CONFERENCE
Recording during the thematic meeting : « Higher algebra, geometry, and topology » the May 06, 2024 at the Centre International de Rencontres Mathématiques (Marseille, France)
Filmmaker : Luca Récanzone
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thank you very much and and uh first of all let me thank the organizers for organizing this this event I I really look forward to this week and it’s it’s great to be back in lumini um so I will be speaking about um certain High structure that appears uh when you study vibrations with the certain Duality properties namely oriented vibrations that have P care Duality fiber um and um the certain High structure governed by conserves leaph complex that you can Define and um basically the sort of the main takeaways that this what I’ll try to convey is that this actually allows us to connect problems in In classical problems in communative ring theory about multiplicative structures on minimal free resolutions and also problems in differential topology such as the problem of putting smooth structures on on a given vibration um so this is I think very much in on the theme of the conference of of sort of high structures in both algebra and geometry and Topol so let me first say what I mean by graph complexes today so let me indicate what graph complex I will be considering um so this is the coner is Le graph complex or rather the sort of the the hairy version of the graph complex um uh and so what is this thing so it’s a uh it’s a certain finite uh uh chain complex and I will be working today over the rational numbers exclusively and so all cool Etc will be with rational coefficients uh so this is a finite chain complex uh and it’s spanned uh by connected graphs and I’m not giving a formal definition here I I will just try to indicate what type of object this is and then uh um um we’ll move on so the connected graphs uh with let me just explain what the parameters here are so n here is for the rank so the first bet number of the graph and then the L here is I have L hairs or legs so these are not this is sort of um um yeah legs labeled uh by the integers one up to L uh and then you have uh internal vertices that are at least trivalent um so the internal vertices um uh or at least Tri valent and then the way I grade this is the homolog iCal degree will be the deviation from Tri valence so trivalent vertices have degree zero four valent vertices have degree one Etc so um okay and then uh there’s a bunch of you know you have uh to to specify signs in the differentials and such you have uh orientation data and this depends on M so M here is just a parity it’s even or odd um so uh and I will not spell out all of the orientation data uh ask me later if you want to know the details uh but this this depends uh on the parity of M and uh you have certain you have certain sort of what do you usually call it the reborn structures so you have cyclic orders of the half edges adjacent to each each um each vertex but um again I will ignore sort of giving the full full details here this is just an indication and then you mod out by uh this module or certain uh Shuffle relations and again I will not spell out formally what I mean I will just try to indicate in an example now what what what kind of things we’re looking at and and then the differential uh Delta is given by uh Edge expansions okay so so so let me just give an example to to show what kind of things uh come come here so this is uh this represents an element in you know with uh rank one and I have two legs labeled by one and two so this is an element here here I have four valent vertex so that deviation from Tri valence is one so the homological degree is one okay uh and if I apply the differential here I’m going to run through all the non I mean Tri valent vertices and explode them to decrease the deviation from Tri valence and the way this works in this example will be I can explode it in two ways and with certain signs that I will also not explain in detail right now so you get these these types of expressions and this is the differential going from here to uh degre Z here okay so this is the kind of GRA graph complex I’m interested in and to be like a little bit more formal if you if you happen to sort of know this this formalism of of modular operads and and fan transforms um so this um GM is if you collect these chain complexes for all n andl this is a modular operad well a twisted modular Opera if m is is odd uh and this is the is the soal fan transform of uh the Le operad and there’s that’s why why it’s called the Le graph complex um okay that’s all I wanted to say about this graph complex um uh uh about the definition of this graph complex but there’s some the main important theorem which which is due to um yeah l m m so it’s a certain parameter that you carry with you it will be the later it will be the dimension of the fiber but for this definition it only depends on the parity so you have an even version you have an odd version that’s all okay doesn’t affect the gradings the way I set it up here um and then um you have this I can squeeze it in here so the sort of original theorem about this graph complex which is Du to conage in the well in the vacuum case when L is zero and then this was extended to the Harry Case by Conant kasabov and vman and then the OD dimensional case was was uh done by lazarev and vov uh it shows that the homity of this graph complex uh computes the rational homology of certain automorphism groups of free groups so out FN so FN is the free group on N generators and then here the subscrip I will explain in a second and with Twisted coefficients in well the determinant representation if m is odd and the trivial representation if if m is is um is even okay and so here um uh if L is zero then this automorphism group is the outer automorphism group of a free group and for L greater than zero uh this is the semidirect product of the automorphism group or the free group acting on FN to the power L minus one where out FN acts diagonally here uh and so this gives you an interpretation of the graphity in terms of you know sort of classical objects in geometric group Theory and I should say that the homology here is non-trivial in general uh but it’s largely unknown so it’s a sort of big open problem to actually compute these homity groups okay so that’s the graph complex I’m interested in in and uh the theorem I want to discuss today is that there’s another appearance of this graph complex in the context of uh High dimensional manifold Theory so um the main theorem let’s called theorem one is the following so suppose I have um P Pi here is an oriented vibration with uh fiber uh Simply Connected [Applause] pity complex M of Dimension M so for instance if you have a closed oriented Simply Connected manifold this is an I mean that will give you an example uh so let’s call these things uh pity vibrations okay so for instance if you have a smooth manifold bundle the these are P Duality vibrations so if I have a such a thing and if I have a graph fromy Clause so gamma in HK of this graph complex then to such to these two objects I can associate um so then there is a [Applause] map of degrees M * n -1 – K which uh is kind of a you can think of as a generalization of fiber integration from the chology of e and as always in this talk will be rational commy uh to the e e tens power that down to the commy of B and um yeah again Q coefficients with certain properties so first of all uh it does extend uh ordinary fiber integration so um [Applause] so it’s ordinary in the sense that if I there’s a if I take a specific class that I’ll call Epsilon L and I do this this map uh to a sequence of L cohomology claes in E this is just the same as taking ordinary integration along the fiber of the product of these classes taking the cup product in the Comm of e uh so for all this is for all C1 CL and the of e no and this is an I mean this will be you know this will come later I mean integration along the fiber is defined whenever you have a Pity fiber that’s maybe your question I mean so you can Define this in terms of the C spectral SE for instance if you happen to have smoothness you can Define it as actual ual integration but this is this is actually not you know actual integration but yes yes I was just about to say it so but I was interrupted that’s okay yeah so so we have this where uh eel is a specific I mean generator for so this it’s this this graph it’s this Tri I mean you take the tree uh one two L and this is a generator for H zero of of the graph complex where you have rank zero and L legs so this uh this um particular graph class this particular graph class recovers fiber integration for you um okay but then the point is you get much more so in fact uh this representation if you like of graph is faithful in the sense that uh if you have a nonzero graph from CLA then this uh this uh fiber integration map uh is nonzero for at least what some choice of qu graduality vibration P so you can always sort of detect graph classes by studying this this this map for suitable choices of P Duality vibration and actually since you asked about smoothness it’s an open problem to uh determine what whether or not one can detect graph classes by using only smooth bundles uh and there are indications to as to saying that that’s might not be the case so so this this sort of picks up perhaps some difference between vibrations and smooth bundles so so we’ll get to that later in the talk okay so let’s that’s the main theorem so I will now sort of discuss how I prove this and uh there are different ways that you might go about trying to prove such a thing and um so I’m going to tell you about the way I prove it and one of the nice features of my proof which is maybe sort of a bit idiosyncratic but one nice feature is that it opens these connections to problems of commutative algebra so I’m going to approach this using rational homotopy Theory and commutative differential graded algebras models for spaces Etc so um so let’s now switch to algebra and Define a corresponding notion there so now my base will be some augmented commutative differential graded algebra over Q uh and I think I will start writing K so to be safe K is always Q if I happen to write K instead of Q um so the definition is that a commutative differential gred algebra morphism from R to S is uh a pcity morphism if and then I mean you sort of want to model the situation above if the fiber which algebraically can be expressed as follows uh I mean the chology of uh yeah of the fiber uh is a Pity algebra that is to say I mean it’s it’s it is an algebra because we’re working with in the land of commutative DG algebras uh and then it makes sense to ask for this algebra to satisfy P Duality uh right and um some of the main examples uh of pcre Duality morphisms I mean we already have one but it’s it’s actually interesting to look at special cases um so uh the first example is uh if you take the unit map of uh differential fors on on on a p creality space and here again I’m not assuming m is smooth so let’s take uh solivan polinomial differential forms with Q coefficients that’s what I mean when I write omega star here um so this is an example of uh uh uh so m is a graduality space uh this is an example of a p Duality morphism um and the H here in this case is of course just the the comy um which satisfies ponre Duality basically by definition of pcre Duality space u and then slightly different example comes from algebra so if you have R uh mapping to R mod I so R is now just an ordinary commutative graded I guess graded commutative netharian K algebra and I is a gorenstein ideal then this is an example of a poity morphism um the chology of the fiber in this case will be the Tor algebra and uh the fact that actually by a theorem of AR of and gorod you can take like I is a gorstein ideal actually if and only if this morphism is a pker duality morphism you can use that as a definition of gorenstein [Music] ideal and there are plenty of such ideas I mean this is something people in commu study a lot um okay and then the third example is well the sort of the you know uh the the one we have up there so if you have a Pity vibration if you look at the induced map on differential forms um uh so so now uh where this is a Pity vibration um yes so those are some examples um and I guess to be safe uh for Simplicity today um I’m going to assume that the spaces I consider are Simply Connected uh you that’s not absolutely necessary but it will you know complicate the exposition if I don’t assume that so so let’s just assume that for today um let’s see and so now if you have this type of uh conc Duality morphism what can you do with it so that’s the first sort of the here um so if R to S is upon cality morphism uh then you can find uh a minimal uh or semi free uh resolution I mean s is in particular you know a differential graded module over over R via this map and so it makes sense to look at coant resolutions of s in the world of differential graded modules over R and R is augmented so it makes sense to talk about minimal resolutions uh so uh you can find such a resolution that admits certain higher algebraic structure oh this is s uh so that admits uh what I call a Pity C infinity algebra structure so I’ll explain what this means in a second so um yes so I mean even in this case I mean where you look at the ordinary mutative rings I mean um this is sort of sort of a classical Pro problem you can always find a minimal resolution of a cyclic model module over over the ring and then you might ask whether there is a multiplication that on that minim resolution and and this says that you can always find a multiplication of this particular type um so let me explain what these terms mean uh so um so uh ie uh so p radiality c infin algebra structure means that you have first of all a c Infinity structure an R linear C Infinity structure so if we have structure Maps mu n from n tensor power of a over R uh like so so this is a c infinity algebra structure uh over R so C infinity algebra structure or some people call it commutative a infinity algebra structure so it’s a uh a coherently homotopy associative um multiplication and C Infinity means that you assume that the multiplication mu2 is actually commutative and then there are higher there are constraints on the higher mu NS uh uh the vanish on shuffles Etc that that that’s sort of uh yeah so that’s what’s the infinity Al is uh and so that’s the first part of the structure you have and then I need to erise this a little bit so um and you also have a non degenerate or linear pairing and and this is the degree shift so this this means that the map is of of of homological degree n minus 2 and then here I have have a map an R linear map of homological degree M and this is supposed to be a non- degenerate uh symmetric or bilinear uh pairing um and such that these two structures are compatible in the sense that uh if you look at the following operation so mu n + 1 hat so this is an operation that takes n +1 arguments from a and it splits out an element of R uh so namely you take you pair a z with mu n applied to A1 uh up to a n now this is an operation with m+ one arguments and this is supposed to be this operation is um invariant up to sign under cyclic permutations of the inputs so this I mean this this this kind of thing has different names many people call this cyclic C infinity algebras or fenus C infinity algebra I prefer concer Duality because it connects to the to the geometry um right so we have this kind of structure and then minimal so minimal I mean I think I maybe said it but let me so means that uh so a is a differential graded R module and it’s minimal that means that when you uh uh um look at it or compos this has has has trivial differential okay so now as a remark this is an extension of two classical um theorems you you can view this as a parameterized version of U kades is sort of minimality theorem but uh with the you know enhanced by by pity so um so this generalizes two classical results uh first of all so this is due to kades with I mean without the P Duality and then an observation of stev uh which was reformulated in terms of cyclic C Infinity albas by losev uh so this is in the case this is in the special case one so this is in I mean in the sort of case of a trivial vibration um so for for M upon graduality space so uh uh it says that you know a in this case will just be the Comm admits uh minimal uh P Duality C infinity algebra structure uh and a see Infinity qua isomorphism uh from from the commity to to to the Drone form so this is the classical result that says that the well the rational homotopy type of any space simply the space m is is recorded by a c infinity algebra structure on the comy and uh the observation of stash and last is that if m is a p quality space then the C infinity algebra structure can be made into a into a p Duality C infinite albra structure in this sense um okay so that’s a special case of this uh this is sort of but but this also specializes in another way so maybe I move over here to explain that so uh another classical result from commutative ring Theory which is due to books bom and isud is the following so this is in the case when when you look at the second case so when you know I is a gorenstein ideal and R is just an ordinary ordinary graded commutative nean ring without any differential so um um uh so this says that the minimal R free resolution a of R Modi admits a commutative and homotopy associative uh pity algebra structure okay so or here is or is ordinary so uh so this this theor specializes to a generalization of this result namely that they only show that it’s homotopy associative so but this theorem is saying that there’s a you know it’s coherently homotopy associative so you can view this result as a sort of a supremum of these two these two observations if you like no it’s just a you have a me2 which is commutative on the nose and which is homotopy associated without any coherent homotopies exactly so it’s sort of navely homotopy associative uh yes but I don’t have time um the proof is not by using homotopy transfer theorem so that’s maybe might interest you it uses it uses I mean techniques from rational homotopy theory about derivation Le algebra models for for spaces and vibrations so it’s it’s a kind of a different type of proof than what you might be used to in this for proving these kinds of theems I’ll happy to tell you more later uh um okay so then give granted the existence of these types of um um resolutions uh we can do the following so so now we can go on and say uh okay so now that we have such resolutions um we can say that well okay so now let’s give ourselves a let’s call it P A pity morphism uh and then let’s give ourselves so every graphology [Applause] class um gives rise to a map of the same degree as before uh in the uh yeah so in the derived category so I’ll just by abuse of notation I’ll just use the same notation so this will be from the s l fold derived tensor product over R going back to R okay so this is a map in the derived category of uh of thegr modules okay um so that’s the theorem so now this is a purely algebraic theorem but it’s it’s it’s specializes to the theorem in the beginning uh by speci iing to uh to this pity morphism and then take homology so to get the map in the first theorem okay uh so this is in fact a better theorem because it’s it gives you a map in the derived category not just on the level of commodity uh and how let let me just indicate the proof um so let’s see where do I want to go um uh [Applause] okay okay so um so the the proof is basically you pick uh P Duality C infinity algebra resolution of s uh over R uh and then you you get a chain map by borrowing coner which’s idea of these he calls them partition functions so we get a chain map from the graph complex uh like this by uh conservat partition functions so conserv use conser used these things to Define Co Cycles on graph complexes Associated to cyclic Infinity albas but that was sort of in an un parameterized setting so so we’re now just extending that to the differential working with differential graded algebras rather than just ordinary fields or or rings and I’ll just explain the idea in in examples so the idea is that uh basically it’s determined by I mean if you know it I mean it’s basically recording the um um um the structure of a I mean of a of a cyclic C Infinity Al right here uh and uh so to the Corollas so for the graphs like this okay so now I have a a three AR operation I need to get a three AR operation on a and what I can do is for the coras I just send them to the I just use the structure maps of the C Infinity structure uh let’s hden back here hope there’re still here yes yeah so I use I use these operations right so uh and then when you have more complicated graphs let me just do one simple example to the idea so if we take this graph with one Loop so this is supposed to give me a two AR operation so if I had two elements of of a I send this to well I I use mu4 hat because I have a four valent vertex and then I plug in a and b in the slots one and two and then I have this edge here so what I do is I plug in the co evaluation as it were of this inner product space over R so I do take the sum over EI and EI dual where uh EI is an R linear basis for for a and uh this EI and this is the Dual basis so I mean we have this inner product and and we can define a dual basis uh uh using this inner product in the usual way okay and then you can imagine doing graphs with several several vertices I mean you write down similar expressions and you just multiply the the contributions for each vertx and every time you see an edge you plug in these Co evaluations you will have lots of indices running but this is there’s a general formula that that sort of you I think you I hope you get get the idea um and this actually defines a chain map here and then we’re basically done because then so in particular uh so if if if gamma is a is a is a is a cycle is a cycle then we get I mean we can just restrict to that cycle here so we get the we get this induced map um so we get yeah we got a answer uh like so and this this represents this this map that I claimed to exist um um um right and I mean a is is a cerent resolution so this does represent the map in the in the derived category so that’s that’s an indication of how the proof goes um and there are some observations to be made here uh um so so the first observation is that this map in the derived category um does not uh depend on the choice of um resolution okay well you need a c Infinity P pity Infinite Algebra resolution but you can imagine having different sort resolutions and the result will not depend on which one you picked okay so and then but then you observe that if a can be chosen to be strictly associative IE all the mu’s are zero for n greater than or equal to 3 that means that we have a strictly associative and commutative Point Duality algebra over R um and then if you think about the definition of this map the tri valent things I mean they use you know they use mu2 but nothing higher whereas the four valent things are higher they use you know mu3 and higher so if if you have a conquer Duality C infinity algebra where these are all zero then anything you try to build out of positive degree graph for graph classes will automatically be zero already on the code chain level um so so in particular this this map will be zero uh for all gamma with positive homological degree so remember that the degree was uh deviation from Tri Valance and so the conclusion then of these two observation is that it it gives you sort of a meaning to this map what does this map and the Comm Clauses used constructed using it what do they what do they do well they obstruct the existence of strictly associative PCR Duality resolutions so if you have a nonzero operation this obstructs uh this existence of strictly associative con Duality algebra resolutions a and also uh minimal or not so um uh and this this is something that now connects two classical problems in in commu to ring Theory so so there one is often concerned with finding multiplicative structures on the minimal resolution you can always find multiplicative structures on some resolution that’s typically non-minimal but now we’re looking at a different type of resolutions we’re looking for pity algebra resolutions and because these Clauses are non zero it shows that you cannot not always do this and so an interpretation of these operations is that they they obstruct that kind of thing so let me elaborate on this point and where do I do that yes so um no sorry I can’t do two at once no no oh sorry yeah that would be a shame um for uh so yes so returning to these these questions so so right so now if I have a fixed pity morphism you can ask the following question so let me now just say what I I just did so so the qu the basic problem that you might ask is does s admit uh uh an associative um minimal ferity resolution over R so this is a question and um we can answer I mean there are answers to this I mean so in special cases this is a question that has been studied so here are some answers so in the case when you look at the I mean the the sort of trivial vibration you can this is precisely the question of whether m is is formal or not so this is yes the answer to this question is yes uh if and only if m is formal over K in the sense of rational homotopy Theory and so in particular it’s the answer is no in general uh so because there are examples of nonformal manifolds and the obstructions we all know that you know obstructions to formality are given by for instance uh Massy operations Etc so this is a classical you know thing to study right uh and this problem specializes to that classical problem it also specializes to this again I I already said it but I mean in this case uh the answer in this case well this was actually conjectured by books bom and Eisenberg they conjectured that the answer was always yes so they conjectured but that conjecture turned out to be to to turn out to be false but it’s it’s true in some cases so it’s true for when the ideal is a complete intersection you can use the cul complex to to write down a quality resolution uh or when when the grade of I is less than or equal to four so the grade is the length of the longest regular sequence in in I and this was observed by by kustin and Miller uh back in the day this is like 80s or so um but then it was proved to be false actually in general so false if uh when grade the grade is is you can find cont examples in every grade five and above and this is this was due to arov and then sasan well not together but she also studed this problem so so here also the answer is no in general um um so the problem that we’re looking at is sort of a variation of these these classical problems in that we do not long no longer require minimality but um we want um so let me just do it like this quicker we require poity so so here’s a question Prime so does s admit uh an associative pity resolution a which I mean minimal or [Applause] not so this is an easier problem um if the minimal resolution has a such a structure then of course the answer to this question is also yes but it could be that the minimal resolution does not admit such a structure but you can find some other resolution and this is exactly the kind of thing that in the absolute case and then for the trivial vibration this is exactly what this this this well-known result of lambre and Stanley so in this case the answer is always yes and this is basically the this theorem of lam and Stanley that says that anytime you have a p Duality space you can find a strictly Associated Point graduality differential graded Alma model for for the the ROM chains of course if the manifold is formal you can take the chology but if it’s not formal you can still do it but you will get some pity algebra that has a differential so it’s not minimal so you can always do it for the trivial vibration so now in in this commutative algebra world this I think is a new problem I don’t I haven’t seen this addressed in the commutative algebra literature so as far as I know this is a new problem and I think it’s an open problem I don’t know the answer I think it would be interesting to address sure all of the questions I talk about here are very nont and have have have have serious content if in the simply contic case but it’s not really a necessary assumption but it will be too technical to go into nons Simply Connected case so I I don’t do that today okay so are you worried about like if I take ordinary like commu rings they’re in degree zero so they’re like not representing Simply Connected things well you can always if you have commu rings for instance you take a polinomial ring a modulos and gorstein IDE well you can grade that by assigning uh positive even degrees to the generators of the polinomial ring and these questions make sense and and sort of yeah it’s just about regrading the classical situation yes uh yes so okay sorry yes sorry yes that that result is for Simply Connected yes yeah okay sorry yes yes yeah no good point so yes yes I stand corrected yes okay um right um no wait sorry I don’t no no I did state in the sorry the theorem says I only state the theorem for Simply Connected fibers but the other spaces I was talking about the other spaces the E and the B so I actually don’t stand corrected okay good uh let’s move on uh so uh in this case a new result which is a consequence of what what I just told you is that um in this General situation the answer is actually no so this is a coroller of what we just discussed um so uh because these graph classes are nonzero in general so so so so so the the lamri Stanley theorem fails for for General vibrations so that’s I think is kind of an interesting outcome how much time do I have left now yeah seven minutes seven minutes okay so um so then the last thing I want to discuss we want to return to this question of smoothness so I want to say a few words about how these classes might know something about whether or not the vibration admits a smooth structure or not so let’s see um okay as you can tell we don’t have these kind of things that my University so so this comes to something that um yeah so this is a sort of uh which I mean this comes to the origin of these classes actually so we discovered this kind kind of structure when we studied chies of various types of wam morphism groups of of manifolds um so I’ll try to give you a sketch of what what what goes on here uh so if we now specialized to smooth bundles and I will also allow uh these generalizations of bundles called block bundles uh I will not have time to explain uh exactly what they are in case you um haven’t seeing them so um um but you can think smooth bundles here so uh in gamma is a graph class and uh to any such data and then I can give myself a sequence of characteristic classes of uh stable Vector bundles then I can Define uh what I call you can call graph Capa claes so these are characteristic classes of of bundles of this type where I take I use this sort of generalized integration along the fiber to Define um uh yeah so so I do this and this gives me Comm classes in the base of the bundle and the remark is that these classes generalized these wellknown Mill Mar classes so when you specialize to ordinary fiber integration that’s exactly how Mill M claes are are defined so the cap of this Epsilon L you remember this generator in zeroth dimensional graphology if I apply that to a sequence of sorry there’s something I’m sorry um I made a mistake here sorry I haven’t told you what these um I meant okay I have this characteristic classes of of vector bottles and I apply them to the fiberwise tangent modle excuse me so um yeah so this yeah sorry so if you have a sequence like this this this recovers the classical mm of claes that use see in in you know characteristic classes of smooth fiber bundles and yeah let let me just sketch what I try I mean this will be very sort of uh brief and maybe mysterious if you haven’t thought thought about these things before but I I I want to convey the sort of general idea hopefully so if we studyed this particular manifold gfold connected sum of SD cross SD and then we uh remove the interior of a an embedded disc here to get the manifold with boundary of sphere and here D should be greater than3 uh and then roughly speaking so here’s a this m m classes are featured in the computation of the stable chology of the dimorphisms of these manifolds du to SAR galatius and Oscar R Williams so let let me draw you the a cartoon picture so so I will be making some you know telling a slight lie here but but to convey the idea um these Capa claes Associated to the universal fiber bundle over over B diff of of this manifold here uh they uh in the stable when you let g go to Infinity so there’s a range of any commical degrees that are that are small compared to the genus where where this is an isomorphism so this is D galatius and rondall Williams um now uh then you can look we what we can look at is the an enlargement of the def morphism Group which is the so-call block dimorphism Group which classifies these so-called block bundles which are more General than orary smooth fiber bundles and there’s a canonical map like so and we can Factor this isomorphism as follows we can include these so we we we observe that these can be viewed as graph Capa classes coming from Zero Dimensional graph modity so these include into uh a bigger polinomial ring generated by well for certain C that I will not spill out exactly what they are but here we take all non-trivial graph claes and roughly speaking the theorem that a theorem that I proved with E mson A couple of years ago is that you can compute the stable chology of this loock ify morphism group in terms of GRA these graph classes so this is so what I want to point out here is that okay so it’s kind of funny because here we don’t see any graph Capa claes but here they all survive so the question is what happens when they with this map so we have this graph cap CL here for some some gamma of of positive degree and and where does it map to I mean they they don’t seem to be necessary to to account for all of the commodity here so it could be that they just map to zero um so this you know spawns some questions so I’ll end just just posing these questions U so now for a general smooth smooth bundle uh could it be that these are all always zero so is this generalized integration along the fiber always zero when you take positive degree graph classes and if so you know a possible reason for them being zero could be that they admit these kind of multiplicative structures on the resolution so so if so does this uh you know the forms on E admit uh an associative P graduality resolution over uh the forms on the base for for smooth bundles so this is very speculative but I mean this these computations which are basically sort of yeah these computations indicate that these graph Capa claes something severe happens to them when you pass from to to to the smooth case um it could be that there’s CR we don’t know that this is the case but that’s maybe the the natural guess uh so these are also some open questions that I think would be very interesting to to answer okay thank you [Applause]