 Hello. Good afternoon. I would like to welcome you to our today's lecture. My name is Anna Wienhardt. I'm the scientific director of the Research Station Geometry and Dynamics and one of the spokespersons of the Excellence Cluster Structures. And this lecture today is part of several events. We have this summer which go around the Poincaré conjecture and they are embedded into activities nationwide in seven different universities in Germany where each university organized events around one of the millennium price problems which were posed by the Clay Foundation 22 years ago. All worth one million dollars and the Poincaré conjecture is up to now still the only one which is solved. As part of the activities here in Heidelberg we have school workshops which already started so for high school pupils we have today's lecture which is more a mathematical lecture for a general interested audience but mainly with a scientific background. And next week on Friday we have a public lecture for the general public given by Sebastian Hendel who comes from Munich. So I'm really happy that so many of you made it here today and so it's my pleasure to introduce our speaker today, Markus Barnagel who is a professor for topology here at Heidelberg University and he would give an introduction to the generalized Poincaré conjecture. Okay Anna thank you very much. Yeah also from me welcome everybody I'm glad you could make it here. So as Anna said our topic is the Poincaré conjecture. This the millennium price that Anna was referring to is specific to dimension three but the Poincaré conjecture well can be phrased in any dimension and there are different aspects of mathematics becoming active in different dimensions I would say in this talk here I will mainly focus on the high-dimensional aspects of it which I mean these aspects sparked very influential developments in high-dimensional topology and our sort of foundation make a foundation for the high-dimensional classification theory of manifolds program which is called surgery and so the techniques that enter in the proof of the Poincaré conjecture in high dimensions form a foundation in general for classifying high-dimensional manifolds. So I'll focus on that mainly although in the end possibly if time remains I can say something more about the special dimensions four and three. Okay so let's see what does the Poincaré conjecture say in the first place. So the goal is to recognize a sphere okay and so you do this by making some local and some global assumptions. So obviously a sphere locally looks like RN so you want to make that local assumption and that leads to the notion of a manifold right. So when we say a space should locally be homeomorphic to Euclidean space RN then that leads us to the well-known notion of an n-dimensional manifold. So usually I will assume manifolds here to be compact. So that's our local hypothesis now the question is what global hypothesis has to be added in order to be able to conclude that the manifold is homeomorphic that is topologically equivalent to a sphere. Well so of course compactness is a global assumption but the other assumption is of a homological nature and it has to do with whether or not you can contract loops inside of the space as we will see right. So that's roughly the statement of the Poincaré conjecture. If you have a compact manifold it's simply connected and has the homology of a sphere then it should be a sphere right that's the okay. So let's see let me start out in low dimensions. So what are some examples of manifolds? Well if n is 0 you just have the point in the connected case and then you have finite sets of points. If n is 1 then the connected compact one-dimensional manifolds are circles and then you can take a finite disjoint union of those if you want for any equal one. So there is not too much to be said here. It gets a little more interesting in dimension 2. So what are some two manifolds compact two manifolds? Well there are the orientable ones and for example we have the two sphere then we have the torus T2 and then you can add a number of holes right you can take a connected sum of several tori and for example if you take a connected sum of two of them then you get a two-dimensional surface of genus 2. Genus counts the number of these holes and so on and in general you get a surface of some genus G orientable and then you have the non orientable surfaces so you have a real projective plane and then you can take connected sums of those and obtain all non orientable compact surfaces. All right good. So in 1895 Poincare wrote a very important paper called Analysis Cetus which was one of the founding papers I would say of topology and specifically algebraic topology and what he did there among other things is he introduced what we call homology. This assigns to a space a sequence of abelian groups denoted hi of the space so these are abelian groups and you have one for each index for each dimension 0 1 2 3 and so on. So these roughly count i-dimensional surfaces in x that do not have a boundary and are sort of closed and finite so compact and they you count them modulo compact i plus one-dimensional surfaces in the space roughly. Okay so let's maybe see some examples well so what what did we have here we had the surface of of genus G the zero of homology is always the rank is always a free abelian group whose rank is given by the number of connected components by the number of path components so if it's connected then the rank is just one and you have Z the first homology is given by a free abelian group of rank 2 G where G is the genus of the surface and H2 is again given by Z and all others vanish so that's what this would look like whereas for instance if you took one of these non non-orientable guys the second homology is zero and alright so now looking at this list one may notice from a certain perspective the following corollary if M2 is a compact two manifold whose homology is isomorphic to the homology of a two-sphere then M is homeomorphic to the two-sphere from from this list and of course this uses the classification of surfaces you have to believe in the classification of surfaces but let's say we do that then this would be a corollary okay and so therefore one could look at such a statement in the next higher dimension which would be three so there are various addendums to this paper analysis C2 those were Poincaré's complements and in the second complements Poincaré says the following so he says if if a compact three manifold has the homology of a three-sphere then it is simply connected and therefore he says and therefore homeomorphic to a sphere to a three-sphere to S3 now this is an interesting statement he says well I don't have time to to work it out here it would take me too far a field but it's a theorem it's true that this that you have this right so now actually he makes two statements right the first of which is wrong the second of which is correct but took more than a hundred years to prove right so so it is in fact not true that such a manifold is simply connected and even if it is it is very complicated to prove the word therefore is kind of funny right that's that took more than a hundred years to figure out that they are for so so alright so let's let's investigate why is the first part of the statement not correct and in fact he realized this quickly and and then wrote about it in a later complements of this analysis Cetus paper he recognized the following example so let's see let me start let's say G let G be a group then you can construct its abelianization so you make it commutative just by dividing out by its commutator subgroup then you can also create some language in group theory a group is called perfect perfect if this abelianization is in fact trivial alright so now let me consider the following example let's look at the group s03 of special orthogonal 3 by 3 matrices and there's a homomorphism from s3 to it of degree 2 so topologically so threes are p3 so this is the universal cover if you're familiar with that with that notion here you can it's in fact a homomorphism of groups you can imagine the three sphere as the group of unit quaternions in quaternionic space h which you identify with r4 right with Euclidean fourth space for space and so inside of s03 there is an interesting subgroup the icosahedron this consists of the symmetries of an icosahedron and but you just consider the orientation preserving ones if you want the reflections also then we can look at the pre-image let's call it i prime as a subgroup of s3 so this group is isomorphic to an alternating group on five objects it has so it has 60 elements it's a group of order 60 and therefore this group here has order 120 let's see so you could also write this group in terms of generators and relations so what's this i prime that's kind of a geometrically you figure out the symmetries and there are two generators let me call them a and b and the fifth power of a and the third power of b are both the square of the product of the two generators and so that is a description that is a presentation of of this group i prime sometimes called the binary icosahedron group so this is the group Poincaré considered and now by our construction this i prime acts as a finite group acting freely on s3 and we can consider the orbit space s3 modulo this group action let's call this sigma 3 now let's think about this space a little bit what does it look like locally we emphasized that locally we want this to look like rn but in fact since this is a finite group acting freely on s3 this quotient map from s3 to the orbit space is what's called a covering map and so locally the quotient looks the same as the space s3 so therefore it is also a manifold and it's clearly compact right so this is a 3 manifold compact 3 manifold the kind of thing we are interested in at the moment and so let's try to compute its homology so first of all the construction use a theory called covering space theory which will tell you what the fundamental group is so the fundamental group of a space what is it that you do if you have a space x you pick a point you fix it you call it the base point you consider loops at this base point and then you consider in fact homotopy classes of such loops so if you can within x continuously deform one loop to another you regard them as being equal and the group operation is just given by concatenating two paths sort of running around the first one twice as fast and then running and immediately running through the second one also twice as fast and then you have a new loop that's the product and it's called the fundamental group and if if this group is the trivial group you call the space simply connected right now what is there and this fundamental group is written as pi 1 of x so what is pi 1 of our 3 manifold here of our sigma 3 well covering space theory due to its very construction will tell us that you get exactly the group whose orbit space you took here so it's the group I prime then there is a general fact an algebraic topology that the first homology of a space is just the abelianization as explained above there of the fundamental group of the fundamental group we just observed is I prime so we have to make this a billion so let's try to compute what this is well if I write this maybe the billion then I have the equations 5a equals 2a plus 2b equals 2b and the only solution to this equation is a equal to 0 that's the only solution therefore we have shown that this group is a perfect group so this abelianization is 0 and therefore the first homology of this is 0 then there is a principle that was also observed by Conqueror which is called Conqueror duality it says in this case we imply that the second homology of this is also 0 and so the homology in degree I of an A manifold is isomorphic to the A minus I co-homology of the space and then here the dimension is 3 so n 3 minus 1 is 2 and so it will lead to the fact this is sort of generically then true that h2 also vanishes and now take taking together this actually means that the homology of our sigma 3 is isomorphic to the homology of an actual three sphere yet sigma 3 is not homeomorphic to a three sphere it's not even homotopy equivalent to it since the fundamental group is a homotopy invariant that three sphere is simply connected but this guy is after all it has a fundamental group of order 120 by our construction so so they're not even this is my symbol for homotopy equivalent so they're not even actually he didn't write it this way I'm writing it like this these descriptions came later Conqueror actually used genius to handle bodies and to do two of those together by means of the diagram called a Hagar diagram but you know this is this is one description of his example so therefore you realize that this is not quite right and there were as I said further complements and then it is fifth component he phrases what has hence become known as the Poincare conjecture is a compact three-manifold known here perhaps we should point out in the case where the dimension is three the assumption of simple connectivity implies automatically the homological data that we discussed right because if it's simply connected then in particular the first homology we vanish and so with the second homology by comparability and therefore have what's called a homology sphere right so if this is if something like this is satisfied if homology is that of a sphere we also say in this homology sphere and if it's homotopy equivalent to a sphere now this statement is very difficult and it took more than a hundred years to prove and eventually it was proved in this dimension three by the proof was completed by heroin but you beat very strongly on a program of Hamilton which involves the rich flow so but that was not obviously historically what happened historically what happened was rather in the 1960s beginning of 1960-61 Snail kind of shocked the mathematical world by all of a sudden showing that in high dimensions in all dimensions five or higher the statement the Poincare conjecture in those high dimensions is in fact true so that dimensions four and three became sort of isolated cases as I said those were then later handled with specific techniques in those dimensions I can maybe discuss this later but what's made sure what's called the high-emotional injection so that's the following statement of if n is a compact simply connected then it's homeomorphic to such a sphere to a standard so when I make this symbol I mean homeomorphic you may wonder so why don't they get different right why don't they get a differentiable kind of identification with the standard smooth answer here so I hope to explain this later the arguments mails argument rates down the fear morphism would be false in general and in fact is connected to the existence of exotic right so Miller discovered for example in dimension seven a smooth manifold which is homeomorphic to a seven-dimensional sphere but not the fear morphic to it and so the statement the fear morphism even assuming smoothness here is in general false and in dimension four this new property conjecture as far as I know is not settled it's open as far as I okay so that's mails theory and in the further course of the lecture that's what I'd like to explain so I'd like to explain more about of course I forgot to say so high dimensional so by which I mean in greater than four dimensions four and three are special but in all dimensions greater equal five we can recognize this here from this kind of data so in the piecewise linear category this is also true it was about the same time proven by stallings at least four dimensions seven or higher five and six was open but then quickly settled by Zeeman actually so that he all these dimensions in PL it's also low and later human established that kind of engulfing methods that stallings had used in the PL context also in a purely topological context and thereby proved the high-dimension and procreate rejection also but I wish to now discuss the following thing let's pose the following question first let me discuss much easier issue which is the following if so we what we say is if M is simply connected and it's a homologist here we claim homeomorphism which is an extremely strong statement and much weaker statement would be to simply say M is homotopy equivalent to this thing right so this is of course a necessary condition to what we prove but it's not perhaps a priori entirely trivial because the homotopy groups of spheres are in fact very complicated so M has to have the same homotopy groups of the sphere yet we only have this kind of homology sphere information together with simple connectivity so it's perhaps maybe not entirely clear that even that statement is true but let me point out that this is this so that's the question why why is that true before we even start to discuss homeomorphism and so an explanation is the following if you have a homology sphere so first of all you can always create a map from M to an sphere because M is a manifold you select a small ball and you collapse the entire outside of the ball then what remains is so then you get a map you map everything outside the ball so that this M will be a small ball but this is topologically the same thing as the N module of its boundary which is an N minus one sphere this is just the same as an insane so you always can make that to a sphere that's not a problem and then let's call the map so you make such a map let's call it F so you want to see it becomes a homotopy privilege so the homological data will tell you that the homology of this map actually vanishes this you get from the homological data and then the simple connectivity is used in applying another theory from algebraic topology called the ravage theory and this use is simply connected to say that the homotopy groups by star of this map vanish and once you have that there's you use a theory called right-hand theory to know that F is a homotopy privilege and then you're finished so I'm just pointing out that there exists some standard tools in algebraic topology that that we certainly guarantee that such an M is a homotopy sphere so that is not the hard part right the hard so the hard part so it is therefore once you know that the manifold is homotopy equivalent to it's a homotopy sphere to then show that it's actually homomorphic that is the hard part that we have to understand so suppose we knew the following knew the following statement let n plus 1 in grade 5 in a compact n plus 1 dimension simply connected manifold with boundary being the disjoint union of two manifolds in dimensional manifolds those manifolds let's say M0 and M1 you have an illusion so here's the picture here is M0 and 1 and n plus 1 manifold compact whose boundary consists of these two manifolds such a thing is called a comportism so suppose we have a comportism between two manifolds and it satisfies these assumptions and moreover the inclusion of say M0 and W is a homotopy equivalence there exists a defiomorphism from W to a cylinder on this A so M0 cross so then you just take M0 make a cylinder on it you want to recognize a cylinder in that fashion from this kind of homotopy data so by this symbol I now mean in fact defiomorphism so suppose we knew this statement then I claim the rest of the conquering projections can easily deduce it from this fact this is in fact Smale's theory right it's called the age comportism theory it's a very famous theory and extremely useful as I said in the beginning I mentioned to apologize the age comportism age because this inclusion is a homotopy equivalence so suppose we knew this then my claim is that the rest and the concrete rejection force is easy because what you can do is just the following thing now suppose in order to prove the general high dimensions that we have our signal in I could just so here's here's the signal is two different points inside of the sphere it's a manifold so the neighborhood looks like a like a disc there's a small neighborhood looks like a disc so let's take such discs one down here and let's remove them so this is maybe maybe I imagine this to be the north pole and this is the south pole and I call this disc dn plus and this one dn minus so you select them and then remove them so now what remains is a compact manifold on the boundary you now have spheres because the boundary of a disc is a sphere that is an n minus one sphere we have a southern sphere and a northern sphere and the thing in the middle that remains that I initially don't know anything about I'll just call w and this is now my w so now let's see what I can apply the theory here well we know that sigma n is simply connected so therefore we can conclude that w is simply connected as well because it has been obtained just by removing two points and since the dimension is high this doesn't change the fundamental group so it's simply connected and the homologic data implies that the homology of w is n minus 1 minus 6 we manage and therefore this w is an h comportism in this sense and then the h comportism theory implies that you must be familiar with the fact d-pheomorphic n minus 1 sphere across an interval now I threw initially these poles away but now I threw them back in right now let's remember that we had these two discs we just threw them back in so what I what I obtain is w union this one this is dn minus dn plus that's now homomorphic well the w is a cylinder now so it's a dn union is n minus 1 across an interval union d plus but if you take an actual cylinder on a sphere and union to discs the result is an n-sphere right so this is just an s-n whereas this is the thing that we start with so the contouring projection is proving right we have a homomorphism from our sigma to a standard n-sphere now let me point out and here is however a point so where so the cylinder is recognized d-pheomorphically but the sphere only up to homeomorphism why is that actually what you do is so if you're morphism and you can say more it's it's the identity on n 0 identified with n 0 cross 0 so but the but the map at one you cannot control that's any few more right that's some defilm morphism about which you know essentially nothing so so so on one end you can actually do with the identity and so the thing will be smooth up here to extend to extend to a homeomorphism doing in the other disk you have to do something technically which is sometimes called the alexandar trick you extend radially the homeomorphism on the boundary onto a homeomorphism between the disks but since you do this radial extension that's of course in general on smooth so you lose the smoothness at exactly this point so at the very last step at one at this point it fails smoothly only at this point and so in fact you can get these exotic spheres by selecting sort of funny give your work phases on boundaries years and then doing two disks together it's in fact how you can make exotic spheres all right so so what I've explained then is that that we are reduced to proving the issue if we want to understand the concrete because you're right so this argument first establishes only six but then you're right technically five would not be covered by what I said and then yes all right so let me say a few words then about the age component and I said this is actually the core of the argument it's not a very simple theory but I'm trying to explain the main points here in the main stage of the proof of the age component of theory but first of all a strong vehicle to establish homeomorphism this is an extremely strong statement so first one has to think about where this is supposed to come from right this bijection this so the idea is to use Morse theory so the idea is use a theory called Morse theory so in particular you select on such a w on an age component and Morse function f that's a function that's a real value smooth function on w and Morse means that there are only well since w is compact there are only finitely many critical points and near each critical points near each critical point locally the function is then given by normal forms you can choose insutable local coordinates a representation of f near a critical point as minus x1 squared minus and so on up to minus xj square plus xj plus 1 square plus 2x10 plus 1 square this j that appears here is called the index of f at this critical point right so there's this notion of index that a Morse function has at a critical point now there are no obstructions to finding Morse functions there always is a Morse function in fact there are many you just pick one at random so to speak and you fix it now if you are extremely lucky then f might not have any critical points now in that case if f has no critical points that the gradient flow would give the desired if your Morse is into the cylinder at your time the problem therefore is of course connected to the critical points right in general you select your Morse function of course have critical points and this will not work but it suggests that perhaps the homotopy theoretic data that we've given ourselves here can be used to illuminate the critical points maybe we can modify successively and inductively the partial f so as to remove all critical points and then in the end when there are no critical points anymore we can just say this and we're finished so let's see so the idea then is you cancel critical points modifying the function until you reach this stage so I would say the key lemma to do this is it is a cancellation lemma which is actually due to Morse it's not due to me this was already observed by Morse so Morse observed the following cancellation so he said so let's look at let's say xj is a critical point of index j index j and xj plus one a critical point of neighboring index j plus one I can cancel those against each other somehow under what condition might this be possible Morse says here is a condition if there is only one flow line connecting xj to xj plus one there exists a Morse function let's say f prime such that the critical points is an f minus these two points can be canceled and have to remove so the picture is like this so the picture you should have in mind perhaps likely something like this might be two critical points of a jc index xj plus one there is a function f increasing in this direction say that's our Morse function okay so there are flow lines and so if if there is exactly one flow line from xj to xj plus one Morse says then you can modify this f up to get an f prime so that things will disappear and here is what he's doing how does Morse know this he says you do the following thing if there is just one flow line and you can select this you can take a small neighborhood of it and in that neighborhood you modify the function as follows modify it like this of course you have to say analytically how to do it I mean it's you have to really do some analysis here but then but it can be resolved and then just have this one line you can modify like this remove these two critical points and outside the function is not changed so all the rest will be the same and so then you can do this idea so the remaining question then is how can we guarantee that there is only one flow line right this you have to somehow arrange from this from from the data that that's given to you here from this homological data you have to somehow guarantee this so a disc here called usually in this business the left hand disc associated to the critical point xj or point left hand disc of j and then there's a right hand disc which you see here the complementary dimension so this is a disc dj and this is a disc it's called the right hand disc and it's a disc of the complementary dimension in w n plus one so that's n plus one and a similar data you have close to xj plus one and here in the picture that is the left hand disc of this critical point and it's a disc of dimension j and these discs have boundaries there because of spheres so the boundary of the right hand disc here associated to the point j is the right hand sphere associated to j that is a sphere of dimension in w n plus j right and and the boundary of the left hand disc of this point is a sphere called the left hand sphere associated to j plus one that is the boundary of a j plus one dimension disc and therefore is a j-dimensional sphere and if we select a level here say y and cut between these points that is to say we look at f inverse walking that this will be a manifold since the point has co-dimension one in r this is an n-dimensional manifold and you can dissect these spheres of complementary dimension in this n manifold you can intersect these and look at the intersection numbers there's an intersection matrix that you get and now you organize this as follows let's say c plus one uh cj plus one of w right and zero let this be uh the free opinion group the uh the critical points of index j okay so you let the critical points of a given index generate a free opinion group that you can do this in any dimension and uh these intersections give you a matrix here which i'll call boundary so this is exactly the intersection matrix given by intersecting these left hand spheres with right hand spheres in this manner and then there would be a cj minus one like this as well but remember that we were inductive so we can by induction already assume that this has been canceled and there are no such critical points anymore now comes in the homological assumption that the homology of w red and zero vanishes which is what we assumed for the h-cropodism theorem but if the homology in this degree of this chain complex should be zero and this group is zero it must mean that this is on two so this is on two this um matrix has full rank and um then by elementary row operations by basis change here you can therefore assume that the matrix has a row which looks like this first entry one and then this one is an algebraic intersection number so now we are almost finished but there's one trick that i still have to tell you and then we are finished i haven't really used the dimension hypothesis have i right i said n greater than 5 but it seems to me this is really a period so it seems to me i have to still say something to make this into a full proof so you see here it is this this is an algebraic intersection number where things are counted algebraically meaning use orientations things might cancel they may not actually intersect geometrically one point because there may be cancellation they were if they actually intersected on this level in exactly one point then i just have one flow line and i would be finished so we somehow have to still say that if i have algebraic intersection number one i can somehow by further modification achieve geometric intersection number one because then i would be finished according to wars so now how do i achieve a geometric intersection of these spheres and as n minus change is exactly one point this will be a proof in theory so let's see so can you hold the pre-image of this value of this intermediate value y and so we have our two spheres complementary dimension which we intersect and the algebraic intersection number is one but we would like them to intersect geometrically just one point so what do we do this is a method used by Whitney and it's often called the Whitney trigger Whitney says the following if you have two intersection points of opposite sign so that they be canceled algebraically then cancel them geometrically by doing the following select the path in one sphere and another path in the other sphere connecting these points since the fundamental proof is trivial and this is where we use this i can find homotopy theoretically a disk a two disk that fills in the loop which you get by going here and then back on the other sphere so you can find homotopy theoretically such a disk because of this condition but the problem is that you can't usually disentangle in general from these spheres or remove data points you want to actually nicely embed it to disk whose interior is destroyed from the spheres and so on and this you can only do if the dimension is high so now the dimension hypothesis enters if any straighter for equal to five then get an embedded d2 and then you can make it sort of disjoint from the spheres if you were in a four manifold if you have two spheres in a four manifold this doesn't work anymore and then you have to do complicated things involving castle handles and infinite towers of handles and whatnot and this is in fact what free-banded it becomes extremely complicated topologically but if there's enough room dimension wise then you can get a nice disk and once you have it you can sort of push along a suitable geography on this sphere to push one sphere off the other one the longest disk and then you're done you've removed these two intersection points and then you do this for all of them until there's only one single point left then you can apply the worst and cancel the critical points and then you're done okay so that finishes the proof of the h-cubanism theory and i hope i've stitched sort of the main points that you need to improve in high dimensions okay thanks very much well thank you very much for this very nice talk and a tour through the high-dimensional primary conductor so we have time for some questions is the topological proof of in high dimension following the same sort of document that we placed in most theory by something else or it's completely different actually um well maybe you can make it I mean historically I briefly mentioned human and stallings and zeeman so stallings idea was somewhat different he used the technique which is called engulfing where you kind of make you have an open set intersecting some polyhedron and then you try to make the set bigger until it swallows the entire polyhedron so there are techniques called engulfing in topology and and stallings use those techniques in the piecewise linear context and then human saw how to extend these engulfing techniques to topological picture and that's how it was proved in higher dimensions and in dimension four the topological concrete the topological concrete which is two more dimensions no more dimensions and in dimension four this was very intricate and fascinating work by Michael Friedman and as I said the the trouble is actually with this witness construction so I mean if you are actually in a four manifold simply connected so you can still homotopy theoretically find the disc the problem is you cannot make it into an embedded disc so nicely disjoint from two spheres so if this is my disc there will be double points that you have problems with but if you have two points that become double points then you can connect them by path forming a loop under this inversion and sort of run the same trick again giving you another disc associated to that arc but then those discs might have the same problem again so you make higher and higher choices of such pieces and you achieve an kind of an infinite tower called a castle handle and then actually this is the skeleton of a castle handle you have to zicken it up a little bit to get an actual castle handle and then that thing is a very this is not smooth anymore it only works topologically and then the game is to kind of shrink it again you have to shrink it and show and this is the main thing that freedmen did that you can that castle handles are standard in fact actual handles and a lot of work of geometric topology from the 1950s and 60s entered into this work of people like beam for instance or bob edwards entered there i think might have freedmen used essentially much input also from from bob edwards work to to achieve this shrinking procedure is there any other question i have one actually many is there any indication that company was thinking about or where a higher dimension version of the country connector because he stated it is that's right of course he had different models from manifolds i mean it was already he talked about polyhedra but of course like he said when he talked about homology and introduced homology he was certainly thinking about higher dimensions as well and in fact if one looks at the paper carefully if i remember correctly it seems to me that he doesn't perhaps specifically say in all statements that he wants them to be limited to three dimensions so i think he had high dimensions in view as well it seems to me so i don't think that there i don't see any other questions or thank you very much again and we hope to see many of you um same day Friday um for the talk by Sebastian thank you