how do you tell them apart. So let me give a list of some ways that classically we get four-manifolds to study. It's also good to have these kind of... I don't know, in mind: have some examples to be thinking about as we go through the talk.
Any questions before I do?
I’m going to keep stopping and asking for questions.
Okay, well, eventually... So the first sort of place we get four-manifolds is some sort of very basic small examples. For example, there's the four-sphere— that's great, there's complex projective space, there's its orientation reverse, and you can take connected sums of these things.
That maybe seems very basic, but in fact, sums of CP2 and CP2 bars play a huge role in the exotic literature. Okay, you can also try to build yourself some four-manifolds out of lower-dimensional manifolds by taking products or bundles.
For example, S2 cross S2, or more generally, products of surfaces are very interesting examples. Or maybe you want to take a manifold cross S1.
Okay, another way you can get some interesting four-manifolds to study is you can ask a geometer to give you some. In particular, complex [Music] surfaces and symplectic manifolds have played a large role.
And those are actually kind of, for the most part, those are our root examples. I'll say a little bit more about mangling them, but this is kind of the natural sources for the most part that were studied classically.
So we also like to take them and kind of Frankenstein them together using cut and paste operations, which is what it sounds like. You have some manifold X, which contains some co-dimension zero submanifold Z, and you're going to cut it out and replace it with something else.
Okay, and then the final construction of manifolds, which I just wanted to sort of mention here—this actually doesn't really play a role in the classical exotic literature—but you can build manifolds out of these really basic building blocks called handles.
And I'll say what those are in the second talk. So that's how we build manifolds.
We also need to be able to tell them apart, and classically, if you want to distinguish manifolds, you have one option: it's gauge theory.
I'm going to say very little about this today. There are like four or more world-class gauge theorists in this room—none of them is me. Very crudely, this is an invariant that counts solutions to a PDE on the manifold.
It’s a very powerful invariant theory; it's not an exaggeration to say that everything we know about exotica we know thanks to gauge theory. So this is how we're going to tell our manifolds apart for the most part, at least classically.
But I just wanted to give you a couple of warnings about gauge theory. The first is that, in general, it’s not possible to compute it explicitly. If you want to build some manifolds and then distinguish them, you need to be able to actually literally make computations and then say, “Well, they don’t match.”
Well, people are able to sort of get around this, but it’s largely by using formal properties. You can sometimes get your hands on things if you have some geometric structure, and we have some understanding of how these invariants behave under gluing.
Okay, so that’s the first warning. The second warning is that gauge theory doesn't work when you have no B2; the invariant just isn't defined.
Invariance—so, in particular, while this theory has taught us a lot and there’s still a lot it can do for us, you’re not going to disprove the Poincaré conjecture using a gauge-theoretic invariant, at least as we know today.
Yeah, good, yeah. The methods are the same; if you’re working with a manifold that has a little bit more structure, in general, it’s just that the structure is going to help you maybe get your hands on this.
Alright, sometimes we can say more about classifying manifolds that have more structure. Among those manifolds which have more structure...
But that’s an aside. Any more questions?
Okay, so that’s what I wanted to say about classical techniques. So now I’m meant to tell you what we know, which is quite a lot. This board is deliberately too small to read—there’s a lot of exotica literature.
We would be here for a long time if I tried to kind of state everything carefully and give you all the names. So let me instead try to give you a sort of tell you what the patterns are on this board.
So how’s the board working? There will be kind of three eras I’ll discuss. In each era, I’ll tell you how small we can get. I'm assuming manifolds are simply connected unless otherwise. I’ll tell you how many smooth structures we can build, who did it, and then these are the two sort of important columns—I’ll try to say something about what the big major developments are, both in constructions and in obstructions.
So the first era in the development of exotica is the Donaldson era in the '80s. It's kicked off with Donaldson’s development of Yang-Mills gauge theory. The examples in this era are for the most part complex, so we start to see non-complex examples computed using gluing formula and cut-and-paste operations toward the end.
That’s kind of era one. Era two is kicked off by the development of the Seiberg-Witten equations in 1995, and there’s an incredible amount of work. Then we don’t actually see any increases or any decreasing in the size of the manifolds we can get during this era.
But I sort of want to emphasize that this chart is problematic in that that question is not the only question you want to know about four-manifold, so I’m only presenting sort of smallness here. There isn’t smallness developments, but in this there’s a ton of new results about smooth four-manifolds, for example questions about minimal genus surfaces representing homology classes and a lot of other things.
So this maybe makes it look unimportant.
The second column?
Oh sorry, this one—no, no, sorry, the first column. Ah, the first one, yeah, great. So this is B2; this is the size of the manifold. So Donaldson’s first invariants have B2 equals 10; in the '80s we got nine, and that’s about it.
In this second Seiberg-Witten era, no development—nothing smaller.
Good, yes. Other questions? Other things you want to be said out loud because you can't see them?
Yeah, what makes it harder to do?
That’s a great question! Yeah, it’s... I think it’s because one of the reasons is because of a lack of small examples. When you look at this list and you say you’re only interested in simply connected, you probably don’t want products of surfaces or things like that.
“Well, you immediately end up here, and these tend to have a lot of B2. And so if you want to have less B2, well, you have to kind of get your scissors out and start trying to put them together in ways where one of them eats some B2 out of the other one, and you end up with very messy objects.”
Yeah, thanks. More questions?
Yeah, yeah, oh yeah, sorry Tom—that's Tom, but nobody can see that anyway.
Thanks!
Okay, so this is the second era, the Seiberg-Witten era in the mid-90s—big development in techniques both constructively and obstructively, but actually no movement on the smallness of the exotica we can produce.
More questions about that era? Anything?
Okay. And then the third era is kicked off in 2005 by a result of John Park, who is rational blowdown, a technique that was developed in the Seiberg-Witten era to get the first improvement in the size of exotica in a long time.
So he gives us exotic simply connected manifolds with B2 equals 8, and that kicked off an arms race. Everybody works on this. There’s a lot of results by these people and probably others in all sorts of permutations that I’m not going to write down.
We start getting smaller and smaller B2, but nobody cares about five, but that’s fine. Until the end of the era is this result—AV made of and Park; they give simply connected B2 equals three exotic manifolds, and that’s the smallest exotica we have for a long, long time.
Yeah, there are a lot of candidates. If you just kind of forget about ever being able to compute a gauge-theoretic invariant for it, you got lots of options.
More questions?
Yeah, yes, exactly—that’s right.
It doesn't; we just, by and large, don't have a more combinatorial invariant, but I'm about to say something about that.
Yeah, more questions?
Okay, let me try to tell you about a couple of more recent results. I’m going to use the same table, but I’m going to write a lot bigger.
Okay, so there are three kind of camps of results recently that I want to say something about. The first is some work from 2021, which really doesn't make things smaller. It’s like 23—the biggest thing on the table yet.
No development in techniques; the development is in the obstruction. So in 2021, we get the first example of a pair of exotic manifolds which are distinguished with something called the slice approach, which is this method of distinguishing manifolds which is a little more convoluted than just saying, “Compute invariants for both; hey, look, they don’t match.”
And this approach... I’ll tell you what it is in just a minute, but let me sort of tell you now that the thing that’s exciting about it is that versions of this approach could possibly work with no B2.
So it was good to know that this approach can work ever, since we’re maybe hopeful that it could work in cool settings later. This is joint work with Tran Manescu and Marco Mangan.
The second set of results that I want to tell you about is from last year, and again, the examples in these results are not particularly small. There’s not really movement here. What’s exciting about this result is the development in the techniques.
So classically, we build exotic manifolds by starting with some big, big geometric stuff and making a hot mess, but we wanted to be a lot more juvenile about our constructions.
So we’re really just going to start with handles—these basic building blocks—we’re going to make a stack of them, and that’s how we’re going to get our manifolds.
So explicit handle constructions, and the reason that we can get away with this is because we’re prepared to make explicit computations of something called the Higgard FL-Mixed Invariant.
So you should not really think that we’re getting away from gauge theory here conjecturally—the Higgard FL-Mixed Invariant is equivalent to the Seiberg-Witten invariants.
So this probably is a gauge-theoretic proof, but we’re accessing these invariants via a pretty different method that allows us to do computations much more explicitly.
This is joint work with Adam Levine and Ty Liman.
Okay, because our constructions are much more explicit, we can monkey around and make other stuff happen, so we’re also able to produce exotic four-manifolds with B2 equals 4—so not the smallest, but pretty small—at the expense of having a little bit of π1 Z2.
But in fact, what’s interesting about these is that their intersection form is definite.
Okay, so that maybe doesn’t mean too much to you—an intersection form, as you know, can be definite or indefinite—and let me sort of assert that all the exotica previously was indefinite.
And you know, there are other things we ask about manifolds besides is there small exotica, and for some of those other questions, you see a difference in the behavior of definite and indefinite manifolds.
So it was maybe reasonable to wonder whether definite manifolds were more rigid. We’re saying here that they’re not.
Okay, and then the most exciting result I’m going to tell you about today, which I wish I could tell you was due to me, but it’s not, is work from last year from Sti Shits and Sabo.
And they’re going to give you exotic manifolds with B2 = 1, π1 Z mod 2, and again definite. And this I think you should think of as the first movement in actually getting smaller exotica in a long time.
Their techniques are going to be a mix from the good old days... Oh, I lost the green. From us. And they compute the Seiberg-Witten variant.
And then the final result that I kind of want to mention in terms of progress in exotica is very, very recent—from earlier this year.
This is a result about exotic manifolds with boundary, which in a sense doesn’t even belong on this table—the with-boundary setting is very different; it doesn’t make sense to compare.
I’ll talk about that a bit more later on. This is work of K Ren, who is a graduate student, and Mike Willis.
And what’s so exciting about their work is that their obstruction is combinatorial, so they use something called a skeletization module, which comes from cavology, and this is the first time that compact exotic manifolds have been produced without gauge theory.
Yeah, no more questions? You can put a description of a manifold into my collaborators and maybe get something here.
More questions?
Yeah, yep, um you look at the universal cover. If the manifolds have non-morphic universal covers, then they’re non-morphing—the universal covers have B2.
Plus good question. Other questions?
Oh yeah, right. So, okay, what I want at the end of the day is the set of smooth manifolds with π1 trivial is injection with anything. And just give me like a reasonable other set that you can relate them to—that would be awesome.
For now, the question that seems to motivate us more is like the question that we can sometimes work on. Apparently, it’s the one where we say like where can I find exotica—sort of a geography question.
More questions?
Yeah, no, it’s bad—the situation is bad. I’m going to try to say something about deeper in the talk about, you know, maybe if you could try to start saying where do all smooth structures come from, but it'll be not at all ready to see what kind of set you're looking for.
So the plan for the rest of the first half is to tell you what this slice approach is. This is an argument that's due to Casson in the '70s, and here’s how Casson says you can maybe try to distinguish manifolds.
So let’s suppose you have a pair of candidates for exotica; you have a pair of homeomorphic smooth four-manifolds. Here they are—that's X, and this is X prime, and they’re homeomorphic.
Alright, so what Casson would like you to do is remove an open neighborhood of a point from both of them. These manifolds now have boundary—they have an S3 boundary component from that ball you removed. They’re still homeomorphic.
And now here’s sort of what Casson wants you to do: he says suppose you could find a knot K in S3 such that two things happen.
One, there is a D2 embedded smoothly in here—in X—such that the boundary of the disk is this knot. So what you should have in mind is that the knot lives in S3, so you could think of it as being in this boundary component right here—that’s K.
And we’re asking that there’s a smooth disk that bounds somewhere in here. And then also, you should want that there does not exist such a disk in X prime where the boundary is K.
If you have both of these things, then that implies that the manifolds are not diffeomorphic. It’s easy to see that the punctured manifold can’t be diffeomorphic. If you had a diffeomorphism, you push this disk across it, and the knot would bound over here, and it extends to the closed things.
So that’s the argument. It’s not quite as direct as “compute the things they don’t match,” but it’s not too bad. So why should you want to do something like this?
Well, for one, this argument can be used to show that R4 is exotic. So this is the non-compact analog of the Poincaré conjecture, and it’s false.
So this argument has teeth, sometimes in a slightly different setting.
Okay, so that sounds kind of good. Then we’d like to know whether you can use this argument to show that S4 is exotic.
And well, we don’t know, but there is an invariant, or there are a few now, actually. There exist invariants in the literature which could do two—they could provide the obstruction here even when X prime is S4.
So, unlike using gauge theory, we could conceivably just prove the Poincaré conjecture like this using tools we have today. So all of that sounds kind of good.
But here’s what’s not so good: it's not easy; it's not at all clear how to sort of set this up—how to find a good K and K.
So if you have even two, even a known exotic pair, it’s not clear how to find a knot which bounds a smooth disk in one of them but not the other—that’s what we did here. It was the first time that this argument had sort of been run in the compact setting.
Okay, question?
So, okay, this is an argument—it works sometimes; it could maybe disprove the Poincaré conjecture; maybe that's something we should be trying to do.
In follow-up work with Chan Manescu, we just sort of tried to think about how you would try to get this running. So we gave a very systematic method for producing four-manifolds X which are homeomorphic to S4, and which come with a K as in the first step.
So it’s this really pretty straightforward construction. It gives you candidates for counterexamples of the Poincaré conjecture, and those candidates come equipped with a knot which we don’t know that it doesn’t bound a disk in B4, but it doesn’t seem to have a good reason to, right?
And the method is really systematic. I personally made a computer spit 3K examples, and I personally am terrible at coding.
So we produced a whole bunch of examples, and some of them were kind of interesting—interesting meaning we get some knots, and there’s something like if the knots have blah property, then the Poincaré conjecture is false, and we just sort of don't know; we fail to show that the knots don’t have that property.
I don't think this was ever meant to be like “aha, these ones are going to disprove the Poincaré conjecture.” It’s more meant to mean somehow the method isn’t kind of trivial out of the gates; like, it does spit some kind of subtle looking stuff.
Okay, since then, it’s been proven that these examples are actually not particularly interesting by my younger sibling, Kai Nakamura, but what Kai did, it really just ruled out the examples we wrote down. It doesn't kill the method—you can build more.
There’s a bazillion ways you can prove build more, but if you want to see some that have been written down, Chan's student Chak has done so.
And you can start to see how hopefully there’s going to be a little bit of a conversation here; then you build examples, rule them out—maybe that tunes the parameters on the construction.
Maybe eventually, you'll show the construction does not disprove the Poincaré conjecture—that would be fine. So that’s sort of the idea.
But of course, given that whatever year it is, not only are we having that conversation; ML is having that conversation.
So there’s work of Gukov, Alersson, Chien, and R, whose name I really can't spell—sorry—who are using ML to both build, so to run our construction, and study the examples it produces.
So the neural net that kind of evaluates them is out; that came out last year, and this one is in progress.
And maybe a comment about these is if the neural nets win, it will really be a proof—they are biding explicit constructions, and the way they study examples, it does spit something verifiable.
That’s what I wanted to say for the first half.
Questions?
Yeah, oh, well, I don't know because they don't— they haven’t produced—they claim they’re producing, but oh, but they're going to be homotopy spheres.
How big are the knots? I don't know—big.
Yeah, so right—good.
So there’s actually a little bit of a convolution, and this is why I kind of hid what I meant by "some of them interesting."
There’s a flip in the logic. What they’re actually going to build is they’re going to spit knots where, if you could prove the knot did bound a disk, then the Poincaré construction is false.
So, what the net actually does is looks for disks.
Oh, they definitely are detected by Cy, oh yeah, definitely are—it's a standard... Do other—yeah, it’s the theorem manifold, which we’ll all learn what that is next time.
Yeah, so far, all of the new approaches are really in proof of concept stage—no new actual phenomena, just like new arguments.
So welcome back for Liisa Pillo's second talk on exotic phenomena in four dimensions.
Hey! Thanks for coming back.
So the plan for the second talk is what I said I was going to do, but not in the order I said I was going to do it. So I’ll spend at least half of the talk talking about where exotica comes from.
I can give you explicit pictures of how you can get exotic manifolds in a pretty high level of generality, and I want to do that because it's very nice; it doesn't have to be sort of mystical.
Then I’ll talk a little bit about how you might try to say how distinct a pair of smooth structures are, and then hopefully, I’ll run out of time, but if I don’t, I’ll tell you a little bit about the work with Ty and Adam.
Okay, so starting off with kind of where exotica comes from. So let me start off with an observation. Let’s suppose you have a pair of a candidate exotic pair, so you have a pair of smooth four-manifolds, and they’re homeomorphic.
The observation is that if X and X prime co-bound a five-manifold W, and so what you should be thinking is that it looks something like this—like maybe that’s X, and here’s X prime—and I’m asking that there’s some five-manifold, which has the two of them as its boundary components.
So if they co-bound a five-manifold like this, and in fact that five-manifold is diffeomorphic to a product, it’s actually just diffeomorphic to X cross I, well then that’s going to imply that my manifolds are themselves diffeomorphic.
If this cobordism is really just a product, then I can kind of think about flowing X along the interval direction, and it'll land very nicely on X prime—that’ll be diffeomorphic.
So, manifolds are diffeomorphic if they co-bound products. And in fact, we’ll tell us that not only do we have this nice manifold running between them, but actually most of it’s boring.
There’s a sub-h-cobordism W prime such that W minus W prime—that part is actually a product.
So there’s this sub-five-manifold here where all the interesting non-product stuff is happening, and this is very boring.
In particular, the fact that we have a product here tells me kind of that this part of X is actually the same as this part of X prime—the difference is like these parts, whatever that means.
And the theorem I really kind of want to spend most of the rest, or this section talking about is something called the cork theorem, which is going to improve this even more.
So the cork theorem was proven by everybody in around 1997, so let me give you some names, but not write them.
Curtis, Sayang, Freedman, Stong, Matviyenko—maybe we should say Kirby and Cass improved a non-compact version of it before all of them were born.
Okay, so this kind of was in—came out of, I don’t know. Anyway, it’s due to a lot of people. What it says is that even better: W prime, this bit that has all the interesting stuff going on, can be taken such that if you look at how W prime intersects X, let’s call that C—and if you look at where W prime intersects X prime, let’s call that C prime—that's here.
We can take W prime so that these are contractible.
So contractible means that their algebraic topology is as simple as possible—so we have the algop of B4.
So all of this put together is telling me that if I have two homeomorphic smooth four-manifolds, then what’s different between them is all packed in this piece, and this piece doesn’t have any algebraic topology.
Let me just write down, to be explicit, a corollary. What we learn here is that if you take X, you remove C, you glue in C prime, what you get is isomorphic to X prime.
Right, we start here, lose this—that’s the same as this.
So these two doohickeys—what was that?—called corks, and this operation of cutting one of them out and gluing in the other is called cork twisting.
Any questions about this statement or anything?
There are several things I think are really surprising about this result.
One of them is, um, that... Well, these corks are pretty interesting objects in their own right.
You know that they can’t be diffeomorphic because, well, if they were, then all of my four-manifolds would be diffeomorphic, which they’re not.
So let me write that down kind of explicitly.
This is a result that was originally proven a little bit earlier in a slightly different form. Let me also quote Ruberman for the form I want.
So what we get from this is that there exists exotic contractable four-manifolds.
Now, let me emphasize that contractable manifolds have boundary. So this is not quite the setting I’ve been talking about kind of all along, but contractable manifolds are very simple—they have essentially no algebraic topology; B2 is zero; everything’s trivial.
So this is very reasonably the with-boundary analog of the Poincaré conjecture, and again, it’s false.
And another thing I think is really surprising about this theorem is that, you know, over here we’re having this really hard time producing exotic with no boundary when there’s very little algebraic topology, but this is telling me that that issue is technical.
Like I can pack the exoticness—and the difference between the two manifolds—into something that has no algebraic topology.
So I’m going to give you fairly explicit examples of exotic contractable manifolds.
Any questions before I do?
Yeah.
Um, yeah, they’ll be integer homology three spheres— they can be as simple as surgery on a knot.
More questions before I try to set this up?
Let me just comment that this is the theorem that Run and Willis re-prove. Technically, they kind of reprove that version of it.
Alright, so to set this up, I have to tell you about handles, and I also have to remember not to write in the shadow.
So help me out with that if it doesn’t go well.
So I’m going to try to do a one... We can see this part of the board great! I’m going to try for definition by picture here.
So please bother me if it’s not clear. Handles are a set of building blocks for building manifolds, and you should really think that they’re kind of a manifold version of a cell complex.
So I think most of us know how to build a cell complex.
And we’re going to do basically the same thing, but all of our cells we’re just going to thicken them up so that everything is consistently dimensioned—whatever I want, in my case four.
So let me start off with a 3D example of building a manifold out of handles.
Let's suppose I want to build a manifold that kind of has one zero cell and one one cell.
So, okay, I might start with a zero cell—there it is, it’s a point.
Okay, but that's like not great as a three-manifold.
So my zero handle will be this three-dimensional thickening of that.
Okay, and okay, now I guess I wanted to add a one cell to this—so you know what that would look like from a cell complex perspective?
It looks something like this: not a three-manifold, but we can get around that by again thickening it up in this case in two more dimensions.
So that's a one cell. This is a one handle.
Okay, in dimension four, it works the same. Let me try to give something of an example.
So let’s suppose I want to build a four-manifold out of a zero handle and a two handle, say.
So what do I do?
I start with my generalization of a zero cell, so there’s my zero cell. I need to thicken this up in four dimensions.
So I can’t draw you a picture of that, but let me just draw you this picture again and declare that this is B4.
Here it is, that’s my zero handle.
Okay, and now I said I want to glue—I want to use a two handle. So that’s going to be this kind of thickening of a two cell.
So, alright, here’s my two cell, and how do you glue a two cell onto a cell complex?
You have to glue it boundary on to whatever you are doing. So, okay, here’s the boundary; that’s an S1 here, so I need to glue this S—I need—yeah, this S1 needs to get glued on here.
The boundary of this B4, well that’s the three-sphere, so I’m looking for an S1 in the three-sphere.
So I’m looking for a knot.
Okay, okay, pick a knot—fine, glue, and then, well, alright, we need to turn the whole thing into a four-manifold by crossing this with another D2.
Technically you need a framing here—we’re not going to worry about that.
And I’m only going to build four-manifolds out of two handles today, so that’s what you need.
Questions about handles in the broad sense?
Okay, let me give you two more quick facts about handles. One is just an even kind of cheaper picture of this.
So a schematic, or an even more schematic, of a four-dimensional two-handle that you’ll see me use a bunch of times is you draw your zero handle like this, so this is my zero handle. It’s B4, and here’s its S3 boundary.
And then I’m going to attach my two-handle to.
Okay, we have some knot in that boundary.
My two-handle is, you know, fundamentally there’s kind of the D2, and my two-handle looks something like this.
Yeah, so I’ll use that representation of a four-ball of the two...of two.
And then maybe the final thing I wanted to say is that it’s a theorem of Morse that all smooth manifolds can be built in this way.
Questions? Anything wrong on the board?
So to build a contractable exotic manifold, I need two handles, and I need one more operation, which is something called carving.
Building manifolds by carving... So let’s suppose I'm given a smooth four-manifold X with a disk—a D2 embedded smoothly in X—such that the boundary of the disk goes into the boundary of the four-manifold.
So you should be picturing something very similar—schematic, something like this: here’s X, here’s its boundary, and we have a disk with B.
Okay, so I want to build a slightly more interesting four-manifold out of this.
And well, not very exciting, I suppose—what we’re going to do is cut it out.
So we’re going to remove an open neighborhood of that disk, and we’ll get something that looks, in the schematic, like I don’t know, like that.
This is X minus the disk. Fine—if I have submanifolds, I can cut them out.
Last thing we need is a lemma—we need to check that we’re kind of all on board—attaching a two-handle and carving out a disk are in a sense kind of dual operations.
Here, we’re adding and thickening up a disk to the outside; here I’m removing a thickened disk from the inside.
And the lemma relates them sort of even further.
So let’s suppose we’re given a disk embedded smoothly—let’s just say in the four-ball—and okay, I want the boundary of the disk again to land in the boundary of the four-ball.
Then there are two manifolds that I can build out of this data. I can consider B4 minus the neighborhood of that disk, and I can also consider B4 union a two-handle glued along the boundary of that disk.
Right? So this and that.
And the statement is not that these manifolds are the same, but that they have the same boundary.
And the proof of this is super cute.
Here's S4, which I can decompose along an S3 into two four-balls.
Let’s suppose that my disk lives in this one, and now I’m just going to repartition S4 where the neighborhood of this disk goes with this.
That’s B4 union a two-handle along the boundary of the disk, and the rest is the four-ball minus the neighborhood of that disk, and their boundaries are just identified by S.
I’m going to go over there.
Okay, so now we’re ready. But really, let me check in: questions or anything or unhappiness with the construction before I do?
Yeah, um, yes let’s say it’s smooth.
It turns out for three-manifolds there’s no difference.
But it comes from you looking smooth, you don’t have to appeal to anything.
More questions?
Alright, so this method I’m going to give for building contractable exotic manifolds is in fact due to Barry Mazur, the number theorist—it was his undergraduate thesis here—but the development of this owes a lot to Sullivan.
And here’s the construction. So first off, if we’re going to build exotica—you know, build the thing, sure they’re homeomorphic, sure they’re not diffeomorphic—for the homeomorphism, we’re always going to ask Freedman what he can do for us.
And what Freedman can do in this setting is he says that if you have any contractable C and C prime that have the same boundary, they’re automatically homeomorphic.
So if I want to build candidates for exotic—contractables, I’ll just build contractables and make them have the same boundary, and that’ll work.
So I’m really in the market for manifolds with the same boundary.
And we just learned how to build manifolds with the same boundary—the good thing about this construction is that they’re super-duper not the same or contractable.
This one has B2; this one has B1—okay, not the same.
But Madur gives us this really beautiful kind of way to use that lemma anyway.
So what he says you should do is consider a link L which has two components, and this link lives in S3, and the link has the following properties:
So one, both components are individually unknots.
So maybe what you have in mind is something like this—maybe that's L1, and maybe L2 is something like that. We can still see getting borderline.
Okay, and I won’t go any lower; it’s the size of it.
Okay, thanks. I will not go smaller.
This is a link—this one's kind of obviously an unknot.
If you squint at this one and you can see it, it’s also an unknot.
Okay, and unknots, they have this nice property which is that they bound disks—they just bound disks in S3.
There’s one—you can push that disk into the four-ball a little bit.
So unknots are also going to bound disks in B4.
So what we can do is construct a pair of manifolds in the following way.
We’ll start with B4. Our link L1, L2 lives in its B, and we can build a manifold here by removing a disk for L2 and attaching a two-handle along L1.
Let’s call that C, and we can build a C prime by doing the same thing in the other order.
So we’ll remove a disk for L1 and attach a disk for L2.
One interpretation of the lemma is that the boundary three-manifold can’t tell if you add a disk or remove a disk, so the boundary just can’t see the difference between these two setups.
So these guys have the same boundary, and as long as you assume that the linking number of L is one, then they are both contractable.
Alright, so I’m not now going to go on to prove that contractable manifolds you build like this aren’t diffeomorphic, but let me tell you that you can.
And in fact, for this literal link that you may or may not be able to see, the contractable manifolds you get are not diffeomorphic.
And in fact, this literal link is the one that was used here, and it’s also the one that these guys redetect.
So, you know, these two manifolds which are defined very simply from a very straightforward link like this—they give you a pair of contractable exotic manifolds.
Okay, and the theorem—the cork theorem—maybe gets like even better.
So this is the last kind of leg of the cork theorem: in fact, you can prove that any corks can be built using a generalization of this construction.
So any exotic pair is related by cutting out and regluing a contractable manifold, and you can assume that those manifolds are built using something like this.
Nope, you might need a bunch of components and then a bunch of components, and maybe you’re not using the four-ball either; maybe you’re using a fixed other contractible.
More questions?
Yeah, um, yeah, there’s some really easy ways to check it—they fact through a little bit of contact or simplicial topology, so they’re gauge-theoretic somewhere in there.
But like if you can draw an Andrew diagram of them where some Thurston Franklin number is positive, then you win.
Something like this...
Let me erase this one; this is the better one.
I’m going to talk about quantifying exoticness.
Anything before I do?
Yeah, so I just erased the theorem that had two attributions here, and it was one relative, one absolute.
Yes?
Um, yeah, it’s not too bad.
Let me at least, it’s not too bad!
So it turns out, this isn’t clear, but when you remove a disk for the unknot, what you get is S1 cross D3.
So you get this thing homotopically equivalent to a circle, and so if you want that dead, you better make that two-cell go around once, and somehow that turns into the linking condition.
Okay, so all exotica comes from cork twisting. Corks can always be taken to come from this construction.
So if you want to know how different two smooth structures are, one way you can try to measure that is by saying how bad does the link that gives the cork have to be.
So a couple of definitions of how bad a link like that might be are—well, okay, let’s suppose we have a candidate for an exotic pair. So X is homeomorphic to X prime, then we’ll say that the complexity of the pair is something like the minimum of the amount of geometric linking of L, taking over all corks which get you from X to X prime.
And we also study a version of distance which is called the stabilization number—that’s going to be the minimum size of the linking needed again over all corks; maybe a normalization for both.
A comment perhaps for people who come from a more high-dimensional perspective: both of these notions of complexity can be defined in terms of the H-cobordism, but I’m defining them kind of discreetly here.
So these definitions are set up so that for a homeomorphic pair, if the complexity or stabilization distance—stabilization number—are zero, well that’s the same thing as the manifolds being diffeomorphic.
Alright, so we certainly know that there are pairs of manifolds where this complexity is positive.
But in fact, it’s a major open question whether it’s ever bigger than one.
Do there exist X, X prime that are above complexity 1?
This one especially is kind of a very famous and apparently very hard question.
No, um, probably of this bigger?
There’s like a bipartite link, you want sub; I definitely want to subtract one.
Yeah, there’s normalization—sure, that’s right.
Subtract one, divide by two, but you know something that’s counting that and then appropriately being what you should be—I think you probably want to divide this by two, this by two also.
More questions?
Okay, um so we are apparently pretty far away from being able to address this, but there have been current developments.
And for manifolds with boundary, we actually can answer this question.
So let me start with the complexity.
This is a theorem—the heavy lifting of this is really from Morgan and Sabo in ’98.
We’re going to need something that’s due to Aqualu Ruberman, and if you want to see it written down, that was done very recently by Roberto L.
So, okay, the statement is that there exist contractables with boundaries where the complexity of that contractable pair is big.
And even more surprising and exciting is a result from S Kang a couple years ago who said that there are such contractables where the stabilization distance equals two.
And this, I think, yeah, this was a very surprising result.
Let me maybe say one more thing that’s a little bit outside the purview of the talk before I kind of move on from complexity, which is another very recent result from John Penglin in 2021 who said that, um, let me say there exist exotic diffeomorphisms—non-manifolds with stabilization of the diffeomorphism of the identity, whatever.
Let me just say stabilization number two.
So I’m not going to make this precise, but you know I said at the beginning there’s a largely parallel story to the one I’m telling for submanifolds and for diffeomorphisms of manifolds, and Junfang solved this problem for diffeomorphisms, and what’s really exciting about it is he’s working with closed manifolds, so he solved the real one.
That’s what I’m going to say about distance or quantification—questions here?
Ask questions?
Yeah, um, if you have a... if you have a spin C struct—if you have different Seiberg-Witten invariants, spin C structures, with a large expected dimension, then you can’t have small complexity, which is going to run you smack into the simple-type conjecture.
But that’s a bound. More questions?
Okay, oh, um, so let me tell you something about the techniques that Ty and I use.
And to start, let me give you a statement that I’ll talk about.
So one of the things we show is that there exist exotic four-manifolds which are homeomorphic but not diffeomorphic, or there exist four-manifolds that are homeomorphic but not diffeomorphic to S1 cross S3 connect some two copies of CP2 bar.
Nope, two copies of CP2 and nine copies of CP2.
Okay, so if you’re kind of keeping track of stats here, this is B2 is 11, so not particularly small, and π1 is Z.
That’s the biggest π1 you’ve seen all day, so this is maybe not a good theorem from the perspective of the table.
This is the one I’m going to tell you about anyway, and the reason I want to tell you about this one is because I think, well, because the proof is really easy and it still captures, I think, most of the ideas and developments that we use with other results to prove this.
We develop a new four-manifold invariant, α; it’s an integer—it’s only defined for four-manifolds that have some B3, that have some second homology.
Let me just say for convenience, B3 is one for today, and this invariant really is new.
It’s provably different from Donaldson, Seiberg-Witten, Higgard, FL-Mex, Bauda, and invariant.
We can prove some other things with it that I won’t write down because I’m not really going to motivate them.
Maybe just give me three sentences out loud, and then I’ll come back to saying things that I’ve backed up or defined.
So some things we can prove with α are that we can build manifolds with non-vanishing α invariant that have contained square zero embedded homologically essential spheres.
That’s something usually gauge-theoretic invariants vanish on, and we can also distinguish four-manifolds that are related by FAL or not, Serrano, and Alexander polynomial one, not using this invariant.
Okay, and unjustified sentences—so I’m going to define α for you and then show you a little bit about how you can expect to literally compute it sometimes.
To do that, I need to give you like the world’s shortest crash course in Higgard homology.
Any questions before we do that?
If you have a buster, then you don't have to do Higgard biology.
Alright, RS, so what is this?
This is an invariant package defined by H [Music] Sabo.
I have it backwards. Oh, okay, great!
Thanks, around—I don’t know, 2002. Everything I’m going to write down is due to them.
For experts, I’m thinking about HF red, and I'm summing over spency structures.
Okay, so what’s the invariant out of the box?
It’s an invariant of a three-manifold, and let’s just say it’s a finely presented module over Z mod 2. It satisfies a TQFT structure.
So cobordisms Z from a three-manifold to some other three-manifold, which, you know, that’s something like this—you have your two three-manifolds, and there’s some four-manifold running between them.
These cobordisms are going to induce maps between CH of Y, HF of Y prime.
In this way, you can think that it gives you a four-manifold invariance. This map is an invariant of the dualism.
Okay, and these maps can be written down sometimes, and one really powerful tool for writing them down is the existence of an exact triangle.
So there exists an, um, let’s see, for Z to handle cobordism, so if this is pretty simple, just start with Y cross I attitude handle—that’s it.
Then you have this exact triangle, which has the map you want in it.
So it’s going to go from HF of Y to HF of Y prime (that’s the map you care about), and then, okay, well there’s something else—let me call it Y sub K; doesn’t matter what it is.
I’ll just tell you that it’s some explicit ancillary three-manifold, and if you built the cobordism, then you know what this three-manifold is, and that’s it— that’s your crash course.
So this α invariant for a four-manifold with a little bit of B3 is defined to be the minimum dimension of Fred of Y for Y generating H3.
So this is a very common way, in general, of getting an invariant of a manifold.
We’re frequently building invariants by saying, “I don’t know—what’s the minimum genus of a surface that represents some class in H2?”
This is an analog of that, because you take a minimum, you get invariance for free—but usually, there’s a cost, and it’s so you can’t compute your thing.
Let me try to convince you that you can sometimes compute your thing in this case.
To state the lemma, let me get a little bit of a picture going. If your manifold has some B3, then it also has B1, so it’s got a little bit of a kind of circle in there somewhere.
So maybe that’s my picture of X, and my generator of H3 maybe looks like that.
Okay, so something I can do, this thing is going to have to be non-separating so I can cut along it, and I get a cism from Y to itself.
And the lemma is that if that cism, if the map associated to it—which goes from HFY to itself—if that's an isomorphism, then Y was minimal.
So you can find minimers.
If you can find cuts where you get nice.
So let me just conclude by saying then how should you build four-manifolds where α of X is whatever you want?
Well, we’re going to work backwards.
You’re going to take a three-manifold Y where the rank of HF of Y is n—it’s whatever you want.
Three manifolds, invariance are reasonably computable, especially if you ask somebody who knows what they’re doing.
So, okay, you can find something like this.
And now we’re going to build a cism Z from Y to itself, so something like this—such that two things: I want the map FC to be an isomorphism, and I want π1 of Z to be trivial.
Then you define X to be Z with the NS glued together, and by setup, you get that α of X is whatever you want it.
Okay, so you can kind of... this is a pretty straightforward way of writing down a four-manifold that has whatever α invariant you want, supposing that you can do this.
So let me just say one thing—maybe in words even about why you can do this, in particular, how are you going to get your hands on this?
These maps are, you know, you can’t necessarily always just write down any eabo is a map.
And well, you can keep control of this by keeping a good handle on this, so if you build your Z by only using two-handle cisms, and you only attach them in ways where this ancillary manifold has no HF, then you just force an isomorphism across the top at every step.
So as long as you build a thing and you’re pretty careful, then you get this.
And I will stop there. Thanks!