4/5 Contact homology and virtual fundamental cycles

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4/5 Contact homology and virtual fundamental cycles
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The goal of this course is to give a construction of contact homology in the sense of Eliashberg--Givental--Hofer. I will begin with an introduction to contact geometry, pseudo-holomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. "virtual fundamental cycles") out of the relevant moduli spaces of pseudo-holomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.
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now do you want to do this and that and the time of the group who were In a letter to the man man and we don't want them to do so you In the world in millions of I start today I want to talk about how to use the machinery that from the previous lectures on to define virtual fundamental cycles which are relevant for contact knowledge eh come so In the 2 lectures again but I defined bunch of atlases much spaces and last lecture I showed how the UCI's Abbas's too and I dunno virtual fundamental cycle on the other 1 today as I say how they use their package all this information together on coherently between the different much less basis of we have something which deserves to be called coherent ritual fundamental cycles on although much less space what so I recall of those all of the relevant objects are indexed and by the set S which is a collection of trees so trees the around there all on connected undirected on every vertex it is all of this is due to the letter brief vertex and has a unique and coming hedge along with with labels itself so the on the vertices such either on the edges of the labeled with Rebe orbits and the vertices hard labeled on both homology classes relative to the apartments Creselious the recall set contact manifold "quotation mark lenses a contact form years while and the set purpose on the streets such something like this it is directed downwards on the edges are labeled with orbits and the other vertices are labeled with the knowledge that live with you Rabe orbits people who didn't want to go on sale in Louisiana will give you a call on his way so that all the whole orbit so I can understand "quotation mark could get close to directive of the it's not necessarily simple could covered with those who have yet comfortable yet will ya so to s member for every variety we have so much less space on em and the which is on summer home off a curves In our cross wide of mobile on the mean that and so is the teaser dual grass of the domain and it but the labels are Duncan compatible with enough so that is on this this streak response to this two-story home after college and the villa he had little did he lose the use of his arms up the parameterization yet and and and we also module caution by the translation our direction differences in class yeah they were so S is not just a set it's it's has been more structured as a category 4 morphisms refuted the prime on office on his own contraction edges so here only 1 educating attract even morphism from this to single protected square sponsor going home Africa's on S also on as some sort vaguely when model structure writing this summer just say has concatenations it's exactly what that means so that the elections would please let it I would love to live in the concatenations together with a bunch of guys trees on news 6 AM matching between the heads of input upper edges some such as the result is connected and and ended and is a treat on you can get mandatory service was given the structure we can write down what the compact location of these much spaces looks like so 1 simply take the Union overall trees tapping and at that eh this this district intakes and 2 counts at all possible to generations of take the can have a 2nd splits and so their for trial Ashford results ended the spaces Compaq right so zone the space and market behaves well under these operations the same thing if you have amorphous this induces an inclusion of compact 5 muscle spaces and if you have a concatenation this induces collapse .period a isn't what it on May the are essentially tautological given definition but I know what I want do serve abstract abstract away these 2 properties and because can turn
out the most somehow most of them objects were going to study and have similar functoriality to to the much space itself so it's confessed module that this is a loss of here yet on the other the selectors issues exacts a little moping about and you could the delay in the last year the freedom co-workers and that you have or you could have a bubble yes Sergio bubbles than that the only thing that really has the change here and is is a definition of the much less space you have to allow bubbles but I won't I will require the bubbles in the in the trees because there that I somehow so finesse as model is From the Texas from this category asked to any symmetrical another category the bump along with for every concatenation natural maps From contends a product of exits to X of the little meaning is little if look I'm not necessarily out ending and is using this to denote the that whatever the operation conference since here it states that its product on the unhcr might have detonated said said yet not the arms race and so so we need the the following the natural for compatibility relations I into the really cool on sickle variant but I mean In the immediate is just a means stopping see faceoff dozens of other women so if you have a 1 with that please check and should on this just means if you have a bunch of treason you can concatenate gunmen in 2 steps sings contending thing all want so examples 1 this the embargo modelers face from past to the category of Compaq tells the spaces with their products much of the structure it is much for essentially the regions later said on last time I defined this set an index at a bar for the inverse of Abbas's on on the quality of the deletion judges region this is going to have to deal with the I'm so Baker a concatenation takes in and so am I I'm leaving something out notation here as I wrote I write just for a collection of trees but really come as part of the data is a matching of the is the landing of the last day of the year there is a lot of won't use that I will find that there is little that they raise their head and that it would release major Lemontvue in dealing with Jews graduate on so let is that fought for every choice of matching he gathered a map like this maybe them to a collection of trees there is this comes a product a lot of money as you need to have major new location for medical insurance he knew where they held with "quotation mark were made available to the meeting it was a huge role in all of the usual yeah yeah that this the concatenated object depends on how it matched and so on and for every trace of matching their Stidham applicants which the yeah I mean the right hand side depends on a matching so of course the matter after all know that with the I mean this is Olivia policy initiated I want you to major the left hand side yes and the left hand side does involve the matching is just the 10th inning the most structure the right here is is subject to this there's no we did the branches will make a major the whole of the life and the challenges that of the With the machines they use their own communities analysts said the future you know because of the holiday usually I guess you could I got it and this is where formalize an extravagant promises of so the last time I defined any From the next set of the of the Alice's and it's a it's also known as module from progress I want to Fox a category of sects this train as much as 1 of the structure and so it when we say explained what was about to this over to recalled that some are we define the something of AST this is an index that just can't accept indexing what I called thickening Dayton's the data that we needed to write down certain transfers the kings of the mental space come and then we defined a vase t to be the district Union overall subtrees so subtrees meaning connected subtrees so that is I choose some sort of vertices I take all the edges incident to 1 of those for season and demanded that the connected so this is a safety and I'm just by definition and it has some it has on this as model structure if you have a man from tee to tee prime than any subtree of the primary polls about subject he said that the correction and summary of the concatenation on any subtree of 1 of the TI's gives you a subtree of concatenated actually have success but also and before giving the the outline of against construction of ritual fundamental sectors adjustable more definition so people favor the object of this category is
effective if only if this compact modular space is effective this is not entered so I reeling in the compact side much less space but not the not compact and mental space it is possible to have a disease much less space and not necessary cut transversely it's a period and in In practice probably on possible perhaps from a given tree whose compact modular space is empty get whose which has not synonymous the boundary Strang tho important ratification here Rice is so henceforth we will Friday at but well actually mean full subcategory of effective objects sizes and presenting would stick ignore those completely and this is needed just 1 so that this category has good finest properties the savior of a given tree can just right down this is the tower of forces us to it's just by splitting many many times and In practice this but doesn't happen for Holdorf occurs because of the compact result and so on we're going to need to do surplus of inductive arguments by induction on on on trees and this if we had and we would want to take advantage of the fact this can happen who will lose their lives it uses them to on so for instance like constructing maps between us models were constructed by induction on the compliments forwarded by so and when you did when you done 1 defined amount tree like this has to be defined for every tree mapping to authorities In image In the end so let me know given the of the construction Level Scotland were going to do so the 1st thing I I wanna do is consider it wasn't related to the Indian virtual fundamental cycles it's 0 so consider matters as models so pretty tricky we have this the thing which I did find last time called ritual coatings on much less space to respect to a given because was on the beach modular space and the and show that for the moment this is cause it's more tonight because it's a perfect check the chains on With coefficients in their orientation not all of this but the maneuver of the patient but we also defined is defined a map on this virtual coaching complex to chains on the contrary state Of the specter space to direct some of all obstruction spaces some fundamental vector basis is it was a Smurfit dizzy they still on the 1 thing will have to do today is so was always last time is redefined this map for any particular T defined map like this which on under these causes morphisms is exactly the ritual it is fundamental cycle this it's simply the definition of a virtual fundamental cycles on this was roughly analogous to defining the Civil War the classic vector bundle by pulling back the top class and always going to do now is and I make sure that this construction is appropriately I'm compatible with all of them ways you can concatenate and map Trieste other retreat ,comma efforts appropriately compatible on the chin level on will conclude that these on this issue would be laden in here you than that for the last year you have available throughout the yourself and others yeah so the result is that we're trying to prove this is something like this so have the any Wilander J. there exists Somerset data of J. none of the on along with for all the elements of that set virtual much like cats With depending on data on such the satisfying these mastered questions the squared equals 0 said on human rights in the Aegean you know someone writing yet so this analogous there also not part of the statement it's nite and part of the statement that the homology that is independent of the choice of little favor is analogous therefore anti-Thaksin practical Buddhism and for us 1 from the family exactly PacifiCorp to capture snow these and I the chain chain home topic and 2 if you take all these together you you can show that independent of of the country and Luis using the head of is young and then whether you can yeah Gordon fan and 4 1 ,comma from awarded toward you of all of these you get variants but but doesn't if the fall from any them but so far long high-level outlines the following week that we define this
collapse last summer defined the Surf for any individual trees and and we have to do now is make it a map that's models come and wants to market test models of the fact that this is a but as models implies the following properties of these virtual fundamental cycles the the the boundaries the given is the sum over all the 1 boundary straddles the corresponding virtual fundamental cycles and virtual fundamental class of foam on the space which is much space of multi-storey Curtis simply the product of classes of the of the each of the 2 so if we can if if 1 defines sort fundamental cycles of these properties 1 very easily defines say but the virtual much like and degrees 0 that that that give the cinematic which itself so this is some of the gold construct cycle satisfying these coherence properties governor .period is just that the coherence properties are encoded in the fact that this is a map of as much right so to make this precise the few tasks we have to do so 1st we have to define both of these complexes as S models and the match between them and 2nd we have to define these on horizontal policies morphisms as causes morphisms of S models that's compatible morphisms and concatenation and then and then we have this matter which is switches will so right now that will salute to the the whole thing show would you it was a long to that we should all his good work on and this is just like the queen of product of like on philology at something like chains and acts cross chains on the chains and amendments the Volvo euros it is viewed as a whole because there's a there's a quotient I this there's a map and its quotient by Zeman about a product of my cover what was his for the ambiguity it at home yeah the for the it's a 1st invalids firm I think that's and so the question help How is Savchenko chains and that's part so on their push for groups for this at least not not not obviously because if you have to pry mapping the tea that gives you the inclusion of embodies a boundary stratum of this and this is really not going in the wrong direction so I would really is a son of connecting home-office if it couldn't have won it really is the code under that plan on-call knowledge on so to write this down the chain level and we need to pick a different model of checks chicken chains so soldiers replaced it was the following year and that he would visit here when the rules relating to the services you yes so this long on total complex His causes more effective to this 200 had see this this is cause it's worth it because this market because you can check the same thing a level achieves just look at the z were inclusion of and bar minus the boundary too and bar from this chief this mischievous on this checkered changes check chance of of the chief on this cause is marketed as complex as a resolution where you take the and everything and attracts some of ziti hasn't he can borrow decried thing of couldn't 1 if you the move was going inside the all this were the latest in a it's it's everything ,comma this is a major swung 1 to To the yes sigh I could've I could have written this also as direct sun just of all of the all of the guys here of chicken chains all of this I would like this because that's what I'm saying about matters but so at home with this model it's clear what the on the push forward maps for on the prime mapping to Tr on because he feared simply see the sub complex for a given on the prime mapping the tears as inside this complex on this realizes all these Benjamin of prior to this and vision imported some complex of this which will single out will cause a few ask the which we
have some Iowa just take so this is a resolution that has taken the constant sections in each of these guys soldiers the complex as he goes to some overall guys couldn't have 1 going to the sum of all guys into set and this raises some complex focus rightist simply take star equals largest takeover degrees are part of each of these so this is somehow In this complex of this generated by characteristic functions "quotation mark close data and the boundary is somehow as as you expected to be the boundary on the boundaries of the of the generator the given the prime is the son of a World Cup mention 1 boundaries to of that the treaty so now was likely due to what we will find a map From she to this complex virtual could change and bar rebounded in there is the same 1 of you it's like to have closest morphism but In order to define virtual muscle accounts all you really care about is evaluating these on virtual fundamental cycles on on 36 functions of Closter with would not serve wing where general Coleman 2 classes on the much less space so so sold this this complex he's enough of a quality that we In it's was difficult a check that there is so that that map from the method best models from Q the joint to Q so secured S under this complex I'm very explicit this differential on 2 is the S model which is just 2 for every tree with the true identity map as that Torrealba action and for concatenations the matters just product the madam as models like this on is exactly the data is going to you on multiplication of the losers in this case you Bloomberg is it's the same it's the same as adjustment Democratic it's this the I felt it is exactly data of upon numbers for those streets hospital dimensions 0 satisfying this summer people 0 4 staff of couldn't mention the subvention 1 so this is some another advertisement for this notion of an S model creditors this this notion as culturally incorporates all the of the relevant data of hold all how all these much less basis for together and this is 1 instance of this effectively we just had this the very country complex which on the stove built from the straddle the modular space in unremarked 1st models it's exactly the same as a virtual martial account satisfying on the creation and some of the others equations encoded the boundary operator mistakes and in the myself what I need to do now world analysis on tell you have defined this as an as much of the findings so we'd like to just take what I wrote but unfortunately the that maps so the match going the wrong direction so far I have so if if a given amount of trees from 250 prime but we get sort there's a conclusion of this modular space into this 1 song hasn't had any party hasn't that was table are the prime on inclusion of Abbas's goes in the other direction when you have this industry the more thickening Dayton's -dash usually don't and forget about all things In this somehow this inclusion it's compatible with about us but not to use on more general so you get the following is a map from the bottom the body we have our competence in this now there's and so on increasing in the inflation of the Atlas causing some officers from an MBA from from fear using just this small subset the conditions and this is the push for now but we want to what to years so unfortunate this and this is no way that we want this matter on but there's no way that the funds canonical because this is a crisis were physicals in the wrong direction the source of the Texas and it we have to take some replacement for that these complexes In need all 0 who is
a bit of a lot so and the whole definition so this new model of curtains were for adults over the Atlas defined as Portland little what's known going on here but me let me say what we should do just in this hemisphere a simple example here I suppose we only have 2 trees TNT prime and suppose we wanted to find out these 2 complex is such that we have this far and a push for now this so obviously I think this would little definitions it doesn't work so and so it makes for a while we we can simply redefine our target claiming this complex on to be the following mapping please take a virtual could change With the status of the early Laurie places by something so we replace it with falling income so you into this matter so I had this we just we do same diagrams this man from this complex this complex is a causes some amorphous so and so at least up the classiest morphism these just cancel our left at this this whole mapping Cohen is simply causes more effective the subcompacts on the other hand this topologically maps from this complex into the map income by inclusion here so we had what we always do years we replace this complex with something which is a bit more complicated on Bowie gain is the strongest push forward maps for In induced by morphisms of trees so there's some complex were just take on people 0 and 2 0 equals P equals the and this then there is virtually no change on members of the Taliban and is the cause more for essentially the same the reasons was across its in here the quotient has had infiltration by I can't intractable constitute intractable as intractable as infiltration is associated grated is is contractual because it's just a direct some of cones of conservation for and from increasing signs of so this is what we think of this as some sort of explosive model for all topical so now we can make a make similar definition yet to the the usually will soon will you and we will involved using the state of our union this is the wrong on these 2 officers you know who you are and you have so many it's even yes the smaller Alice's cut up yet it still gets full-fledged recovery In the we should use it to you the addition of militants some of the earlier this year the year prices so we can make a similar definition of fought for this complex the this chain has has a similar deficiencies found as model for basically the same reason and you want and fix it up in exactly the same way as defined chains and Tom strength of the his by definition some polar bear war-battered again some appropriate topical of these guys but again it's just it discloses more Africans but the of administrators himself on what all of them now is for ever actually defined this as an S model we have to change the definition slightly to something plus isomorphic Tallwood wanted such expectations that this is actually an S model and now on this again we had to replace it with something else to make it as well so it's all changed annotation slightly but not now these on redress for their defined but the said and now I would like to start on defining ,comma these concerts morphism mission because its markets he should because it saw fit to Checotah so instead to that he construct it's not
the as models from this join ask to so nicely going to construct this questions morphism on the chin level rather construct this map on the channel worry think of this as on the complex on generated by characteristic functions of the staff so although we want this map dissatisfied forgiven here in such a manner still the name W really class direct some form of otherwise this income so thought this is basically on politically class here so this it is really the the horn here in the office the category punters From the category from the S 2 tourists so this is a natural transformation functions we can restricted to just trees over tea and you get a class here simply just after taking the action that was this just has a bunch of this is just a freak Free generated by John G. statement at the time this is what you now if we look at what this this complex actually this is because it's it to check the chains on embarked he not relative to anything basically this as we ended up checker chains on each of Estrada on each of the open Strider and glue them together in the right way there's a there's a boundary operator on here I didn't step in the right way To get checker chains on the whole thing some without him and and we really require from the fossil W coincides with what the class called the the function was and what what this essentially says is that NW underscores morphism here W. corresponds to this 1 this inclusion or you can serve as complex of and that is still to construct a man on constructive W 5 induction plan it s so what I mean by that so far the 132 and analysts were going to say that he wanted to stay the T 2 if and only if and this some concatenations TI I'm pleased to 1 and all yeah isomorphic to some tonight on and a man from the concatenation to teachers so picture the following you have a treaty to this amassed 232 from some tree like this and on and we the 2nd half and we think of this as a concatenation 2nd cat out of these 2 pieces 681 associated with future reintegrated into the on known this case Fraser claimed that the we could s with the on relations this is the post-Sept so this requires power In our assumption ah ah ah restrictions to effective trees because otherwise you can do something Buffalo let's let's see what sell so reflect so transitivity this plot not difficult this holds general on the effects of this trip by definition always 51 by concatenating 1 with itself and that's no good at all the real thing to check is on the symmetry as usual when your defining a poster the that most subtle thing the check 2 nd cemeteries fails if we just consider arbitrary tree for the following reasons just consider some the tree like this has a morphism and provinces serve mess around the labels so it may be relabeled this sale To some homology Class AA and labeled as such a bias B & B so this boy
concluded here is that this treaty with the label B is less than to this treaty with the label Armed with his true any a and B and so and so have these relations but for Frannie indeed the homology and that's discovered the breaks as long as the home of cheese is non-zero this can't happen no if we only restrict to effective trees so this this holds assuming the line and 2 the effective fight said my 1st spaces for both nominated 1 way of thinking about this is that if you had if you have a non-trivial for all the EP could do this with T not equal the T 1 what you have is a concatenation the nontrivial concatenations 4 where 41 itself appears as well the boundary strand and Miss Woods vary wildly contradict compactness he'd have some space and where your space cross Cross another space is a boundary stratum of itself From Compagnie also tells us that this partial was set use well-founded II there's no I'm the decreasing strictly decreasing chain like that and this is the property that we need to do Newman induction talks about uses aside explains lightning-fast constructed by induction you may not be the inventive step on worship the district boundaries patent extension problem listings but so so here's the complex where we want W To Live and we want and we already have defined W on all of those trees which restrict requested the less likely than men so the company restricted to on the price of crude dimension I know this because it's worth it to Texas Chainsaw Mama the this is just causes Norfolk to check the chains on the boundary Ahmadi this essential restriction the homology class of thirst W here promises to to 1 and we want to the whole across of W here article 1 is now the existence of a lift simply follows from the fact that this is the subjective and those homology crassly want from maps to the amount she class of assessing were so it the Memorial somehow we can we can construct maps the out of out of this thing come in any particular desired help homology grass as long as the homology presses I'm satisfied some reasonable coherence condition written that 3 is we need to understand this on this question more forcing him to the source he says that 1 thing that we can strengthen a map of its models the cold but we know from this test model of a two-year posted to step 3 we want work underscores US market this because it's not because police would really like is a map grew at a map to the to that as models he wishes the trivial structure maps and we like to constructed by induction like before on but this construction is going to fail for the following reasons yeah fails because the SES models not co-favorite Sony fine what it means for a friend as small just because I vibrant and and you'll see why we need a little more collaborative so this module of the ACS is called the "quotation mark vibrant but only at 1st I want all of the concatenation maps the isomorphism and 2nd I want to do go for maximal rex with respect composition that is he with a single Vertex the number vertices of the 1 the following is injected to take Co Ltd Texas the primed for all of the new crime over 30 top of the dimension of 1 which should injected into the pockets of the but it if we
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limit but not a whole article in that so we don't know anything about this critical mass we want to do is defined the find this guy here so we find it's too frightened of tea such that the injection here and this is a subjective process sources on the
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Exco fiber the simply take to be the the mapping cylinder of of the script the music the of the world yes for the rest of it's over for furthermore this is even the cofounder replacement functor for heavier a map of S models from X to Y induces amount from Mexico fabric 1 Kiefer "quotation mark so that's we need that right so too has the finds this government defined on this matter so so-called snap picking so we will define his back he was forced to find the actions of at the inductive steps so that is we need to extend the following month limits from primary to squander this co-firing guy injects into contrivance on time space the it's already defined as 2 Q here and 1 extended so now here it's not as simple as just saying as the saying arms there's always a commuter homology work for OK so in order to guarantee that the extending the existence of an extension on sums of elementary ,comma logical algebra 4 tells us that 1 we can extend it practice if and only if because the on them in the city but we need some cake stand if on 1 so it the commutes right so story this is concentrated agree 0 concentrate degrees 0 this it's a mystery and and that is the problem of soul only need to commit it's 0 and we need H sub minus 1 of this column equals 0 that's because this is a little needed this is another thing which look for some reason didn't come up in the southern part inductive argument so but both of these are not obvious so if you pick any particular T prime it's clear that this commutes on homology we have I mean this is isomorphic is even less Moffat disease 1 ends moreover that these these isomorphisms camps are compatible with this man once take the calamity From there could be H H minus 1 there could also be extra 8 0 which is not generated In the news on the image of 8 0 author of these cats so this is I support the problems on think key observation here is that if since since this is called hybrid refuse cover performance again the Catholic :colon so this calamity start of governments the squad this is this cause it's worth it To the whole topic :colon Over for century you will what has come to you the Commission so this this a map here and the summit here and 1 extended its on and moreover wrote I want the action at the there's specified map I want this to be on homology it's also the fact that and all the maps I'm from Seiko firebrand of T 2 Seacove happened a few prime are injected in implies that this column is actually the same as the home of topical and the whole happy color is isn't that none of the topical it is invariant under cause amorphous themselves and each of these are causes for Africa's even we just get the homotopical disease which is I'm just changes on on the nerves of of this category S Over tea or rather than the full subcategory of stuff which is governments at least once so this looks like some complicated answer on that for whatever reason things were covering nicely because all we'll would dislodge of energy and negative whole call homology and unnecessary the 6 it's all season seen from this description of things that that out for the image of H 0 the individual guys here's rejects on 8 0 of this just that push forwards the points but for from points on to Jackson on 8 0 of all of the above the whole complex and it's negative all of about space the Islamic is conditions and we conclude that there exists I'm 1 so this is almost almost all you want for more than 1 and last thing to so we meet and you remember I already marked remarked that match the best models from the sky to the sky is exactly the date of virtual model accounts and we almost have a match like that you have a map from here to here which we can compose like this and if this map produced in the opposite direction would be done instead we have live so we have to make a list but we tell them sadly this again construct topping told us by induction of either that there there's no requirement on W. told other than as this diagram can and somehow the fact this is "quotation mark it's as a means means any list despite a little dog 1 the original the law of the land and on either I think so I mean as as long as I made the definition of contrivance to correctly it it it should be just general properties no yeah dead at that and more moreover somehow the effect even make this list just comes from them the copper corresponding fact for chain complexes so the at the end of the they need me to list so have called him at all if you join access the prime still couldn't 1 this injects so-so here now work were again using the fact that the skies itself is coal-fired this injects into 2 of the best teams we have a map from curiosity here is the strike chains see this is a suggestive causes morphism and this Is the map W. told of the defined so far give us a map to the colt vibrant replaced knowing what is a map officer now In this situation we have Michael vibration collaboration and the subjective standard and a secret vibration to general properties that lifts like so this is always exist services last lifting property in protective mobile structure take it on non-native agreed to Jenkins the subtleties that In the fighters but week 2 sources of there's a lot of much algebra here in fact that's mostly what happened there was a logical algebra and it's possible that all of this could be the I have ever tried to do everything service concretely as possible as it's probable that I'm using certain other machinery holes can be reduced to a few lines but that on kg extended those 2 lines in understood actually mean it would be something something like so I now what let me just say we exactly what the senators investments there so some talent data data it is the choice of the but we knew and that told us as as we constructed so that there are a few requirements on this map had to induces below the canonical morphism beat on homology this matter had to correspond to the canonical its office in between this and check chains and the diagram had to commit the souls of those of those conditions I'm and and we also saw so we the truth about that but there's always exists today that this this is 1 of the In now given but element Interpol's data we get this models from few enjoying S you namely on the composed with the Beatles but which as a remark before is exactly the choices of virtual model accounts for only for trees Trenton 0 such that found the compatible with concatenation hands on this summer is there on business and operating in mn these 2 statements of on we just interpret this virtual count on as 0 if if he has spiritual dimension which is not Fretilin mistresses this is the publisher of the final step in the previous there Gray so the next time 1 wonders a lot of analysis so 1 of them things which I I haven't discussed so far gluing and that specialty next time the good thing is inevitably part of any on construction of the floor from homologs :colon variants but we need to show Gloucester next time is that on when you have much less space of home a curves on that the increased for the most expensive Holdorf occurs relevant for contact homology and if the moduli spaces transfers than locally it's mauled by manifold both corners only it's basically the the product of a space of gloom primaries and the kernel of the 1 nearest operator few and now