Lisa Piccirillo: Exotic Phenomena in dimension 4

[Music] um it's my pleasure to introduce the second speaker for today uh liisa pillo from um MIT and UT tin Lisa pillo got her PhD from UT tin after a an undergrad um at Boston College and after that she has continued to alternate between Boston and Texas um she is most famous for solving the Conway not problem and she has been awarded the Mariam meakan New Frontiers prize uh the clay research Fellowship a SLO Fellowship among other Awards um so it's my pleasure to introduce her and her talk on she would speak on exotic phenomena

in dimension four thank you hi yeah thanks so much uh for the introduction and for coming um I was very pleased to be invited to speak here I was also a little bit surprised because um every time I've spoken at Harvard before I've really just given an awful talk so um hopefully today we're going to break the curse um but also you've been warned uh so okay so so I'm a low dimensional topologist so 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 um okay but if you're going to do that you you probably first first need to specify a little bit more carefully What you mean by manifolds uh so you probably should you should declare what dimension you're going to be working in uh but you also probably should declare what kind of manifold you want to study uh historically there's there's kind of three types of manifolds that that people are are interested in and 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 top To ological manifold today so that's kind of no extras but we also study smooth manifolds where we're going to just require that uh in addition the transition functions r c Infinity um and also historically people are interested in something called a PL manifold uh but I'm not going to tell you what that is today uh reason being is that in dimension four it's the same As smooth so it won't really come up uh okay let me also just make a couple of um convenience assumptions today all of my manifolds are

going to be oriented and unless I explicitly say otherwise Compact and uh without boundary so these are the kind of manifolds maybe that I'll 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 uh tell you what we know in other dimensions so in the sort of classical low Dimensions Dimensions 1 two and three um maybe I don't know uh check we have classification theorems um in dimension three that was uh not easy um but that's a story you've heard

before and not the one I'm I'm going to tell today um Let me let me also just tell you that there's a theorem um in dimension three Of Moise that uh there's no difference between smooth and topological manifolds in low Dimensions okay uh the classical High Dimensions so greater than or equal to five um in in high Dimensions there's something called the surgery program uh so or surgery Theory uh and this is a theory that was developed in the 60s uh by a whole bunch of folks so uh let me let me call out um

Router novakov Sullivan and wall in particular and um in a slogan what you can think is that surgery theory is going to reduce the classification of manifolds to problems in algebraic topology um I want to comment here that I I think there is a misconception I I had this misconception for a long time that because surgery Theory exists the classification of high dimensional manifolds is kind of done um that's not True it's kind of not my problem anymore um but it's it's not known uh in general so so there are things that we know for

example Simply Connected smooth five manifolds are classified um but there's lots of things we don't know for example the smooth pay conjecture is open in dimension 126 um okay um and in high Dimensions um as as perhaps you know um we can have a difference between the Smoooth and topological categories that's originally work of Milner okay so that brings us to Dimension WR let me start off by talking about the topological category um where uh groundbreaking work of fredman from ad2 tells us um well one way to phrase it is that surgery Theory works topologically

it works when the Fundamental group is uh what he calls good so sometimes this Machinery uh will will let you get classifications in Dimension four for topological manifolds um and and he carries this out um for example he gives a classification of Simply Connected topological for manifolds so the set of X4 top with uh no Pi 1 fredman tells us that this is injection with some other set of algebraic dudads that are that are more Well understood so it's in B with this set of pairs um there's a form from ZN to Z uh which

is whatever unim modular bilinear symmetric and there's an integer in Z so so right the theorem and and I wanted to kind of state this to see what classifications um 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 but anyway there's a big algebraic set and and you say that um all of these correspond to a unique manold oh let me um interject here with just a couple definitions for people who don't do um topology or geometry

all the time um this this Pi 1 thing I keep referencing this is the fundamental group uh it's not important that you know what that is today uh just know that it's a a group that you can associate to a manifold is kind of a Primary invariant for them uh and this thing right here I'll mention a lot as well this is called the uh intersection form again it doesn't matter that you know what that that is um but it's a form which is an invariant of a manifold um and fredman tells us that for

Simply Connected topological manifolds it's like the invariant basically um if you know what this means the intersection form is the cup product on H 2 okay any questions so far this is kind of a board on everything I I'm not going to talk about um let me also say that I really like to get a lot of questions so so what I really want to try to tell you about today is um what we know about smooth for manifolds which is nothing Um we have no classification theorems we have no conjectured classification theorems um 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 um we're going to say that a smooth for manifold X is exotic if there exists another smooth Form manifold X Prime which is homeomorphic but not diff amorphic text in this language uh the smooth four-dimensional ponre conjecture says that uh S4 is not exotic um but in fact we we know from uh

Donaldson's work in the early 80s that there do exist exotic form manifolds and these two uh statements together serve to motivate um the question That'll sort of guide The Talk today which is uh which smooth for manifolds are exotic um and since we're maybe particularly interested in in one day eventually understanding the ponre conjecture maybe we're particularly interested in understanding which small for manifolds are exotic where small will maybe be taken to mean today that the fundamental group is Trivial uh Donaldson's examples have That uh 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 uh invariant that I'm going to sort of impose upon you today this is the second Betty number uh 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 uh B2 of the four sphere is zero so this is the question that I'm Going to use to to guide The Talk today um the plan is to tell you classically uh what are

the techniques and what do we know uh then I'm supposed to I guess tell you about current developments so what do we know and how do we know it um but then what I really want to talk about and what I'll spend a lot of the second talk on is um 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 questions okay so um before I move on let me just

um 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 um but it it's really not the Only question in smooth for manifold topology um there there's a lot of things we want to know about uh for manifolds and we are also interested in things other than the manifolds themselves for example sub manifolds particularly in codimension 2 that's kind of higher not Theory um we're also interested in dimorphisms of manifolds and um for these two types of objects

submanifolds and endomorphisms the story I'm going to tell today uh by and large Has a parallel story um and in fact both of those stories are are very active right now as well uh and maybe the final uh comment is that uh from now on unless I explicitly say otherwise all manifolds are smooth and Simply Connected okay so classically how do you build Exotica so um this is really kind of a three-step Process the first step uh is to give yourself some candidates so build a pair of smooth smooth for manifolds x and x Prime

which you know you think reasonably might be homeomorphic but not diffeomorphic uh step two show their homeomorphic this is really riveting stuff uh step three you know where this is going uh okay but I wanted to point this out because I'm going to do it a lot uh it'll give me a little structure and I Also wanted to uh sort of sort of point out that this is basically under control by Freedman um so really uh what you need to do is you build yourself some some manifolds and you just get the intersection form right

and then fredman says you're good to go here and now you just need to distinguish them okay 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 um give a list of uh some ways that classically we get we get four manifolds to study and it's also good to have these kind of I don't know in mind have some examples to be thinking about like as we go through the talk any questions before I do I'm G to

keep stopping and asking for questions okay well event entally Um so the first um sort of place we we get for manifold is is some sort of very basic small examples uh for example uh there's the for sphere that's great there's complex projective space uh there's its orientation reverse and you can take connected sums of these things uh that maybe seems very like very basic but in fact um sums of cp2 and cp2 bars play a huge role in the Exotic literature uh okay you can also try to build yourself some four manifolds out of

lower dimensional manifolds by taking products or bundles um for example S2 cross S2 or more generally products of surfaces uh are very interesting examples or or maybe you want to take uh a manifold cross S1 okay uh another way you can get some Interesting form manal folds to study is you can ask a geometer to give you some and in particular uh complex [Music] surfaces and simplec manifolds have played a large role [Music] um and those are actually kind of for the most part those are our like root examples uh I'll say a little bit

more about about mangling them but this is uh this is kind of the the natural sources For the most part that were studied classically um so we also like to take them and kind of Frankenstein them together using cut and paste operations which is uh what it sounds like you have some manifold X where which contains some co-dimension zero sub manifold 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 I just Wanted sort of mention here this actually um this is

a way most people work with manifolds on a day-to-day basis but it actually didn't doesn't really play a role in the classical uh exotic literature but you can build manifolds out of these really basic building blocks called handles um and I'll say what those are in the second t uh boards so that's how we build manifolds uh 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 um so I'm going to say very little about this today there are uh like

four or more worldclass gauge theorists in this room um none of them is me uh very crudely this is uh an invariant that counts solutions to a pde on the manifold uh it's a very powerful uh Invariant Theory uh it's not an exaggeration to say that everything we know about Exotica we know thanks to guge Theory um so so this is this is how we're going to tell our manifolds apart for the most part at least classically um but I just wanted to give you a couple of warning ings about gauge Theory the first is

that in general it's not possible to compute it explicitly and you know 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 um so well people are able to sort of get around this but it's it's largely by using formal properties you can sometimes get your hands on things if you have some geometric structure um and we have some understanding of of some of how G how these invariance behave under gluing okay so that's the first warning the

second warning is that gauge Theory Doesn't work um when you have no B2 the invariant just isn't defined um invariance uh so in particular uh while this this this theory is has taught us a lot and and there's still a lot it can do for us uh you're not going to disprove the the P conjecture uh using a gug theoretic invariant at least as we know today um yeah good yeah Um the methods are the same if 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 your hands on this all right sometimes we can say more about classifying manifolds that have more structure among those manifolds which have more structure but that's question more questions okay so that's what I wanted to say about um classical techniques um So now I meant to tell you what we know which is quite a lot uh this board is deliberately too small to read um there's a lot of Exotica literature uh we would be here for a long time if I tried to kind of state everything

carefully and give you all the names so uh let me instead uh try to give you like uh tell you what the patterns are on this board uh so so how's the board working Uh there'll be kind of three eras I'll discuss in each era I'll tell you how small we can get I'm assuming manifold are Simply Connected unless otherwise I'll tell you how many smooth structures we can build who did it um 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 uh obstructions so so the first era in in development of of Exotica is is The Donaldson era in the 80s it's kicked off with Donaldson's development of of yangang Mills gauge Theory uh the examples in this t uh 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 towards the end that's kind of Arrow one era two uh is kicked off by the development of the Cyber Wht equations in uh 1995 and there's an incredible amount of

work uh then we don't actually see any increases or any decreasing in the the size of the manifolds we can get during this era um but I sort of want to emphasize that this this chart is is is problematic in that that question is is not the only question you want to know about about for manifold so I'm only presenting uh sort of smallness here there isn't smallness developments but but in this There there's a ton of um new results about smooth form manifolds for example questions about minimal genus surfaces representing homology classes and a

lot of other things so um this maybe makes makes it look an important the second column oh sorry this one no no sorry the First Column ah the first one yeah great so this is B2 um so this is the size of the manifold so so Donaldson's in first invariance Have B2 equals 10 uh in the 80s we got nine uh and that's about it in this second cyberg Wht era no development nothing smaller um 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 a that's a great question yeah it's it's um I think it's because but one of the one of the reasons is because of a lack of small Examples um when you uh you look at this list and and you say you're only interested in Simply Connected so you probably don't want products of surfaces or things like that um well you immediately end up here and these tend to have a lot of B2 uh 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 like one of them eats some B2 out of the other one And you end up with very messy objects um yeah thanks more questions yeah yeah oh yeah sorry Tom uh that's Tom but nobody can see that anyway uh thanks okay so so this is the second era uh the cyberg Wht era in the mid 90s uh big development in techniques both constructively and obstructively but actually no movement on the smallness of The exotico we can produce more questions about that era anything okay and then

the third era is kicked off in 2005 uh by uh a result of jungle park who is rational blowdown a technique that was developed in the Cyber Wht era um to get the the first Improvement in the size of Exotica uh in a long time so he gives us exotic Simply Connected manifold with B2 equals 8 and that like kicked off an arms race everybody works on this um 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 um we start getting smaller

and smaller B2 but nobody cares about five but that's fine um until the end of the era is this result um a v made of and park so they give Simply Connected be 2 equals three exotic manifolds and that's the smallest Exotica we have for a long long time yeah uh there are a lot of candidates um If if you just kind of forget about ever being able to compute a gauge theoretic invariant for it you got lots of options um more questions yeah yes exactly that's right yeah um it doesn't uh we just um

by and large don't have a more combinatorial invariant but I'm about to say something About that yeah more questions okay so let me Tred 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's three kind of camps of results uh recently that I want to say something about the first is some work uh from 2021 um which really doesn't make things Smaller it's like 23 the biggest biggest thing on the table yet um no development in techniques the

development is in the obstruction so in 2021 we we get the first example of a pair of exotic manifolds which are distinguished with something called the slic approach uh which is this method of distinguishing manifolds which is a little more uh convoluted than just saying like compute invariance for both Hey look they don't match um and this approach I'll tell you what it is in just a minute but uh let me sort of tell you now that that the thing that's exciting about it is that um versions of this approach could possibly work with nob2

uh so it was good to know that this approach like can work ever um since since we're maybe hopeful that that it could work in in cool settings later um and that's joint work with uh Tran Manescu and Marco mangon uh the second set of results uh that I want to tell you about is uh from last year and again um the examples in these results are not particularly small there not really movement here um what's what's exciting about uh this result uh 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 um but we Wanted to be a lot more juvenile about our constructions so we're really just going to start with with handles with 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 hagard FL Mixon Variant so you should not really think that we are uh getting away from

Gauge Theory here conjecturally the higar FL mixed invariant is equivalent to the cybert invariance so this probably is a gauge theoretic proof um 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 Lavine and Ty liman okay because our constructions are Much more explicit uh we can monkey around and make other stuff happen so we're also able to produce uh exotic four manifolds with B2 equals 4 so not the smallest but pretty small at the expense of having a little bit

of Pi 1 Z 2 uh but in fact what's interesting about these is that their intersection form is definite um okay so so that maybe doesn't mean too much to you um an intersection form as you know can be Definite or indefinite and let me sort of assert that all the Exotica uh previous ly was indefinite and you know there's other things we ask about manifolds whether then is there small Exotica and and for some of those other questions you see a a difference between the behavior of definite and indefinite manifolds so it was maybe

reasonable to to wonder whether definite manifolds were more rigid um we're saying here that they're Not okay and then the most exciting uh result I'm going to tell you about today which I uh wish I could tell you was due to me but but it's not um is work from last year from STI shits and Sabo uh and they're going to give you exotic manifolds with B2 = 1 Pi 1 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 uh their techniques are going to be a mix from uh the good old days uh oh I lost the green and from us and they compute the Cyber grd variant and then the final result that I I kind of want to mention um in terms of progress in in Exotica um is very very recent uh from earlier this year um this is a result about exotic manifolds with boundary which in a sense doesn't even Belong on this table um the with boundary setting is very different it doesn't make sense to compare um I'll talk

about that a bit more later on um this is work of K Ren who is a graduate student and Mike Willis uh and what's so exciting about their work is that their obstruction is combinatorial so they use something called a ske lasagna module which comes from cavology um and this is the first time That uh compact exotic manifolds have been produced without gauge Theory questions yeah no more questions you can put a description of a manifold into my collaborators and maybe get something here um more questions yeah y yep um you look at the universal

cover uh if if if the manifolds have a non Morphic Universal covers and they're Nom morphing the universal covers have B2 Plus good question other question oh yeah um right so okay what i' want at the end of the day is the set of smooth manifolds with pi 1 trivial is injection with anything and just give me like a reasonable other set that you can you can relate them to that would be awesome um for now The the question that seems to motivate us more or like the question that we can sometimes work on apparently

is the one where we say like where can I find Exotica sort of a 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 um 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 um to tell you uh what this license approach is this is an argument

that's du to kassin in the 70s and uh here's here's how Caston says you can maybe try to distinguish manifolds so let's suppose um you have a pair of candidates for Exotica so you have a pair of homeomorphic smooth form manifolds uh here they are that's X and this is X Prime and they homeomorphic all right so what C would like you to do is uh remove an open neighborhood of a point from both of them uh these manifolds now have boundary they have an S3 boundary component from that ball you removed um they're still

Homeomorphic and now here's here's sort of what what Cen wants you to do he he says suppose you could find a not K in S3 uh such that two things happen one um there is a D2 embedded smoothly in in here in xir uh such that the boundary of the dis is this notot so what you should have in mind is that you're not it 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 disc that abounds somewhere in here

and then also um you should want that there does not exist such a dis in X Prime uh where the boundary is K uh if you have both of these things then that implies that the manifolds are not diffeomorphic uh it's easy to see that The punctured manifold can't be diffeomorphic if you had a theomorphism you push this disc across it and the knot would bound over here and it extends to the close things so that's the argument it's um you know it's not quite as direct as like comput the things they don't match um

but it's so not too bad so why should or any questions about the approach or anything else thank [Music] you more questions more mistakes on the board okay so so why should you want to do something like this uh well for one uh this argument uh can be used to show that R4 is exotic so this is the non-compact analog of the ponre conjecture and it's false so so this argument has teeth uh sometimes in a slightly different setting uh okay so so that sounds kind Of good um then we'd like to know whether you

can use this argument to show that S4 is exotic um and well we don't know but there is an invariant or there's a few now actually there exist invariance in the literature which could uh do two they could provide the obstruction here H even when X Prime is S4 so unlike uh using gauge Theory uh we could conceivably just prove the ponre Conjecture like this using tools we have today so all of that sounds kind of good um but here's what's not so good um it's it's it's not easy it's not at all clear um

how to sort of set this up um how to find a good KN K so if you have um even two even a known exotic pair it's not it's not clear how to find a not which bounds a smooth disc in one of them but not the other and that that's what we what we did Here it was the first time that that this argument had sort of been run in the compax setting okay question so okay this is an argument it works sometimes uh it could maybe disprove the ponre conjecture uh maybe that's something

we should be trying to do uh so in follow-up work with uh Chan manasu um well we just sort of tried to think about how you would try to get This running um so we we we gave a very systematic method for producing uh 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 counter examples of the P conjecture and those candidates come equipped with a knot which we don't know that it doesn't bound a

dis in B4 But it doesn't you know seem to have a good reason to right and the method is is really systematic um I personally made a computer spit 3K examples and I personally am terrible at coding um so we produced a whole bunch of examples and uh some of them were kind of interesting interesting meaning we we you know we have we get some knots and there's something like if the knots have Blah property then the punk Ray conjecture is false and we just sort of don't know we fail to show that the Nots

don't have that property um I don't think this was ever meant to be like aha these ones are going to do prove the pcre conjecture it's more meant to to mean somehow the method isn't kind of trivial out of the gates like it does spit some kind of subtle looking stuff um okay since then um it's been Proven that uh these examples are actually not particularly interesting uh by my uh younger sibling Kai Nakamura um but what Kai did it it really just ruled out like the examples we wrote down it doesn't kill the method

you can build more um there's a bazillion ways you can prove build more but um if you want to see some that have been written down chian student chak has done so um and you can you can start to see How like hopefully there's going to be a little bit of a 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 is not going to disb the ponre conjecture that would be fine um so that's that's sort of the idea um but

of course um given that it's whatever year it is um not only are we having that conversation um ml is having that conversation so uh There's work of gukov alerson uh Chien and uh R whose name I really can't spell sorry uh who are using ml to both build so to run our construction and study the examples it produces so uh the neural net that kind of evaluates them is out that came out last year and this one is in progress um and maybe a comment about these is um if the if the neural net

if If if the neural Nets win it will really be a proof um they are biding explicit constructions and the way they stud stud examples it it does spit something verifiable that's um that's what I wanted to say for the first half um questions yeah oh well uh I don't know because yeah they don't they haven't produced they they claim they're producing um but oh but they're going to be homotopy Spheres uh how big are the knots I don't know big uh yeah so right good so there's actually a little bit of a 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 was did bound a disc then the park R Construction is false so what they act what the net actually does is looks for Discs oh they definitely are detected by by Cy oh yeah definitely are it's a stand standard do other yeah it's the theer 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 argument so welcome back for Lisa Pico's second talk on exotic phenomena in four dimensions hey uh thanks for coming back Um So the plan for uh the second talk is uh what I said I was going to do but not in the order I said I was going to do it um so I'll spend uh at least half of the talk talking about um where Exotica comes from I can give you explicit pictures of um how

you can get exotic manifolds um in a pretty high level of generality and I I want to do that um because it's it's very nice it doesn't have to be sort of mystical um then I'll talk a little bit about how you might Try to uh say how how distinct a pair of smooth structures are and then hopefully I'll run out of time um but if I don't I'll tell you a little bit about the work with tyen atom okay so um starting off with kind of where where Exotica comes from so um let me

start off with an observation let's suppose you have uh a pair of a candidate exotic pair so you have a pair of smooth for manifolds and they Homeomorphic the observation is that if x an X Prime uh co- bounded a five manifold W and so what you should be thinking is it that it looks something like this like maybe that's X and here's X Prime and I'm asking that they that there's some five manifold which has the two of them as its boundary components so if they c-bound a five manifold like this and in fact

that five manifold is diffeomorphic to a product It's actually just theomorphic to X cross I um well then that's going to imply that my manifolds are themselves dimorphic if this cobordism is 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 a dorph so so manifolds are diffeomorphic if they co-bound products uh Um and in fact uh Cada for Exotica do co-bound something that's very close to a product uh so this is a theorem of wall uh he

tells us that such four manifolds are going to c-bound something called an H cobordism so an H cobordism is not literally a product but you should think it's a it's a five manifold that has the algebraic topology of a product okay so you have your candidate exotic pair you have this cobordism Which algebraically is pretty boring by the way more formally um what this means is that the boundary inclusions are homotopia equivalences uh but you you have your two manifolds they co-bound this like algebraically very nice five manifold and if that five manifold were actually

just literally product then they would be the same so there's a sense in which that five manifold is capturing how how different the manifolds are like the the failure of That thing to be a product is what's generating uh what's generating the difference in these smooth structures um and we can actually make that even kind of more uh explicit and a little more four-dimensional uh so so let me kind of get a picture going so what's our agorism look like it's like there's X here's X Prime uh we have have something running between them which

is maybe like a bit weird Okay um so in fact while tells us That not only do we have this nice manifold running between them but actually um most of it's boring uh there's a sub AG 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 here and this is very boring and in particular like 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 okay and and the theorem I actually really kind of want to spend most of the rest or this this section talking about is something called the cork theorem uh which is going to improve this even more um so the cork theorem was proven by everybody in uh roundabout 1997 so let Me give you some names but not write them um Curtis Sayang fredman stong mat matv Baka maybe we should say Kirby and Cass improved a non-compact version of it like before all

of them were born um okay so so this kind of was in came out of I don't know anyway it's due to a lot of people um what it says is that um even better W Prime this bit that has all the interesting stuff going on uh can be taken such that uh if you look at how W Prime intersects X so let's call that c that's right here 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 contractable

so contractable means that their algebraic topology is as simple as possible so we have the algop of B4 so kind of all of this put together Is telling me that if I have two homeomorphic smooth for manifolds then what's different between them it's all packed in this piece and this piece doesn't have any algebraic topology um let me just write down uh to be explicit a COR uh what we learn here is that if you take X you remove C you glue in C Prime what you get isomorphic to X Prime right we start here

lose this that's the same as this that um so these two uh Doohickeys reps are called corks uh and this operation of cutting one of them out and gluing in the other is called cork twisting um any questions about this statement or anything so there there's several things I think are is really really surprising about this result um one of them is is that um well these corks are are pretty interesting objects in their own right Um you know 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 um so so let me write that down kind of explicitly um this is a result that was originally proven a little bit earlier in a slightly different form let me also quote o rubberman for the form I want um so what what we get from this is that um there exists exotic Contractable for manifolds now let me emphasize that contractable manifolds have boundary so this is not uh quite the setting I've been talking about kind of all along um but contractable Mana folds are very simple they have essentially

no algebra apology B2 is zero everything's everything's trivial so um this is uh very reasonably the with boundary analog of the punk Ray conjecture and uh again it's false And another thing I think is really surprising about about this theorem is it um is is that you know over here we're having this really hard time producing exotico with no boundary when there's very little algebraic topology but this is telling me that that that issue is is technical like I can pack the exoticness between like the difference between the two manifolds into something that has no

algebraic topology so so somehow it's not the lack Of algebraic topology that's that's giving us the issue here so I'm going to um give you fairly explicitly uh examples of exotic contractable manifolds um any questions before I do yeah um yeah they'll be integer homology three spheres um they can be as simple as surgery on a knot more Questions before I I try to set this set this up try to explicitly build some of these let me just comment that um this is the theorem that run in Willis reproof technically they kind of reprove that

version of it all right so to set this up I have to tell you about handles and I also have to remember not to write in the shadow um so help me out with that if it doesn't go well So I'm going to try to do a one we can we can see this part of the board great I'm going to try for definition um by picture here uh so please bother me if it's it's not clear uh so so handles are a set of building blocks for for building manifolds and and uh you should

really think that they're kind of a a manifold version of a cell complex so I think most of us know how to build a a cell complex um and we're going to to do the 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 um 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 uh a zero cell there it is it's a point um okay but that's like not great as a Three manifold so um my zero handle will be this threedimensional thickening of that okay um and okay now I guess I wanted to add a one cell to this so you know what would that 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 um in dimension four works the same let me try to give something of an example so let's suppose I want to build a a a four manifold out of a zero handle and a two handle say so so what do I do I start with you know my generalization of zero cell so there's my zero cell I need to thicken this up in in four dimensions so I I can't draw you a picture of that but let me just draw you this picture again and declare that

that's B4 uh4 there it is that's my Zero handle okay and now I said I want to glue I want to use it two handle so that's going to be this kind of thickening of a two cell so all right 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 uh okay here's the boundary that's an S1 here so I need to glue this 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 um okay okay pick a knot fine glue and then well all right we need to turn the whole thing into a for manifold by Crossing this with another D2 technically you need a framing here we're not going to worry about that um and I'm only going to build four manifolds out of two handles today so That's what you need questions about you know Handles in the kind of broad sense okay let

me give you two more um quick facts about handles one is just a uh even kind of Cheaper picture of this um so a schematic or an even more schematic of of a four-dimensional two handle that you'll see me use a bunch of times is um 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 in that boundary my two handle is you know fundamentally there's kind of the the

D2 and my two handle you know looks something like this yeah so I'll use that representation of a four ball of the two of two and then maybe um the final thing uh I wanted to say is that it's theorem of Morse that all smooth manifolds can be built in this Way questions anything wrong on the board so to build a to build a contractable uh exotic manifolds I need uh two handles and I need one more operation uh which is something called carving so um building manifolds by carving so let's suppose I'm given a

form man x uh smooth um with a a disc a D2 embedded smoothly in X such that the Boundary of the dis goes into the boundary of format so you should be picturing something very similar schematic something like this here's X here's it boundary and we have uh a dis with b um okay so I want to build a slightly more interesting for manifold out of this um 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 disc and we'll

get something that looks in the schematic like I don't know like that this is xus fine if I have sub manifolds I can cut them out um last thing we need is a Lemma need to check that um we're kind of all on board um attaching a two handle and carving out a disc are in a sense kind of dual operations here we're adding and thickened up disc to the outside here I'm removing a thickened up Disc from the inside um and the LMA uh relates them sort of even further so let's suppose we're given

uh a disc embedded smoothly let's just say in the four ball um and okay I want the boundary of the disc again to land in the boundary of fourball then uh there's two manifolds that I can build out of this data I can consider B4 minus the neighborhood of that disc and I can also consider B4 Union a two Handle glued along the boundary of that disc 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 um here's S4 which I can decompose along an S3 into two four balls let's suppose that my disc lives in this one and now I'm just going to Repartition S4 um where the neighborhood of this dis goes with this that's before Union a two handle along the boundary of the disc and the rest is the four ball minus the neighborhood of that disc and their boundaries are just identified by S I'm going to go over there okay um so now we're ready Um but really let me check in um questions or anything or unhappiness

with the construction before I do yeah um yes let's say it's smooth it turns out for for three manifolds there's no difference um but it comes to you looking smooth you don't have to appeal to anything more questions all right so um this this method I'm going to give for building Contractable Exotica is in fact due to Barry Maser the number theorist it was his undergraduate thesis here um uh but the development of this has owes a lot to to Salon um and and here's the the construction so first off if we're going to build

uh Exotica you know build the thing sure they're homeomorphic sure they're not typomorphic for the homeomorphism we're always going to ask fredman what he can do for us and what fredman can do in This setting is he says that um if you have any contractable C and C Prime that have the same boundary are automatically homeomorphic so if I want to build candidates for exotic contract bues I'll just build contracta bles and make them have the same boundary and that'll that'll work so I'm really in the market for manifolds with the same boundary and uh

We we just learned um how to build manifolds with the same boundary uh which which is good except that these manifolds are super duper not the same or contractable um this one has B2 this one has B1 uh okay not the same um but uh but they merer gives us this really beautiful um kind of uh way to to use that LMA anyway so what he says you should do is H consider a link l uh which has two components uh and this link lives in S3 Uh and the link has the following properties so

one uh both components are individually on Knots so maybe what you have in mind is uh something like this maybe that's L1 and maybe L2 is like something like that we can still see getting borderline okay and I won't go any lower it's the size of it okay uh thanks I will not go Smaller this is a link uh this one's kind of obviously an unot if you squinted this one and you can see it um it's also an unot okay and unot uh they have this nice property which is that they bound discs they

just down discs in S3 there's one um you can push that disc into the four ball a little bit so un knots are also going to Mound discs in B4 uh so what we can do is construct a pair of manifolds uh 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 disc 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 disc for L1 and attach a disc for L2 one interpretation of the Lemma is That the boundary 3 manifold can't tell if you add a disc or remove a disc 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 um all right so I'm I'm not now GNA go on to to prove that contractable manifolds you build like this aren't uh Diffeomorphic um but let

me tell you that you can uh and in fact for this literal link that you may or may not be able to see um the contract manifolds you get are not diffeomorphic and in fact this literal link is the one that was used um here and it's also the one that uh these guys redetect uh so you know this kind of uh these two manifolds which are defined very simply from from a very Straightforward link like this um well they give you they give you a pair of contractable exotic manifolds um okay and the theorem

uh that things get theorem the cork theorem maybe gets like even better um so this is the last kind of leg of the cork theorem in fact um we you can prove that um any corks can be built using a generalization of this Construction So any exotic pair are related by cutting out and regluing a contractable manifolds and you can assume that those manifolds are built using something like this uh 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 um they they fact through a little bit of contact or simplec topology so they gauge theoretic somewhere in there um but like if you can draw aandre and diagram of them where some thirst and Benin 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 um let me at least it's not too bad so it it turns out this isn't clear but when you remove a disc for the unot what you get is um S1 cross D3 so you get this thing's homotopia equivalent to a circle and so if you want that dead um you better make that that sort of 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 of how bad a link like that might be are um well okay let's suppose uh we have a Candidate for an exotic pair so X is homeomorphic to X Prime then we'll say that the the complexity of the pair is something like uh the minimum of the amount of geometric linking of L uh taking over all

corks which get you from X to X Prime uh and uh we're also we also study a version of of distance which is called the stabilization Number that's going to be the minimum size of the linking need again over uh all works maybe a normalization for both um a comment perhaps for people who come from a more High dimensional perspective both of these Notions of complexity can be uh defined in terms of the H goris but I'm defining them kind of discreetly here so um these definitions are set up so that um for a homeomorphic

pair um if The complexity or or stabilization distance stabilization number are zero well that's the same thing as the manifolds being diffeomorphic all right so so we certainly know that there are um pairs of manifold where this complexity is positive um but in fact it's a major open question uh whether it's ever bigger than one um do there exist x x Prime that's Above complexity or um and this one especially um is kind of a very famous and and apparently very hard question no um probably of of this bigger there's like a bipartite link you

want sub I definitely want to subtract one yeah yeah there's normalization sure that's right yeah subract one divide by two but 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 um but uh there have been current developments and um so for manifolds with boundary we actually can answer this question um so me start with the complexity um this is a

theorem the heavy lifting of this is Really from Morgan and Sabo in 98 uh we're going to need something that's due to aqualu rubberman and if you want to see it written down that was done very recently by Roberto L uh so okay the statement is that there exist contracta bles so with boundaries where the complexity of that contractable pair is big uh and even more surprising and and exciting is a result from s Kang a couple years ago who said that there are Such contractable where the stabilization distance equals two um 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 uh which is uh another very recent result from John penglin 2021 who said that um for let me say there exists exotic Diffeomorphisms so non manifolds with uh stabilization of of the defom morphism 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 sub manifolds and for dimorphisms of manifolds and and junfang solved this problem for dimorphisms and what's really exciting about it is he's Working with closed manifolds so he solved the the real one that's um that's what I'm going to say about distance or quantification uh questions here ask questions yeah um if you have a if you have have a spin C struct if you if you have different cyberg written variants in 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 um but that's an that's a bound more questions okay oh um so let me tell you something about um the techniques that tyam and I use um and to start let me give you um a statement that I'll talk about uh so one of the things we show is that there exist exotic uh four manifolds which are uh Homeomorphic but not diffeomorphic or there exists 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 uh keeping track of stats here uh this is B2 is 11 so not particularly small and Pi 1 is z uh that's the biggest Pi 1 you've seen All day so this is maybe not um not a good theorem from the perspective of the table um this is the one I'm going to tell you about anyway um 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

the kind of ideas and developments that we use um with other results to prove this uh we develop a new for manifold Invariant uh Alpha it's an integer it's only defined uh for four manifolds that have some some B3 that have some second homology let me just say um for convenience uh B3 is one for today um and this invariant really is is new um it's provably different from Donaldson cyber Wht heg Mex bauda and variant um 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 um three sentences out loud and then I'll come back to saying things that I've backed up or defined um so some things we can prove with Alpha are that uh we can we can build uh manifolds with non- Vanishing Alpha invariant that have a contained Square zero embedded homologically essential spheres um that's something usually gauge theoretic invariance vanish on um and we can also Distinguish for manifolds that are related by FAL or not serg on and Alexander polinomial one not using this invariant okay and and unjustified sentences so uh

I'm going to Define Alpha for you and then show you a little bit about how uh you can expect to actually literally compute it sometimes um to do that I need to give you like the world's shortest crash course in heg homology um any questions before we do that if You f a buster then you don't have to do Hager phology all right RS uh so what is what is this uh 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 um for experts I'm thinking about HF red and I'm summing over spency structures um okay so what's the invariant out of the box um it's an varant of a three manifold and it's a let's just say it's a finely presented module over Z mod 2 it satisfies a tqft structure so um 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 uh these cobordisms are going to

induce Maps between Che of Y HF of Y Prime um in this way you can think that it gives you a four manifold invariance this map is an invariant of the cism okay and um these Maps can be written down sometimes um and one really powerful tool for writing them down is is the existence of a of an exact triangle so there exists A uh so 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 uh then 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 uh me call it y sub K doesn't matter what it is uh I'll just tell you that it's some explicit Ancillary three manifolds um and if you if you built the cobordism then then you know what this three manifold is and that's it that's your your crash course um so um this Alpha invariant for a four manifold with a little bit of

B B3 is defined to be the minimum uh dimension of Fred of Y for y generating H3 so this is a very common way in General of of getting a invariant of a manifold we're frequently building invariant 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 um because you take a minimum you get invariance for free um but usually there's a cost and it's so you can't compute your thingy um let me try to convince you that you can sometimes compute

your thingy in this case um to State dilemma let me um let Me get a a little bit of a picture going if your if your manifold has some B3 then uh 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 of H3 maybe looks like that uh okay so something I can do uh 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 minim ERS um if you can find Cuts where you get nice so let me just um conclude by saying then how uh should you build uh for manifolds where Alpha of X is whatever you want Um well we're going to work backwards um you're going to take a three manifold Y where the rank of HF of Y is n it's whatever you want um three

manifolds uh in variance are reasonably computable especially if you ask somebody who knows what they're doing um so okay you can find something like this um and now we're going to build a cism z from y to itself so something like this uh such that two things um I want The map FC to be an isomorphism and I want pi 10 of Z to be trivial then uh you define X to be Z with the NS glued together and um by setup you get that Alpha of X is whatever you want it okay so you

can kind of very well this is this is a pretty straightforward way of of of writing down a for manifold that has whatever Alpha in variant you want supposing that You can do this uh 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 um well you can you can keep control of this by 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 um so as long as you build a thing and you're pretty careful um then you get this and I will stop there thanks okay