2/3 Classical transversality methods in SFT

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2/3 Classical transversality methods in SFT
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Wendl, Chris
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Institut des Hautes Études Scientifiques (IHÉS)
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In this talk I will discuss two transversality results that are standard but perhaps not so widely understood: (1) Dragnev's theorem that somewhere injective curves in symplectizations are regular for generic translation-invariant J, and (2) my theorem on automatic transversality in 4-dimensional symplectic cobordisms (which generalizes earlier results for closed curves by Gromov, Hofer-Lizan-Sikorav and Ivashkovich-Shevchishin). The common feature of these two theorems is that both can be proved by considering the restriction of the usual linearized Cauchy-Riemann operator to the "generalized normal bundle" of a (not necessarily immersed) holomorphic curve.
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the as the tend to be in you nothing can but at the end last time I let myself the task of proving a theorem that you can achieve Travis salad force impractical for whole of occurs in some practical Buddhism for generic J let me repeat the statement quickly so the setting is the completion of its impractical moralism which has stable Hambletonian structures at the positive and negative and and I've fixed sum open subsets which I said if the compact SUVs an open bounded subset and let me introduce some new notation slightly better than what I did last time at St. J. thinks is just some choice of almost complex structures some sometimes most complex structure and a special class I defined compatible with like for many the stable Hamilton instructors and now the statement too says that there exists I guess I'll the word co-leaders and everyone else they call some said J you rags sitting in the space of all almost complex structures compatible with the data which also match a traces a outside of this subset you such that all let's for all day in this special classes all Moffat curves in the modular space defined by J with the invective points in the sense that you are regular OK so somebody asks a question last time whether a whether I really need to say but the currency is not a multiple cover or somewhere injected or just that there exists an injectors .period in the perturbation domain yield and the answer is all those things are equivalent in this situation but I'm skating in this way because in the proof that's precisely what we need we need the assistance of an injective point you get smacked into the perturbation in other situations where those conditions on all equivalent still this is what you need I think so I've got a few choices of the technicalities when I go about something like this I 1st need to decide but In what space exactly of almost complex structures do I want to consider perturbations and while there's a natural space in the picture which is the wrong 1 because what I'm really interested of course is this space of almost complex structures that I assume moral smooth with the Cincinnati topology on it and you can call that a fresh manifold at least in the year "quotation mark and sing and when NEW apologies the W hat of course is not compact and not so easy to define I'd about norms or even semi norms on a non-combat domain but of course I'm only allowing these perturbations to be non-trivial outside of this compact subset inside compact subset such easy enough to define sending arms and I can call this object here officially manifold but facing manifolds are not very useful to me because the implicit function candlestick-holders still fairness and hold there's a whole there's a lot of analytical results I need which I can use so really I need symbolic manifold and was so invective .period is right if you understand that part work by signing injectors .period whose image is contained in the subset you that's the assumption I I'm not saying all curves of the property and saying I'm only considering careers that have come so when you apply the spirit of course it's going to miss some cards right that the typical applied is you to let you be the Compaq particle is not including the cylindrical and then it doesn't say anything about curves that are contained entirely innocent people but will have another term for that In a little bit so I had to make a choice a symbolic manifold almost comic structures and do explained 1 possible trace yesterday which is instead of looking at the smooth almost comic structures asked them to be of Class C for some finite that's it good choice but I'm a little bit allergic to it myself just because if I can write down the nominee Ramon operator for a J which is not smooth then the operator on the of the section of the boss they fumble will also not be smart about finding many derivatives sorts mail requires my maps to have at least some number of derivatives I'd have to look it up to find out how many and these details I don't really want to worry about if I have a choice I'd rather have everything in the picture smooth except for of course the maps themselves are going to be a of class but likely to be smooth so that my nonlinear said section smooth so I'm going to adopt flows solution to that which is to define the following objects so let's this another almost complex structure that will call the reference to In this space as well and matching J 6 outside and it is going to consider for the moment
almost complex factors that have perturbations of Jarrell in
some precise sense so the other pieces that I need to fix is set of positive numbers help did not collectively by epsilon is a sequence sequences converting to 0 and the the final day of someone to be the face of all thanks a lot shares rest of some y taking X imagine experts to be the exponential map on the manifold complex structures on each tangent space that's something you can define however you like it is a manifold so that's fine why I consider to be in the tangent space to that fresh manifold and also vanishing outside literally this just means it's it's a smooth section Of the bundle whose fighters are the tangent space is to be manifolds of complex structures on the 10th of spaces and things move section but I'm also going to require that its finite in this so-called flourished c epsilon norm but there are signs that sun from 0 to infinity of Epsilon and times see and so I wanted to be fighting secondly require that its small some constant so there's an exercise 1 needs to do and it's a standard 1 but you can find in various places originally in summer flowers original papers but if you choose this sequence of epsilon converting 2 0 fast enough then this norm not only defines a bonnet space but it defines 1 which is large enough to contain smoothed bond functions supported an arbitrarily small the neighborhoods of .period that's the key thing we need to know about this so that defines bionic space of wise and just exponentiation back from Gera we get a bonnet manifold of David Obama manifold justifying 1 shot on the spot manifold so it's a small 1 and this is in some sense not any kind of global construction I cannot get anywhere near a dense set of jails with this but I can get Jason the spot early close to J. Rafsanjani ref to start with what was already for now the universal modular space and the at is going to be the set of parents you where did last day excellent space and do the whole Norfolk for days and also I wanted to satisfy the extra conditions that you it's somewhere objective remains EU has an invective .period perhaps to the perturbation menu so that's traditionally called universal model a space I think given the way we're doing it it's not a terribly good name because it is not universal in the sense of being able to consider all possible jails that we might be interested were only looking at there in some neighborhoods of chair and in fact it's it's not just a smooth neighborhood it's a neighborhood of Jerez in a very peculiar topology on this basis smooth function at the end of the day and is going to care about the fact that I can get arbitrary the Infiniti Costa Jarrell by objects in this space thank you and the last year while the leaders of in the candidates face a complex such literally it's uh it's a section of the and amorphous and bundle which empty commutes with the and this year and will be the morning graphic Victory Monument and measures no 1 integrated at the supermarket I think so an important detail there is of course the thing is only nontrivial in this context subset of the other questions In of book presumably yes it was only half of the year as a difficult question I II also would not say that I really know how to think about apology the main things we need to know are again the spaces large enough to contain bomb functions of small support that's said that's obvious Lamb of that but has pretty quick proof to the floor otherwise you just need know that this space includes continuously and infinite With the ordination frantic quality it was the 1st the
face what's the use of according to the survey this was the reason for the working with the kgs also prefer refinery to do that and I think they sell but I think that we have a minimum sentence saying things like that you want to a decision on the field India wants to ensure that here yeah it's or Omega compatibility is its simplicity here when I said that while the loss of tangent space this this so that is that we knew algebraic condition on life .period what we have to right the reason I didn't use fix here is that if I did when I will approved in the and is that a small perturbation of J. fixed but 2 I would like to actually know that given any In my admissible says I can perturb about 1 To achieve the data that is arbitrary set for the conditions of stay so hopefully were all on the same page now we have this functional analytic set up that we introduced last time we know how to understand the moduli space for more curves locally as the 0 said of a coterie operator possibly divided by some symmetry actions that's just write that down in this case so the result comparable to what I ended with last time is we can say the neighborhood the element let's say you not is the believed to hold across the complex structure Jane art and the targets almost complex structures called capital J so that defines the elements of the universal modular space and neighborhood that element is not going to be imaginative correspondence with the neighborhood of the equivalence classes so this neighborhood of Little J not enough capital not in some 0 sent a section divided by the automorphisms review not where this the section I'm talking about maps so again I have my tightened oversight of the finite dimensional space a manifold of complex structures on the domain dramatizing the neighborhood enticements based please my bonnet manifold of maps of some Sobolev class from Sigma .period into W hat and then I have day Absalon that's the 1 element that I didn't have this picture before mapping to some minor space you can infer from this picture what the fighters of the phonics based on those was to be a plane right on the map for instance little jail you begin to see you 1st victory 52 a little so that again as a smooth section of some fumble it is smooth because all the capital Jesus moved in the capital days were not smooth than I would have a bit more of a headache to figure out exactly how different boulders section because that's where introduced this year on the rest of the world the parentheses there was utilized by the fire divided 0 this section by so this is a list of some diffeomorphisms of the domain which act on all 3 of these while doesn't do anything to big javard acts on the stoop the same way that we had yesterday so I mean I can actually contaminate that from the picture here because actually you know somewhere injected so I can get rid of the autumn of his material he died in the world manifolds at this point even talking about simple concept of delivery the duration of the building yes that start with the church absolutely any right and stars in it so this is just shorthand notation is not just a map you it's an element of this model space which includes the date of the domain the swimming complex factors absolutely allowed to vary throughout this picture so there section and of course I would like to know whether the universal model face is smooth it's going to be a Slavonic manifold if the legalization of that section is always rejected and that's the main thing 1 has approving any
argument of the sort and then we already had a little bit from news about this yesterday on why you have to prove it and why goes wrong sometimes specifically if you don't have the the existence of invective .period so let's talk about that that's right down the line as Asian affairs it's going to take a triple why Ada capital Y Tu well differentiating that thing with respect to lower Jr gives us jail to you while differentiating respect you gives us the usual because you remind them when they arise because you operate associated to you which and is going to you by DU and then we initially have differentiation back to capital J against dive team of day 7 the 1st 2 terms in this expression together are precisely the operator that I wanted to be subjective when I talked about regulatory yesterday so there these 2 surjective operator if and only if you sent home regular that was a definition which I didn't really write down but I implied and I have this additional terms the advantage of which is why is now varying in an infinite dimensional vector space so it makes it a lot more plausible that you could force the operator to be always so how do we see if it is that's claims this thing is rejected and but maybe a little bit more specific about my functional analytic sitting at this point the last time I said that the maps you are and symbolic manifold of Master Class WPP and I didn't specify KNP except to say that Kate you greater than 2 that's of 11 betting tells you this tendency in Europe but for convenience right now just to make my life easier I'm going to cables so let's say this this operators really going to take the tangent space to to tighten the slices plus some class some space of W 1 key the class sections of New Star TW plus the tangent space to be I'm going to take those 2 sections of fast healthy I need the exponential weights as home would point out to me if I don't see myself but the subject of the exponential which should be discussed but I don't have time for it in his talks with leaders should be happy to do that in the discussion was the government the definition of this space when we had this kind this asymptotic condition on the behavior of of sections that said it's not just that they are of classes W K P on the cylinder but the 2 Delta last time section is of course that you can be the forces exponential decay the Delta itself is just some small positive numbers that research as long it's small enough you're not losing any home off crops by imposing this condition so I I'm very fond of the philosophy by the way that nothing you do in this whole story should ever really depend on your choice of solar space as long as you satisfy the conditions for the inclusion to see 0 in some larger different abilities she wanted so I don't like making this choice but it's only the 1st step in the proof if you want to them through the same thing is true the wkt to the view him as 1 piece that becomes easy if you have already you can prove inductively is going to do this case now so it is not subjective then we can say there exists some element of the duel the Elke Delta here that violates the whole image so the image is killed by some non-zero element of sulfate individual al-Qaeda leaders quick digression I don't really like this distinction people like to make between analysts anthropologists and whatever else people want to call themselves partly because I don't personally feel like I'm either an analyst sought apologists have no idea what I am but and most people think of me as an analyst I guess that they aren't they often ask me the following question why do you need to worry about Sobolev spaces In particular in this whole picture of why can't you just say Let's consider maps of Class C K isn't that the get you reasonable bond manifold it does give you reasonable bond manifold of course reason number 1 why we might for the follows faces is that we have certain standard estimates for coach Ramon operators that working full of spaces and this is what tells you that the years whereas operators from home OK those restaurants also do do exist for instance and holders spaces so you can't not talk about the cable CKD for some positive Alpha and I don't know how to prove the estimates there but they do exist you get the Fed property so why don't you use that is not still somehow simpler than solid basis well the answer to that is what I'm about to Of course we know what the dual space of Al-Qaeda's In a very concrete wh and that's not true for holders face the answer is known but it's not pleasant the dual LP is a perfectly pleasant object I can write it down concretely so this is the step where it really pays off actually even I'm aware of what 1 of the mid-Nineties papers but hopefully contained within you use holders spaces for most of the way and precisely at this step the switch to solve all of space so I've got this element of the duel now the duel is just L Q With
exponential weight might 4 .period reduce plus 1 other people's 1 and saying that violates everything in the image needs like to write that is 3 separate conditions in facts depending on whether I operate with this thing on any of the 3 variables so it means 1st of all that if I take any Javed to you of why and integrate that against the state of Gujarat abbreviated all too apparent that's the pairings Elke without you but that's going to be 0 for all the life of same thing will be true for operating the DU Ada at the thereof and it's wife to J. kerosene will also consider again this fall this for all white so that's what comes as saying that the state of violates the image now the 2nd condition in particular you can interpret that as saying that the data is a weak solution of another countryman equation before the formal adjoint the you and you can replace the you on the right hand side with its formal adjoint and it's that paired with all it is 0 that means that the former agent and I read of now there's various regularity results that tell you anything annihilated by a court on operator even if it's just a week solution of Class 2 is going to be smooth and even better solutions of course among equations if they have zeros anywhere zeros are isolated unless the whole thing is review the similarity principles so that the 2nd equation really says formalizing the start of the equals 0 so by regularity predates the Infiniti and has only isolated themselves from me so I don't know yet absolutely certain the background of this is a very general fact that for any coach fury among untied operator there is a formal and giant which has continued to question now what about the 2nd thing well remember why is living in infant dimensional space which is large enough to contain bond functions with small support so I can explain that fact answer let's choose an objective .period whose image is contained in the federation domain and now I can choose wine to be something that is non-zero only on a neighborhood of that .period inside the perturbation domain but I can make fairly arbitrary choices about in a small neighborhood in particular I can choose the wife to be supported near such Sept . 1 this pairing is positive but it vanishes everywhere else and that's my contradiction because I'm able to do this because I know in fact I should also add this without loss of generality by moving this point is that went a little bit I can also assume that gene is not 0 right because the only isolated series of fatal what I got from condition of the I want it well 1 injectors .period implies a neighborhood consisting of injected it's it's an open condition so having Madison a small neighborhood certainly if this is time of war in the number of people in the world and yes it is that you can find wives taking any non-zero elected to any other nuns it's a slightly not obvious fact given that I want white to be also tension to the space of compatible complex structure so that again is a linear algebra 1 of the means to be I think there is approved there is a proof that is a proven my lecture notes as well which is you copied from her bosses partly because it's 1 of these it's 1 of these proves that 1 1 can write it down very short and still get no insight from don't know why it's true but it's true only if the on the excellent question actions expecting understand the question I didn't use can I didn't use condition 1 in proof which kind suggests that I didn't use the title flights at all that's where condition 1 came from that's that's small 1 tension to the right side of but this does not mean that the title is sliced isn't somehow import I was able to achieve Kansas al-Attiyah well for the universal space without worrying about that on the other hand it's still true that there are there exists jail market curves which are stable which have positive indexed so they exists for generic big Jake but not for generic Littlejohn will only exist for certain choices of little and you have to allow that to be arbitrary in order to find those because if you fix the conformal structure the domain you get a problem that may have negative index in those things will not exist that's really is not so much a transit solidly question but it's an index question my 2nd remark is I will have occasion to use condition 1 in some form later in the stock if I don't get
to do so the we will not cannot done with money from get-go so I could the universal modulates basis smooth what was going on you well I could write down the dual as some kind of section of a bundle of very concrete thing instead of an element of the abstract tools faces some of this section and then I used to sell to parents the crux of the argument involves it's all too apparent section and that depends on having a precise picture of what with the doors that because in the end it wasn't that the argument I made was .period wise he was not some abstract Hilbert space in a product something in the reminds the this thing is about manifold now we look at the projection of that today on taking care of your to just Jr and aside snails and tells me that some generalized sense of almost every day will be regular values so there exists a coal major subsets let's call it a the epsilon rags In the Epsilon such that for all the danger songwriter and if you did want to this universal model is based on particular you has injected point in the preservation of mainland years writing and in particular the modular space with respect to J. itself is a smooth manifold so that's great it's a corollary of that is that the set of regular days in this larger spaces move almost complex structures that Max is matched J fixed outside you is dead that's media the because I've achieved regularity for Jason Day Epsilon which are arbitrarily seemed fairly close to J. raft and Jay wrath was arbitrary that's why I chose J. references effects so now this is going I have density of the the regular almost complex structures I think Cottonwood tell me I should be satisfied but I'm not and tho you're going to yet so suicide sites male depends on the fact that indeed this map from projections for home that's equivalent to the fact that the could you remind operators that sets the I disagree with captions philosophy and in a certain sense that results it's certainly not true that I should always be satisfied to get in nonempty said a regular complex structures and maybe that's as if I only care about defining a theory that I want to prove invariant but if I can also want to compute that theory I need something more right because usually I can make some specific choices of Jafar retired and able to count all the whole month because they care about and course that day is not going to be generic so I need to make a generic perturbation of it and I need to be assured that I can do such a perturbation that specific jails where I understand everything and obtain something regular so density is at least what I need I would like a little bit more fact because it's often true that I want to take some space of almost complex structures that have proven a regular in a certain sense and not intersected it with another space as regular by a slightly worse with different conditions think sometimes I want to do some countable intersection of those and while even the the intersection of 2 dead subsets can be and so I'd actually like to know that I get a call so countable intersection open themselves now can get from here to there there's a trick for this that was introduced following the White House and it also appears in the book by and the mark using it for different purposes because they've got finitely differential almost complex structures instead I'm trying to get from the excellent the Infiniti you can apply for the same way I think people often don't realize just how widely applicable this is and I'm going to do kind of lazy version of it based on the fact that we know the space of Holdorf has a nice compact I think 1 can be less lazy and and formulate conditions that in some sense equivalent to that requiring less knowledge I used to see the following list let and the good of the team in the space of all curves you in a modular space which have they injected points in the federation domain and now for its natural number defines a subset of the whole modular space them and they have to be the space of all curves with the property that the distance in some metric that I'm not going to specify but I know where to look up the proof that it exists distance from you too the compliment Of the good space in the compact 5 model so compact divide modular space minus and good I want that distance to be ready for equal to 1 of the so in order to earn
1 can make this precise To do it 1 has to look at the appendix of the SST compactness paper where they proves that the topology defined on the SST compact application has a metric so distance with respect to that matter so this obviously is a subset of and good if I take arbitrarily and large and I can exhaust and good with this countable Union of the subsets but also the subsidies compact it's a close subset the compact metric space and even compact and slightly stronger sense because this matter ,comma compact 5 modular space that depends continuously on Jan so I can also move J. around and if Jay sequencer data encourages I can also get convergence of sequences to order in the sequence of spaces so concentrated that to get the best and most of them all it follows from the definition of him because distances from their positive so let's define Geragos tend to be simply space of all Jason my admissible class such there all curves you in em M the suspected JI regular so
that some set of almost complex structures it's a subset of
the space a smooth almost comfort structures that have the sea infinity topology on it now the fact that this substantive knowledge compact tells me that this space of regular almost complete
structures that only cares about that is open and also densities the argument adjusted above I can perturb any almost complex structure to 1 that achieves regularity for all of anger much of the substance Due to the above stance all I have to do now is to stay the space and looking for a called year-end you it is accountable intersection these and and then and I that's accountable intersections opened substance so the ranch that I will give you on this topic is that the Flossie Seattle space is a very nice object but you shouldn't take it too seriously my personal opinion is it should never appear in the statement of any fear we should appear in 1 house and it should appear improve but just about any theory you can you can prove about the first year conservation can also prove he Infiniti using this it was the 1st time in a row in London yes indeed I mean that the argument went somewhere less the same way you want city regular ,comma do you think that way you won't you which lining up Saunders this annotation representing a space so I guess the key point is is the claim that this set of so it follows what I see here is that the set of genes that achieved Francis holiday for me is dense in the Infiniti as a consequence of the fact that it's also a bear subset of Chancellor this is the fact that day long has a continuous inclusion into the and so once again density in their files the government has been in the the president finding home that that so that when it comes to the rights of the the highest because of that and then later in the the issue is 1 of my constituents so yeah open that 1st of all this is compact because it's close some sort of a compact Spain now I suppose that this thing were not open what would that mean we have we've got some J. that achieves Francis solidly for every curve in this space but we can also find a sequence of curves a sequence of Jay's converging Georgia they don't perceive has a salary for every curve and corresponding status here but now compactness tells you as the days converged the use in these spaces also have a conversion subsequent and they're non-regular curves converting to a regular curve that was not possible because regularities and open conditions surjectivity of riverfront home operators that's another news that the did you believe up to this point so I do as we get we get call major injury Epsilon which employs Danson's Epsilon and that employ them since the infinity just because Absalon includes continues indistinct and think have to all the arrival of the new executive J. rests the manager of that that by anyone so exactly the wet weather this proves it your original J. refuge an approximate J. arbitrary Wellenstein said he was with the regulatory so and Jarrett was arbitrary so that that implies that the regular things and dance and dancing that is actually stronger conditions and what we need them here that was the end of a sequence in MNJ will not anything modal because it is positive distance away from the noble part of the compact model of what you know no that cannot be good consists only of somewhere injected on a stable Africa I mean it's it's it really seems like cheating by using the knowledge that this compact location exist just saying let's stay of finite distance of positive distance away from the bad part of the investigation I don't have to wear was assumed on degenerate reworked its the have to fix more than a year thank you so this is OK well no what actually has a good reason why I won't me what I really should do is due this whole argument just looking at home occurs in a fixed genius and fixed relative homology class and then say having proven we have a commuter subset for that genius and that logic must now take the countable intersection of all those choices if you're just looking at and we see that they said that the policy of giving blood was this is a good idea of some not the president had said that included both for will be successful you know this is what you actually see when people uses in the literature and I'm using this formulation as a shortcut but when you write this down in the literature usually people write down a list of conditions that say things like that the C 1 normal of deeper as the super-normal DU is bounded by 1 over and bound by and also some condition about distances you have X minus you of why divided by distance from next why should be within bounds determined by them so you stay away from the most recover curves and you need something to keep you keep you posted assistance away from normal process while going and also with the write-down conditions like that which should do everything for you without having to 1st prove Gromov compactness rest the compactness which have rights that the argument in fact did you can do this in for Klosters reading do almost complex manifold certain not tamed by syntactic structure you don't have to prove compactness 1st you just have these conditions validity regularity tell C-1 bonds will give you compactness and so forth OK it has 1 last time was injected in like this so if the claim is this if I have a sequence of J converging to some particular J in the sea topology it also covers and Infiniti politics so so as soon as you know you can approximate whatever 1 arbitrary Wellington you have a that's also true in the and the development of legend the the people of that was the right thing that will be used whereas life right now let's look at that again so why still will cover I said Susan injected .period and we make this perturbations of Lewis we define in some fashion on a neighborhood in view of the sea now I want to do that such that this pointwise pairing becomes positive in a neighborhood of the and I want to also know that it's not going to be positive in any other neighborhood somewhere else which would happen if you had another point going through Uzi that's why fails multiple comes in you know I can't conclude anything unless I really know that this is the only place where you was going through this region How much time to actually have a a know Narciso I would I would argue that this it has been a sort of hybrid lecture and discussion sessions and the afternoon discussion session should be the thanking you want to highlight not used condition 1 of the organizers the bank the matter which was 1 of them so let me at least right on the statement what still need to get to the next year or so hopefully we're happy now with it has a solid if a simple currency gains and practical borders and should not be included work so In order to actually do as a team we also need To understand cameras solidly for persons infected patients so that the translation invariant J here is the fear so let's say now and this is a new clothes or dimensional manifold we've got a stable Hamilton construction the induces a titling distribution CSI and radioactive fuel are H and a member of the House from that a special class of stock translation invariant almost complex structures associated to the 120 sable Hamilton instructions so there 3 says that there exists another season the call J. it's a coal meager some said creates such that for all Jason Jarrett all day long curves with as it and that our someone active and not everywhere tangent reckon so unpacked a little bit of but were working In our on so the trivial electrical Buddhism which means our map can be written in terms of real value function and a map from a punctured Riemann surface into em colors you are you and there really to pointed difficulty about this hero 1st the perturbations I'm making for J are no longer really local because I require daily to be translation invariant so any perturbation I make near a particular point is going to affect what J is everywhere up and down from there so that problem turns out to be relatively easy to solve as a technical and which appears In 1 of them mid-Nineties is leave the papers again that says if you the somewhere objective than the ejected .period other than that to and from the point of acceptance to the other dimension manifold are also dance I think nothing to prove that actually have not yet managed to understand the proof because I haven't sat down with the long enough but I can tell you where to look it up if you want to look it up a lot of money the I always assumed find and if you want to hear about infinite energy cousins but positions you have to speak with Joel Fish and not with me so that's 1 thing so I'm I'm I'm not going to be too worried about the translation invariant as long as I know that the map to to end itself has injected .period then I can imagine I could do the same trick I did with the perturbation just supported near that point and that's not going to mess anything up but what I'm worried about is the fact that I remember what are the conditions that these days must satisfy well for 1 thing Jane maps the unit vector in the art direction to the radioactive fuel and the invective field is not ever-changing in this picture that's fixed by the stable how-to instructions started with the only thing I'm allowed to vary it's hard to match the hydroplane distribution to itself so I'm doing perturbations of J on a sub bundle of real Co dimension to and I need to know that those perturbations are going to be a large enough space to achieve gender salary and you might notice from the statement of my theory that if you're not careful this isn't true there are in fact for certain stable management structures J. Hallmark curves that will not be made regular by any provision of this sort just quickly write-down example and then at lunchtime but say and his S 1 time W were again W again a close inflected manifold now with this but make it simpler than the rest of the the day of service as 1 times W and St. 2 years according an S-1 only the area farms and said and this is the the floor homology example that I did yesterday but just with the two-dimensional case so too do floor homology in a two-dimensional manifold you write down the stable Hamilton and structure of the form big omega equals only plus some Hambletonian terms you to take the house and the 0 for now I don't DT so it means that CSI here is the full foliations by the services sector many fine perturbing J in this translation during class and really only perturbing what Jay does on that foliage so that means I'm looking at it the dependent families follows complex structures on Sigma for all such traces there exists Jabal Moffat curves that looks like map similar to Parkinson's one-time I his to constant constant and these will sometimes have negative index I exercise lunchtime convince yourself that if you were really allowed to take a fully generic perturbation Jr you would kill all these curves if the name have the genus is large because of negative index so it means the perturbations in this translation and the aircraft are not large enough to actually the chief Kansas salary for everything they don't for these codes so that's why I have that extra conditions you have to just pay attention to president Ottawa attendant CSI now for the case that most of us care about most excise a contact structure and it's easy to guarantee that your career will never be everywhere attendance success so I will work continue with this in the afternoon the victory in the war many cylindrical qualities the this great discrimination tightly cylindrical all of its functions approach retrieval all 0 them but he was did you don't want to he was 1 of the things I wasn't going to I wasn't going to commit myself to a statement like that but I think you do 1 is already a contradiction in the world OK and I just 1 b as


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