3/5 Contact homology and virtual fundamental cycles

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3/5 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 Bill good and the good of the I'm moving up in the man I'm a member of my book my new huh I understand that my mom and and my good Mai-Mai I think I'm going to have an the president of the good but in Alexa let me begin by just reviewing where we are in the whole construction so are we start with some so much less space for system of modular space this was a pseudo curse and then others much less for collection much spaces we construct some extra data which called and was and then who would like to do this and from there was that construct and construct the ritual fundamental cycle on the modular space or on the verge of fundamental cycles on the system of much space no 1 at 1 aspect of this important aspect of these 2 arrows is that both are canonical to recall last time I described the construction of this analysis on the modular space is relevant for contact acknowledging but and if you understand what what's going on in the construction you see that doesn't actually use anything special about on being the situation contact not you really just applies to any sort of reasonable modular space of all Africa and moreover over what it produces isn't without switched independence on making any extra choices its canonical not all acknowledge I don't mean unique aiming for she could you could find some other way of defining and Persepolis I might just mean that in the process of making this definition I didn't have to make any arbitrary choices and similarly on the many ways you could try to go from possessed of virtual fundamental cycle on blanket described way which doesn't involve making the actual choices it's comical USA Today on why do all out give you complete definition of the status and so on as as you pay no there there are lots of other things 1 can try to put here In particular structures the others as well which are determined to last time come and solve all chase a what what some corporate power plants of features of this definition of this analysis traffic make this is revert recalls to the the midpoint of the shortest path between the elected all the fine and was analysis of where some of the good features definition which enable this canonical constructions and on all tell you how to construct a virtual fundamental cycle out of person moreover it will be a canonical construction so I mean all all just a literally just write down in terms of the charts of inputs Alice 8 chain map which represents the virtual fundamentals the 1st let me just remind you the basic idea we have some much less space thanks to its operation horrible said just compact Comstock well we fired the missile reduction consists of the following such consists of an open subset you Altha inside acts and the consists of on the chart which local X often referred to as the 2nd the health of vacant while choice space and embedding of you into exile for as under the
0 set of some functions to rectify that fired the missile the specs good so you coast of sulfur University right no the reason and we we want equal access with charts like this is because the tells of household the ritual fundamental cycle should be so I namely if you take if they offer itself as a manifold and you perturbed S to be transfers to 0 then 0 set is the manifold somehow lying close to ax and that manifold at least locally over you offer represents the virtual fundamental cycle of another the and the key problem where we have to solve in a 2nd error is to figure out how the on glued together these local virtual fundamental cycles into the global virtual fundamental cycles overall tax do just given these this local data and certain on local compatibility between the director let me down on the undefined for you what compressor Abbas's precisely banks always become had post off and today is unjust some some set usually infects set index of these charts still and that was paid a context of virtual dimension the consists of a bunch of a bunch of data subject to punitive actions no so much interest in the 1st the political covering groups and Alpha Four offered in a these are finite groups and just notation for finite subset of of a rather gamma I know this from notation covering groups than have structured spaces in the office for open a user finite dimensional vector spaces and can be eye-opening interests on the offer profit some more moreover on gamma alpha the facts only nearly on Monday to the office of 5 dimensional representation of thirdly on the season can places and survived the suggests how staff topological spaces for all "quotation mark for final subsets of and on the original modular space access is identified with the thickening on correspond to the empty so this can be better data and the only way it tonight interacts with With that the space eggs were interested in this through this identification the 1st there are pretty she maps as Alltel projects and outside to the Alpha for all but the continuous maps 2nd there are footprints near you by J. hindered open subsets for high contend J. contained in a 6 there are these on footprint maps see I J From 0 set of S J might and Jack 2 you J much of the damage and yet of
course demo yeah my acts on this as a president Ghanaian veritable subset and he just save gamma Alpha for In Jeremiah side prices of folk-rock maps on finally last piece of data is the regular locus fell XI is a much less space on its contrary to what I've drawn here it's not necessarily everywhere cut transversely on itself and so we have to mark gimme subset its rates cut transversely research are going to be there at bunch of requirements for this data but only just paused and somehow related back to this picture and to make sure all on the same page so we have X because we have example because that was just a single charts Was that they have space X where is the same as something said there's a space the figured much less space on terrorism out from a cell phone products to be awful all right X alphabetic mediation releases the brackets around the office I won't In inside except for this this is 0 set a for inverse of 0 and this footprint map identifies review map from from a cell phone interested 0 in action which has its own business office and onto some open set which is you and the set off on inside so this is exactly what we have agreed on the other hand realize that threat explains imitation was this and we defined itself as well as for the S 5 completed direct some of us off 4 of the time a little bit of ambiguity in this temptation because on there are many functions club called itself 1 because the various possibilities for I hear that and so so that's why I have to specify the domain here was that the direct some of all that a selfless for offered in jail and not and look at the effects of that on a mixture prices of this the picture I get on 1 office just a single has just a single element on which is the more lesser level rose over here the differences only here are required ex-officer be manifold here that that that requirement had we haven't seen a requirement that little will come later come and finally given the status of the heart and in this picturesque assumed chemical is trivial but of course the gem of a controversial similar picture their actions here there's was not this is variants and I know this is not isomorphism but just to quotient I can offer is a gamble for cover right the actions for this data the so there are lots of these on items but most of them as you'll see what I gather topological and and they are so in in any particular a friend at the close fathers just some universal construction of an impasse that was fun the restored much less space for the curtains and when you do that construction all of almost all these actions are trivial consequences of the definition but only if you think you really have to the Czech to forward I saw it go up of appear 6 times to resist some some at home but now let me be more for precise ,comma actually demands that but this is a highly amorphous after cautioning that so this this is 1 of the actions which is simple requires little bit argument in practice not not so serious but there's something there that Jay accelerating contained in generating 7 of entertainment facts freely here of another finally some items which which are reported so this would be open as the locus where this much less faces transfer should be an open subset 2nd Round as from X J 2 major unless I should be locally modeled on this projection Over the 1st image of the regular workers it's in particular ways me if I take J. equals size the target is is is the point 0 vector space I think Jake alive is tells me that the regular locusts is a manifold of dimensions the In the summer just as I write this the next devices as a special case of excellent XI raid is a metaphor of mention the Post luncheons Of and moreover whenever I have under some 2 X XI XJ and absolute .period in XI which I Club which I declared to be alone transfers Don when you that point has been cut outside XJ the corresponding section by the by the section of the section is transfers over over the of women prices so here only demanding on topological conditions is locally model topologically here this is a topological mindful that mentioned the customer to I what this yet course there's there's no there's also an analogous definition on analogous notion of an implicit also this instruction you could require things to be smooth what the point here is that for the 2nd arrow we don't need a smooth structure and the giver definition of the BSE Verizon of apology about what actually use any smooth structure on these thickened transfers of spaces and his 1st arrow is
easier if we don't want half to construct this instruction on this spaces of these actions require some arguments and practice finally last text images also unimportant 2 he said at this stage of the reunion on the transfers "quotation mark so if I look at the transfers of XI rag this gives me this gives me the buyer is some subset of techniques text and the set by this and the subsets better cover all ex-ante said this is basically the same somehow we have enough charge to cover all of all VAX with With of with transfers because of the Steelers I hope of recovering the ritual fundamental cycle if I think much less spaces the locus River transfers but doesn't cover politics also requiring here and specialists this looks like a a lot of actions but so I suggest in practice we have this canonical the procedure for for constructing 1 of these and in practice the really only 2 things which require require some sort of the argument the 1st as this if the show that this procedure gives you enough charts this in the particular case of comfort quality and usually in general this amounts to is just saying that any curve on a new much less space the but can you commit stabilize at by on intersecting with transfers devices at some of the ingredient which goes into this phone 5 is actually In addition races 5 is not actually that hot summer features some something about comparing from what the politics 4 8 9 but this is somehow announced to there and so if you if you 8 9 near Point on where were the domain of your your whole market curve has no nodes on the it's easy assists some of its policies an implicit there into functions therefore ballot manifolds on near neighborhood of an old Kirby have to work harder this is this is this is where where the analysis comes in verifying so I'm just briefly this example he should keep in mind is the fact is just some the modular space of perhaps you go where you equals 0 the exercise is some spaces of maps you along with choices would be offered in the office profit by such that the 1st you satisfies some open conditions and they are helpful for all fell in iron and gold you plus the equalizer so all this is just over a cartoon version of the definition I gave them and the previous lecture Tom but just even given justice cartoon version you can check most of these actions things like this Hargis a topological on excise radio here is as locusts where XII is cut transversely and has trivial isotropic that all morphism group of Of that the particular curve with whatever marking and we've chosen addition here is a little some remarks about his definition from the city is in practice the set of all they can't I call the elements of a thickening Dayton's is some universal set for all possible choices you might make of the candidate Tom so it's the very intense sat uncomfortably in that sense 7 1 thing to point out its it somehow it's it's not and completely triviality to formulate a definition which allows you to make they can spaces for arbitrary subsets of the candidate Romareda if effects that it raises said all the conditions so I could demanding here but I I can construct the 2nd much less spaces even while I take to thickening games which are very badly non transfers to each other In the 2nd simultaneously with 2 devices which are turned transfers to each other and the fact that we can the formula definition which allows allows the surf universal construction taking into account everything at once is not entirely trivial and that's tomorrow nite it's not obvious and other construction services as possible both other notions about this on another sir the difference between this definition and and others is that the XII the little 1 1 is that XI ragged you might you might want to require that this excise transverse everywhere on for nontrivial fitting belongs with the can at least at least some then and then we go changes much saw that this is this is this tempting to conclude this is Maxim because Of because of this picture right here X X or Y jurors transfers everywhere this is the case with the most intuitive on the other hand this requirement makes it much harder to construct implicit and so on if it is indeed doesn't let us do lots of other things we wanted to solve all give examples of this we don't want that to include the sexy also more were generally this is excise are treated on equal footing on independently of the cardinality so in other definitions that both similar this 1 you have 1 for formalism for helped ex ante said car outside XL and give another formalism for excised from X J 1 excised moment on and this somehow leads to serve duplication of the they have to check on so this is what I think good simplification another the reason why the went but both of these points are very important on interfaces construction of the product with us so 5 space accident was was a space while of the I can construct and was outside products with and accept a distraught unity on namely the they what West sex crosswise thickened by Hyundai prime is just the product of a vacant lots spaces and radio Locus Publishing only 1 Crystal Palace possible it's XII grade plus 1 as of now we we see
immediately that on this this construction on even even if he started with in this analysis satisfying this property the route the product doesn't necessarily sets for this problem because is 1 of because I
just union I promptly nonempty but 1 of 105 had and could be empty but it's also criminalists this somehow completely completely fails if if we don't treat excise on equal footing cracks and the environment the so some people find this notation quite strange they use the sanitation for the modular space and that they cannot spaces but I think this is a good justification for what that's natural attrition and that in this particular British always finally this is the definition for the let us spaces without boundaries I am required thinking 3 mouthfuls Of course there natural on natural generalization of this inclusive and this analysis with of space acts would close subset knows its boundary on the cult's boundary this notion of an implicit on this space which just consists of thickening inside thickening 3 we specify close subset and then all of the actions of the same except on the local model we allows now what we now levels of mouthful to boundary as as long the model and In the local model and have them the boundary of XI rank as a manifold was boundary should coincide with the source of with the could be expected and you can do the same thing with more general stratification it's just a matter of required that the stratification be compatible with the footprint maps and and specifying direct local model that you this creates so yes the last time I played a constructive in this analysis like this on a monthly basis for cover logic so it's in our town where Sarkozy the definitions West time whom I gave you the construction now argue the reconstruction the virtual fundamental cycle so as I have said earlier Wednesday interviews just the right down a map chain which represents this virtual fundamental cycle on the channel and on this this constructional being the canonical in particular its it's functorial with respect to conclusions of boundary accretions of other strata but in it was at this isn't it as a result all work well on all fours Larkham ,comma school theories were really needed Darius and coherent ritual fundamental cycles over a collection of many of its but many them much less that would be associated so X will be a complex health the space and a the emphasis was on acts I'm going to assume for convenience store the EIS is finite right now this is mostly for annotation of convenience on every everything I'm saying serve works for works for instance a simply buys at the indirect limit over find subsets I'm so this isn't it the big assumption plus choosing from a subset of a given Palace isn't very much of it is very much richer so was the 1st thing we have to get out of the way his orientation so if we have that was next we get an orientation chief all acts banks indeed it depends of course crucially that the that's here in stock at a given point P and X In given as follows so which is a list he told us in the next 5 Reagan and then the stock at a particular point is all for the orientation chief of accelerating relates a manifold sensible funded station chief to stock he told tested was on the orientation model of EIA duel finally taken variants on the river the stabilizer of Peat told us under the action again so we say this hapless is locally oriented ball if and only if this invariance is always no answer Of orientation models are always isomorphic dizzy and on threat orientation model of a vector space is a freezer Marshall of covering 1 comment using a generator that copy of z is is equal choosing orientation but that's orientation models are call for this action clues despite his by signed so far if generators trivial or a referral of all border guards have local or instability but if not we need imposes conditions Pakistan was satisfied for tumultuous spaces were interested but so henceforth will just have assessment the Sumption that locally oriented so
this is this is just a local systems Imax in on and of course the virtual fundamental cycle will be homology twisted my arm twisted by the duel of the orientation the we thought the goal of this we have seen machinery while it will be to the to the finding of fault structure in the middle of the year this is obviously machinery will consist of the following so we will define 1st a chain complex which also called virtual coaching his contacts with respect to any it's just notation also this is really the main thing I'll say is what this complex will come with the natural push forward not to chains on ,comma ,comma stakes of the and it will be economically close market to Czech coach Alex a total services all construct a after making definition orientations and now I'm going to leave orientations out of annotation really I should twist this provided dual orientation chief of eat and I should also mention that this should have a good local corporations and orientation she found but simplicity imitation it when they meet so what defined why is this the virtual fundamental cycle so this can complex is very easy to understand the that town's spaces a this is just a year is a find the selectors based on this is the sphere and solicitous concentrated in degree the so concentrated in degree the we have up the cause morphism is a map here start II what will we have is so you get a cycle you know that acts were in the duel of Chapko chains authorities agreed here this is the integrated the officers Chapko changed acts and have this map what this is is simply an element in the duel I'll check check and agreed side this is a checkered change and recall is this this is a limiting directly over open covers of coaching the nurse now what is the chain level do all of this to note the chair Michael bite on see upper body and called the Murad chains this is used much less than chains and spend most of most of the on construction just in the in the Czech world but it's I among the final answer can be understood this way still retains his son also duel checked chains which is the same as that derived in the selected over open covers of the commentary of them there so this is somehow the right way of doing it homology amateur accents you can you can also take the homology of of of the nerve and take just the usual inversion of multivendor on get something called technology but it's not the Sonic Cruiser up consider this is better and yet so far probably wondering why we have to use all these exotic amounting call multifarious possibilities that the trading exotic is not meant in the technical sense and mean it's these Strange called multifarious also the reasons the following so the ritual fundamental cycle unfortunately does not lie in general that does not lie in singular technology from falling very simple example so it's take take space W the Warsaw circle that's cut inside are 2 of 5 function which is 0 on the inside on and so they give an inside pass on the outside as Argentine are clearly on the virtual fundamentals
cyclists they should be nontrivial goes around the analysts those are the list if you push it forward to their regular neighborhood which is home in Norfolk and analysts and debts the non-zero unfortunately H 1 W singular homology is the the author of Virtual fundamental cycle never ,comma in Indian usual homology on but seeing around each 1 is is indeed Zeitz says as you think that generated by God so so now Our goal now is to is to to find work this object defined this diagram that'll give forests this virtual communities on the train Salemme on Monday 1st just say I hardly this offer single-track but that's already on the obvious and then generalize from a single chart to multiple charts so the bank single Singletary so so I consider the following diagrams so slick chains this right so we recall that on the mention of XL Dragon is the 1st convention of the authority is the virtual dimension it's now we have changed the text of the ad itself already reduced taxes considered the changes versus recall affects offered Reagan at the manifold and map the alpha and acts simply deserves have this month the global fight emission reduction so discussing how how this works in this in the global cakes 1st we I will assume that takes office is from cut transversely so and numbers put forward map as cellphone which forward going to chains and he offer both at 0 that Huntley said offer M which as set for years Of course it's worth it to see in the greedy and right now but rally there's a consensus morphism between this and check on Czech artifacts so then would get the out like this and I claim this this is the virtual fundamentals 3 even so what was just discussed the Smith case 1st Where were exodus assistant section maybe even transfers to 0 not entirely obvious even in that case it is true it's and then live this search and this is the virtual fundamental cycle it is it is very similar in fact basically equivalent to the assertion that on the 2 standard ways of defining the or the classic vector bundle are at the same namely you can even take a generic section at the fundamental class 0 set rather the point duel of of the fund on the class of the 0 so generic section or on you can pull back the time class from the fire this definition of the BSE is said to have somewhere to hold polling at the contrast this kid is going to become and on In is this this is a relative of the class and the usual sort perturbation picturing perturbed as take that fundamental class of 0 set is analogous to that of the corresponding definition following class so I'm in this case all we get that this is a definition of the and and agrees with the definition of that the yes and in this case this this complex right here this is the virtual functions have so this is this diagram I wrote down what exactly what we wanted personnel in the safe harbor the generalize this to multiple charts began planning just do this very explicitly in a single case of of of 2 charts and an overlap chart although although what I what what we do for this will somehow exhibit all the complexities necessary to to deal with the general kicks limited droplet of these things so here's part 1st base acts and 1 shot the Alpha chart looks like this have a manifold soon and although not terrific innings and freighter everywhere manifold effects offer to the same as Excel for regulated districts except for Reagan and from this function from X operated a cell phone to the offer 0 said is identified with some open subset of X so if
1 charge it is the same as what happened over here except of just just takes place over some of pieces None of us most have similar thing happening on them selects his x Gatorade not exactly the same situation as man as beta invader and as favor conditions 0 his beta consent X X is covered by you all 4 of you there now they are
Alaskans another overturning the the alpha-beta chart and indeed as it should somehow this alpha-beta chart is exactly what we need to glue together the local virtual fundamental cycle From the offer chart to the home local virtual fundamentals like the from the bitter is the thanks of a very travel quite but some high-dimensional manifold looking something like this some of the data it has 2 functions and sell them to the Office State of to eBay if I look at just a 0 set of asshole thought that gives me an open subset of the ex-patriot a look at the 0 7 a spate gives me open subsets of the Excel the chart this summer we will cut transversely Sonics alpha-beta 2 sections both of them the transfers to 0 and suffered cuts out it's a beta rather open substance there has been a cuts acts open subsidies you know these never as Alpha direct some Spader this is now an open subset of acts namely the footprint of the sex of a charges that you're the a section of the footprints of the offer Chapman bigotry that's not necessarily transfers itself has been cut out transfer some awful but the submanifolds don't necessarily innocent transfers of various sections of the Act or get rid of up the practices of the picture tell said how long the glued together on the verge of fundamentals but the 1st try is the following following so it's take then again changed the mentioned he offered on Excel for as before the Exxon change expected no so why are we doing this secretly Visa's closets market too this gives me Czech coach Engine Company support on the footprint of the charter gives me check the chance with conference on the footprint of the beta traps now we have to do this together using information from the Alpha chart the there's always take mapping cut they will get changed the past mentioned he often made up of Vader threat that bag that's the victim but also as a drawn here is the marketing complexes all I need you take the cup now says cause it's worth it to cause this market to check on the square checkered chains on the intersection nothing code which is because more of it to check chains and acts can publicly supported by critics as compared to the same so this town looks very good I'm as a replacement 4 this complex unfortunately defining this map on the chain level is rather cumbersome 1 soldier soldiers and 4 really what the intuitively what this map should be the cap particle physics was With power there is rapid snap but as been a pullback upon on that amount make making this unconscionable is for making work 10 levels possible but In the leads to mass so In sounds so it so we don't want this Monday something will bid on the difference it's problem was with this construction basically is that the dimensions of differ from between the cities but between these charts so on the fix that was just take the offer charts and crosses with the officers here speculation statistics this will just take Of all the chart across it with the data and will take their charts and cross over the office now we think the same dimensions and things should go much better so so we continuing this example so virtual coaches we define fall start this far to the right to change the course to mentioned the offer better on stated X offered across the bay the chart and then you take the complement of Textron's 0 direct competition like this and we also want to change on that the he also across text beta cells in the company's history and leaving room for for something in the fuel used to glue them together this has included 7 were going to do a defamation to the normal cup of why alpha-beta beautiful space look at facts in the text of beta be offered in the office the data and data and the and someone 1 such steps as often acts as the folk as beta of access what lies the debate this space has a natural factors 0 1 particularly a family spaces over 0 1 cent to understand what it is and just how you would respond so if he is strictly between 0 1 the I only thing divide by teeth and you can divide the 1 include the E eBay they're determining if we backs itself tea between 0 and 1 of the fibers just of it the people 0 is still divide here but now this equation says that as Alva must be 0 and the offer can be anything so did the offer across the 0 7 sulfur which is the debated charter or rather subset of the chart similar refer to equals 1 and you also forbade crossing flies and cold
deformation to the normal Cone and why is that well To think think that the following way as he goes to 0 for blinds were blowing up along the submanifolds you made alpha-beta so they have a mouthful the couple beta manifold you might 0 7 that up and the goes to 0 we block along the normal direction so that won't actually get a T T 0 all that's left is the total space along bond this is so this is the the motivation for this construction is that it we somehow like the picketed good neighborhood division canonical choice however there is a canonical defamation this comical defamation from acts of abated that this whole special normal missed and requires the chooses to be on the the ground instead instead choosing a matter to be remembered we on the fund family and containing about the rights of use space itself knowledge of why alpha-beta belted the conquest of text from 0 4 0 4 2 0 1 complement and now conclude together again using a map income Construction Co I use you call for a proper burial invader complement structural so snaps in here to buy inclusion maps the here alone by people's water and sewer here with the coffin this whole complex now is Friday mission virtual quick-change contacts with respect to staff as before write-downs isomorphism his checkered chains on the offer this is just the transplant invader persist chance when you offer effect beta called comfort support and this is consistent with the Czech so safe I'm a little bit more about how to make this causing isomorphism on precise 1st right so I had to find another ritual coaching complex this canonical push for not from this 2 chains on the town's face of the alpha-beta namely please take the position of obvious but for all the spaces and agrees on what we believe in inclusions of these on so take that push for another hearing 0 here gives us this so that was the definition of perjury charges just for illustration purposes let me write down the definition In general for general texts and a defiant Mitchell coaching interacts with respect to play is the following so the somewhat for the threat from over chains followed by subsets of any of the changes of dimensions -minus please put the cost of a of the set in a field this is so complicated the definition which is a natural generalization of of what I wrote before just the only you have to do this the figure out how to organize Everything when you have come at many tracks the relative to complement of locusts where because crop that is the definition it is so this is some rather complicated chain complex some what would what will you gain here tho is that this definition of the ritual fundamental cycle once functorial than requiring choices and 2nd it works some without having a smooth structure I'm thinking about us basis and since week just using algebraic topology 2 to define things the other the market is that this is somehow service home atop a of change complexes and so what was the deal ,comma typical innit while still allow times when you are trying to find the virtual fundamental cycle he takes on the history union of chicken much less space may be that the region divide the cynical but this is every day money up includes relation his very drastic review because because the answer simple questions about the topological spaces on the rate of basic properties which which are require certain of work to verify and and insurance and in some cases they just aren't true if you do this 1st analysis that might fail to be host of transport so In this an example :colon just who thinks together nicely the defense instead do the whole topics glowing the happy color so automatically has good properties and used certain you'll have to worry about this and this is sort of this is an example ,comma topical so you could take this construction and just forget about change just to do the same 2nd do at levels spaces glue together the spaces according to this simplicial diagram new partner just this little space and automatically 1 has to properties that it has the properties you want to use it to the final version of fundamental cycle on the New York after the mess around with the analysts adjust
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tells you is that they have this roundabout and unchanged and the sense the ritual honorable cycle back to cover short of what the so this is important for example humor even the most basic settings were so approve go off wooden variants are independent of the most complex structure taken the path of almost comic structuring monthly specials boundaries 1 1 boundary being that much less space complex structure number 0 and that abound in the amount of space stable master complex number 1 of extraction of 1 and this packets of the toll space given over to fundamental cycle whose boundaries the difference with the 2 sides on is game gives you the the independence of debt on a bit more and sophisticated is that you can this on this construction somehow we did everything at the Chernobyl so far on this this diagram allows us to to have a notion of particular curent collection of virtual fundamental cycles over a bunch of there much less basis which which is what we need to do for and this is also extends for the case with our men hold of corners I just knew that drive bigger diagrams right so so finally let me go call on to say this is also is also called for a public yes 2 spaces you can look at we really wanted this of nice definition of the product is toothless analysis so we can make a construction like this this is it's worth it to check chance to act goal boundary and it could have crisis morphism here to check the chain's contracts past why rebounded and his candidates again some chief theoretical considerations which are used to define the presence of a mail served this compatibility of virtual fundamental cycles with of products that this gives you the right acts landslide for its next 4 at this compatibility on the chain of old products and is also crucial for four-floor quality and because you Gables Taylor the virtual final cycle the boundary of the month in the threat abandoned Vermont spaces is expresses a product rather much less pleasing the virtual fundamental cycle of the boundaries that product of those fertile for possible solitude people have so far on this completes the redefinition of using machinery on next time I will I'm show had applied this and to construct a virtual fundamental cycles for come technology and given them and do given the Alice's unlike instructors must also review a little bit that the definition of the exits