5/5 Contact homology and virtual fundamental cycles

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5/5 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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map or local army officer during a mail from this too here while respecting specifications and landing on the regular so this Tom right into the German homey morphism that means it covers an open neighborhood of x 0 and since lands and Bahrain that means and contains an open neighborhood of x 0 in Rhodesia and is open and to since assist you forever it's as tho we have to show it's so so 1st set clear ideas on domain stabilization so home we really want to that was the purpose of this step Over a neighborhood of this . 6 0 in the much less space we want to their fix canonical coordinates on domain the problem is before this and this these these 2 components but there they are by whole Moffat with cylinder in the plane respectively but the not canonically self and we have to get rid of this regret this extra freedom disengage fix In order to be able to produce local bowl on parameters for the much less space this cell will do is fixed .period skew I can see 0 and who are invariant devices DI inside our crosswise on such that you 0 is transfers to the acts you up I don't care whether it's transfers to DI anywhere else just at that to us such this holds and c 0 equipped with these points is stable it also has the support here's the really see zeros already equipped with on its marking by the positive and negative function then so yet so it has to stabilize the look something like this since the I and you are beaten so now since you zeros transfers to the eye to eye this property Is this preserved and the small deformations of the home of a curve that is over an open neighborhood of x 0 you corresponding points in July in in the domain of all those other home and therefore but you've done this gage fixing over underground and so 2nd introduced include co-workers on target this is something that will still go forward growing problem of all :colon element of this the state's including prominent on we define this include targets for the obtained from 6 0 by truncating containing the ends in the horror the and stands look like 0 inventory crosswise to trumpet them 2 0 it would come from swine the same thing for negative and then including fire this the 2nd so thanks but other described as to whether he was the the idea you ask for and so were latest credit it's not entirely accurate so I suppose for taking this curve so we have x 0 6 4 copies of our wide let me just draw and has distraught factor looks something like this in supposing glue all of the parameters using exactly this recipe I get the this 2nd extra hot X she had done the success I've had this 1st piece 2nd piece I good both of these here to get something sorted 9 houses like this Of the fallout of the Sept . 1 1 House of target lead the service of an canonical way of disclosing a backer 2 . so as X again the ex-chief of course again just that a big union copies of our cross but somehow this particular description is the 1 it was the review year yes
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positive items this fundamental after estimates of but also that this is the fines for us always somewhat spaces in this Over over on each of the glued courtesy subject and why it would give it suffices to take people's 2 for this is that it's come this argument we don't need any harm anything else but of course we we could call me everything and anything this is true for any pianist range those on Wall both of you not for this is Chris Waddle has this blog post about this a little of the need for the "quotation mark whether at any point you need to use this the people the sofa from her for this it's it's really not not not needed on I think that was the point yeah we somehow given all the opera estimates which which you know which which sometimes require people with which up Bodmin maybe so maybe they'll never required equals 1 we somewhere directed me can you pickles 1 things necessary the solutions to some of the injuries were the main square requirements I would be here I think it really doesn't mean people that equals the really fine I played you already know you're only dealing with curves which you've already prove after estimates for and me we take a could can take a very large yeah I think the difference between peoples yeah the weight of the issue with those the you might take over the In the last few years I year it is a killed large right of isolated should summarize since we reviewed various his saying the wrong thing so it fervor for this for this schooling argument all the curves are we have were lurches doing this argument after we've proved all the upper estimates and we know our curves are very highly different Seibel just fix a large on Kansas case largely can take people's to there were 1 maybe we the people if you wish to hold 4 this was the year the the island regions and beginning I mean I don't know anything about but this the a little it's unusual to structure will use it to make calls you it would the world is used to reduce size of the well them you know the rules of origin the price also millionaire what we have readily 1 understands the modular space of whole market curves near precluded and so on what we do is of any kind such matters exponential of some exponential of some effective field and on the most complex structure on the domain although actually were Nike's all vector fields so we have to preserve this condition of intersecting D I X Q I still only conservative of space where sea of Q I'd estimate tens of space at the eye and we also may consider some space Delaware this was the same conditions at Shui double was so we have this point futile prime now and then In order to gage fixed the domain again we we have to take that the specified to redouble prime
on opery specify which you double prime on domains of all the Kurds in the modular space near near x 0 so won Wednesday's you double prime is that the inverse image 1st tried in the image of the all of this space under the exponential because regret that as a local devices the inverse image of that under the map will be treatable the story of a vector field under most complex structure and from vector field and almost comic structure we can ask followed bar and this section I J on the on the source policy were varying jailers and I think if he raised a banner day with the idea was this from set of almost complex structures on the on the soft yeah never lived almost comic strip targets always fixed so I want to be Rachel is this now as the so selected takes vector field and almost public structure I'll do know by J. supply on there so sold reading so X U G see that's the map on we take differential take the on the whole Moffat part suspected this J. while most complex structure and jail target this the section Of over CG not of this bundle but rather this it is in the the sections W. payments when Delta sections this part all of all maker 0 1 CG with the chair while most complex structure and with X U G C pullback of Extended Fund so to play here on Billy's parallel transport he promised part is Carol transport along part use some some fixed identification that this doesn't does really matters some complex anew no so apparel transfer for 4 8 j Woodmere connection so many of them there was a tense Republican sense of jailing your connection on no 1 except those are across states on variance to the new connections just for to be safe and I attend services just fixed identification that doesn't really matter what what we do here this summer any map which depend witches see linear and smooth and depends mostly on as the identity for clues alive on the undone with the definition because we also this other component with target so he added and so what is with this component telling us this tells us that vertical positions of all of the the images of all these Cuvee Proton's the Trenberth constraints we did this gage fixing words were now interested in in maps which map QB crimes to the corresponding zeros so we take the next day you'd see see holes in the prime of why so read it to the crime of why because at this point in In this season In C 0 mcg really depends on almost complex after 1 yeah I was going say Q the promise a section of the universal curve so they have this and then we come to this project are the settlement of our cross that dark and we we take we do this for all the ins and outs of all the QB crimes so now with this complicated definitions on for much less space yeah near x 0 the Naxera are much less space to the union over G end of and I'm writing like this but of course I really just mean small small Chrysler's due to the germ of identification on of and this local this gives a like a local snapped from 4 M bar too the space :colon premise but this is not all canonical mappings
depended on traces of of devices and sections QB press and so tight as far as I know all of this should should get works for ratios another comment about the case equals 1 versus larger case should apologize for a string up this discussion 1st time on yeah this this operator on what makes sense seems to make sense for any doubt any K 1 in any peace between 1 and and if you have case 1 and p 2 worlds smaller or if it to and he equal to what I guess we'll see about that and then sections here not continuous which is all but disconcerting on that yet race of course does not and of course yes the need you need this right because this we can do this expansion Texas was only makes sense for on indicate these 3 April 2 and the greater than 1 work April 1 and the greater than to press the differential does not make sense for any and but in takeaways from the and we action so for the glowing Arkansas doing right now I was just whole Norfolk with adjustable forfeiture curve equation but we could really do with any and any of these that we want to it's certain somehow the Fed home analysis is a bit easier With the people's too on and but now when we do this the same thing for the thick and much less spaces it's really important to be able to use large cash because there's an extra terms investors part of the equation the thickening terms and depends on the places where this exponentiation curve intersect certain devices In this mismatch I'm from the space succeed 2 the CG given by take the inverse image under this exponentiation map of affixed devices of this map is a regularity class something depending on cat something seek a minus 2 months of the 1 the United smooth map we can apply inverse function there and to understand that 0 said we can be large a lot of of work so you all you want you she did not the you have to go yes yes I mean if a thank you great so another few fundamental estimates which could spend up collected in the calculations for but not from Saudi just right on something which 1 can prove about this matter so 1st of all estimated the norm of 2 0 0 sold them off with a lot of 0 that is so this measures how far the printed matter is from being whole markets refusing using the law understood 0 as the justices hyper-real although the first one comment about the norms I mean you can define like norms with like actual explicit formulas including the trivialization of of these bundles and in the end but but all that really matters is the Comanche ability class the unified commands a growing class of these norms uniform in jail take big norms which is controlled the commenter Billy constant which doesn't depend on GEN the fine to the point is just to have estimates which are true for all the business and why the grantee fixed gene under this Miss North defined by this weight is controls the norm defined by not doing anyway but 1 that is used to goes to 0 per cent not uniformly commentary also the tune it's little 1 form part of the skies 2 parts the 1 former member of the 0 1 4 vertical Coronet part of someone from part of of active 0 is supported in this region he recalled a definition that was just saying Where Eugene not we're not J. Holt Norfolk on was a splicing together the whole Moffat diet trivial solution the only place that could it's might not be homework because in the screen part of the spice
on on an elevator now tower just use on the exponential convergence of unity to trivial cylinder and the fact that Delta is less than that I don't so or we have a waiting function which is divest that but this difference that we cut off following In the case by even faster than children's topless lost the party and then the 2nd part for this part is just the statement that the prime of CG 2 0 approaches the descent both are this DVD crime Caesar J 0 on just because as as chief of the 0 because the sanctions I got into this so there was some calculation to do but this is so the other thing we really have give a key part of understanding the 0 set of FG of 0 near sea ,comma Y equals 0 is the falling the contract estimates and what the should be really thought of as a uniform bound on this this modules of smoothness of attempting something on basically saying at this in like the efficicacy 3 map from prospects fast-paced and moreover were giving an effective bound on its C 3 Norm independent of G all I know we actually don't need to be C 3 we needed to be we need something considerably weaker it's it's and what we need is this the proposition for the House take derivatives the derivative and 0 is close to the Martin was a derivative of its 0 minus the derivative people and there astounded by constant said so some statement which is very similar to stay derivatives the uniformly for leased 4 C all is capable of at least 4 condition no 1 is really only needed when you have the thickening terms in a sexual devices for for for this guy's it's no should be true for any case also I won't professor good even so now we come to power which has and most of all of the content and services to construct a down the right inverse so so let on the GE the note this derivative of FG at 0 on the mission there in the direction of a 2nd argument the best was all it is sorry about this is terrible notation it's Zeta and overhears a and C are both elements of this hope the new world order the yes and miss this notation pyramid of measurements taken at different 0 is an element of we really 0 ,comma 0 KSE usually CSE ,comma y traders they become "quotation mark president of the state as soon as the driven that act 0 ,comma 0 this operation so it's a linear map from the source to the top of the market but the derivative of FGD at so stingy so-called organized by assumption in 0 is subjective on fully want 2 shows that DG is also rejected on although we that more clearly show that 1 although they write verse for D 0 constructor right the strategic and and it'll it'll have to keep properties 1 is that on 1 at the exact given the Chretien version for the 0 we will construction security here she this is the right verse GE such that huge and have uniformly bounded independent of GE and the image of G for this and "quotation mark varies continuously in safe precisely what for them then in in what sense varies continuously in gene that means it the spaces don't very nicely in terms of gene so it's not really is this knowledge well defined the obvious notion of various continuously but also need I need to see what I mean by that process later it said so on this construction is is really a very hands-on construction given given the writing verse the 0 we simply sort of down of the right inverse for the 1st 4 DG so that we have the key diagrams the following people look at this and say 0 see you I didn't in the kind of job so this gets parallel transported with W K 2 Celtic Sea 0 and flattened guy again with this map which would map which at its peak Texas traced to secrecy and finally phobia calibration that takes this and that satisfies this extra conditions at 2 the double Pratt supply RTS posts are as but I know we have all these In the nearest operators this beginning of transport there's a man in the other direction :colon break and this is the same as they walked right again I would be in the calibrate the racial calibration services that this is something extra which have had which has happened on 4 continents tumultuous which doesn't have a need to happen on when you gluing just nodes and Agrofert theory or even Hamilton the theory on yet the pointers you know we between 10 service this condition ornithologist sketched the definitions of these maps the final great so here have seen 0 and here we have seen the so did I. the gloom man so for glued so including map exposed to give-and-take section over C 0 and give a section of CG so it going to do credit off here cough and a cough here so these at for the sequences series of complicated operation so this is going to happen at something like or 1 3rd in the region where where we've already made attributes under and then there some some come with a school some of them on this map of income because another map break is as we pull back we cut off at the center I thought that was cancel the way we can cut off and for the glue that is not really well use whatever come from the way we cut off for the break that this is crucial peace come on so forceful has happened center and secondly meet these these color functions have to be symmetric that is when you if you break mn serve take the son of fibers back down should get the same thing is this cut functions 1 minus this kind of function all of us and finding finally finding calibration mapped so here's because how picture love scene genes amazes vector field which is just constant combat call off so we look at this ,comma function so this is kind off High Times the so here's you double prime and this is vector field so it can calibrate on simply adds cracked multiple see all of this so that this condition projection about director Phil becomes now 1 observer both of these maps all 3 of these maps of the accident all the maps and diagrams are we are uniformly bounded for for good off check for cover the calibration of 2 it's important here on the Delta is positive for this to be for this to be uniformly bounded In the portend a Q U double Prime Minister in the exact center of the Of the next phase of the plan with a maximum weight so this constant vector field here on which is In the colonel d up the the operator as cost-effective field in here the usual Norris has constant or 1 but but some are not the right not to think of it and we should think of it in waited and In the wake nor does the norm Of this constant vector field looks something like this it reaches a maximum at versus a non-Muslim accident this 1 In in in the center both Of the sky too he double and then decays exponentially as we go 1st to set boundary of and this is somehow white wine is an OK thing to do ,comma cut off here because whatever CSE is at that you double prime it's very small it's like to the minus one-half capital Delta times the norm of CSE so this thing that we added is actually very small and a and the what so he faces key diagram does not commuter but almost commutes and misses basically the property we need to use it to construct this down the dreaded workers so he digressed from he meets the high so statement is following the 1st repair will transport the statement very simple commentator the lapse approaches 0 in March so severely 0 B statements as just 0 a 2nd statement concerns the bottom this goes to 0 and finally the statement further for the middle squares will be complicated because and bragged on opposite directions on and that the statement that you can is both efficient and can actually be as the following this norm is little told 1 times more data only where break quite a agency quoted DE Reggie C ,comma than what a lot of fun treatment of doing the sort of the 2 an the 2 upper squares here sir combined into until 1 I'm history comments not considered this flattening I you 0 party on but I think it's a good serve intermediate Parkinson's divides 2 constituents he said switches were different it's also a very important that we coming off this happening in the in the region where you 0 jeez is a trivial cylinder which makes the estimate much the maker doesn't make all that much easier than this there is less to worry about we would like to hold to the book he was also the year for refer all yet so it's all the housework so so see also csce lies here and it was here and but not to the same thing here an understatement when you it's a them this way it certainly takes see this way all of this that not to the same thing here and statement that we'd take see around this way it's recalls service strange sounding statement by I'll know any of the formulation which is both correct and simpler the state me that you could ask that like I don't like going this way and going this way I really close for the century for each of you here it's not true or it's not enough to show what we once 1 of the 2 I also on behalf of all so mean many of the same words like this is true that mean when 1 can some explicit just explicit calculation on for this point is again just you you 0 did the decays exponentially on too trivial cylinder and I the only really differ over the part where you cut off you 0 2 travails on the on and over that part on now we have assessed that that use Iraq which real cylinder for at least the exponentially with any Delta strictly than Delta Gamma and are chosen Delta's less than gamma so so we have a system on the calibration won the point is just that this vector field the SCG is it is in the kernel of full nearest operators just corresponds to translate into real sold a rotating materials on the on and off they would cut off all but the point is that cut off happened really far away from Q G and so the nor that use product will expand this year the list of services for small again like the demand something comes Delta so it's crucial that Delta's positive from following well I finally this such this yes men may be made of sponsors such amenities in explicitly analyze what's going on .period is somehow that unfold the weight but the key point is somehow to compare weights on the 0 you have weights which look like this and CGU social like this and what he turned to Chapter of the error terminus difference turns out to be on something supported here this type of region but the normally care about is this norm a balance on its Norman in the snow and so that the fact that the ratio between these 2 norms responsible and 0 and justice the case analysis justifying use abounded Friday versus itself what I mean varying continuously and G this is server crucial for the event will not be continuous in this this image really has the very continuously Angie persists and and this is somehow a nontrivial state because the space glowing promises to look something like this and In continuity surfeit for this glowing primary and the school imprimatur complementary so Sony continues GE itself will just 6 some linear map from the source of the UK the Delta To some vector space given by like a Dennis on evaluating at some points sofas so for any any substrates on such that the L 0 restricted to a colonel prompted the 0 isn't my small the fix some points and look at this evaluation map Christopher C. pick some pointers evaluation that is is injected on the colonel curls fundamental so you can certainly do this exam .period on the domain C 0 substances Ganges fight dimensional can do this common and closing down by suitable subsidies the still it's actually on itself this of course gives us a corresponding math LG on on the domain of etched in operating began just no Our going to fix on right 6 Q 0 with which is the writing verse so the 0 0 steadily and the image of Q 0 is the colonel of El 0 Sincell 0 restricted to the kernel of the 0 1 is an isomorphism on the kernel all zeros a compliment for the of the 0 and given futuristic complement determined that the just probably term uniquely in just about every close complement is this continues map of the closed on racial this the slammer implies moment diagram implies all this guy GE which by definition is power creation composed of gluing composed power transfer composed of 2 0 "quotation mark transport and breaking it it is the approximate break inverse 4 you meaning that this difference DG in G Minor said there is less than 1 in fact the goes to 0 as he goes to 0 I thought I mean what happens if we fix our are right in verse In the year 2 0 and we define our approximate reimbursed by just going around and that's that's the thing Of course it is a diagram commuted on those on this would be a true writing versus a profit connectivity gives us that some approximate the inverse Pompano on Wikinews the standard geometric series 2 2 in the office just get true writing verse future with the same covered by construction the image of T G is contained in the colonel Kovalchuk From the space just falls by definition we so T followers boy passed it is some loss of maps are we apply to 0 and we went in the the cradle of L 0 every would've been definition of power transport calibration the somehow only happening on the on the next these evaluation points is for sufficiently small Jia not in the next themselves this not sort composition commutes with the Elf this is our conclusion right now an index calculation shows that this is a nice office so when contracted the index of DG and it's what will be the same as the 0 on so and so 1 concludes that this is an equality population I'm I guess really fervor combined with this races so we end of really we also get up the colonel precluding maps colonel :colon isomorphism from Colonel the 0 2 the :colon the picture of Stephen by simply applying these maps and taking the projection onto the colonel induced by this particular justified this is just projection off of chronology election finally we can the the cleanup which gives us so we want to do so so the founded right injury Terrence local diffeomorphisms between the Kremlin of the DG and the image of 0 so this is basically the implicit function there all had implicitly also of course using crucially in here goes the estimate of enormous Beth FG of of 0 come and the mountains of this writing verse and the quadratic estimates so the picture is like this colonel the chief on the image of queued uses that complemented hands there's some manifold here which is f g inverse of 0 on the map here is just projection it was in the kitchen so this is the man and this gives us a local diffeomorphisms rejection right so the moreover this is valid in a ball of some fixed size independent energy and that's the through the boundedness it is important so so combining these so for combining we get combining for 4 different values of GE forget this map from the past cruelty 0 2 and far X using a chosen ice more efficient use of of the crime up to now claims the is 1 continuous 2 objective meaning its restriction to some neighborhood of 0 is injected and surjective meaning justices rejected means that on the face of a germ means that on for every neighborhood of 0 In the source said there exists a neighborhood of x 0 on the target but which is covered by this map restricted the neighborhood at the source always the least through properties taken together applied at the local army morphism since that sources local plastic on so was the content here on surjective leaves the content here is using these he operated estimates of offerors Huskies under about next which are C 0 close to trivial so others injectivity on essentially follows from the definition of this stuff was talking about gage fixing at the beginning of the continuity of the contents Is that will somehow this in the image of kinky on the various continuously on and what I mean by that is just the image of is given by the the crime of this matter algae which hello how how redefined varies continuously ship which we should be satisfied by this evaluation appoints map on the content here basically is that we have as much from Colonel teaching to pray this is the right we have scrawled glowing from curled a 0 rolled each canvas 10 and ended the images the and image should be prescribed so this this is approved thanks for seems