2/5 Contact homology and virtual fundamental cycles

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2/5 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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been a good and the move move to a new book to her and we higher a number man and we don't want this we will you you can call in and I think 0 0 begin by are reviewing sir the problems of our time I'm reviewing the set the explained last so our graphics contact known for wine one-dimensional on prefix contact for members so that just means kernel of land succeed and In fact of exceeds going to be call oriented which means we actually care about the size of London on hand I kind technology is going to be a form of of worst block analogy for the action functional and from the space of troops in wide and of such associates to give move and the pinnacle of of the contact now the critical points this action functional a precisely on the periodic orbits of of the radioactive fuel followed so recall that this if we have ever the contact for Monday then we get the specter field Haaland parade vector field as defined by the properties that landed on this 1 and the land contrasted with Arlinda 0 it now since we wanna do Moss floor homology for this action functional In particularly want some sort of reasonable Moss complex and we need to assume that all the critical points on Monday January so this set recalled the set-piece criticism all proponents of the hardware digital no and there's a there's a natural petition of P into 2 sets called good Ray Warburton that work so what grave orbits gamma is called good if and only if the actions of the wider the covering multiplicity on the orientation life of gamma is trivial so I'll give some definition of the orientation line later on but for the for the present purposes this this is what the definition is the the gamma This is the covering multiplicity of gamma the rule against untrammelled tries to rape orbits on but they can be mostly covered Sociedad go around 10 times the cover multiplicity at were but he could associates on the so-called orientation line of gamma it's it's a free the model of 1 this is just copy the images but without the economical choice of a generator and that the choice of generators I want to be thought of as the trust orientation for that report that on this set that elegant in usual mysterious that 1 to 2 critical point usually associate on the orientation line of the negative against space of the head they're here that the Hessian at a given Raber retention of this action functional has infinite dimensional negative impact of the negative can spaces on but nevertheless the there's a well-defined way regularizing the determined to get a well-defined orientation and and this is the construction of the orientation lines functorial in the raid were in particular if you have something which is default mostly mostly covered in the rotation by 2 pi over the induces action on orientation on the river was called good reactions to so clearly the only ray of Orbitz which have the possibility of being bad are ones which are even more will come so that he is good union people so now the worst complex long into the at least 1 on as a vector space is the 3 Super commuters she also from generated by these reports the good replica 2 supercomputer that has the usual meaning a B and C B except they and the commute when both are offered every every Ray Warburton envisioned having orientation rhinoceros apparently even even some notes the differential on this complex so the final differentially need to fix almost complex structure J on our crosswise on foreign properties change should be on variants J. so these vessels according on are j of DS which we are Orlando and J. should send the contact playing to itself and be compatible with the land so the contact condition the fact that succeeds a contact for means that the land that is a as a syntactic months C and so you can the notion of compatible most complex structure to really all you have to choose as the restriction of jade exceeds and then the 2 properties define uniquely on and some petition right so now I'm given almost complex structure like this all we can define a differential of the differential of sort of Moss for differential
on this complex counting gradients directories of a recluse that's thus we think of counting is defined as follows :colon depends on Jay so it it's on this is an algebra and differential is going to on Saturday at the derivation satisfies the light to and in particular the baby the data is being less the began with such so specify the differential I'd have to tell you what is on the engine the free algebra Sosebee some all subsets of good rapport following suit the space of all Norfolk curves it's like this the next 1 in our cross gamma were positive and this asymptotic to orbit help plus a native ancestors had to its calendar and this should be misses the coefficient of governments again Gomez a set of of rib orbits islands just up the product and industrial I know for certain that certain reasons I also have to divide by the cover multiplicity of getting plus because of the modulus basis and consider going to get to the heart of a specified markers if you know what that means and this is this is what's 1 over the job it's so the reason we need to work over Q instead of overseas is for 1 reason these these coefficients on this much less spaces can also in general some are sufficiently Stacchi customers in orbit .period should have the account together usually only rational numbers so now the goal is is to define is defined on these numbers much like counts such that we get a well-defined homology theory all of this look as White said so far is teacher Elliott forgiven Hoeffer isn't the definition of Conte ecology and that's what I want is to describe how the make this definition complete by the findings these numbers show me just status of there so for all while the Jr the post on there exists a set of data think functor of this triple and for all elements of this set of numbers such that these critical 0 and and moreover this virtual model account is the count expect if and forests transfers but so far amenities right now you may notice since their might and say anything about the results of the halt you get from being independent of the choice of landed J or the worse in particular little so on the analogous fearing for on the culprit is a matter for them an exact particleboard isn't between 2 contact unfolds and there's an analogous theorem for counting home of a curves in 1 tremor defamation Of of the exact impractical petitions from 1 simplistic 1 contact novel to another on and taken together all of these I'm counting results you can use them together to show that on this the homology you get come technology of why is just an invariant of the contact structure doesn't depend on land the JFA on but but that's not part of this statement is something which follows from from the statements for come the curtains and isn't and the musical competitions the other side I should now tell you through that more specifically what it what compact allocations and I'm using this as much less the only curves and John are from 1 level 1 story curves like this In our costs what what the General Union to together that's only that only the top stratum of this compact and much less space and in order to define his virtual fundamental cycles you really need to understand the structure of the full compact vacation so the Commission's statement about so purchase recalled the usual more theory of the function on a manifold we have some much less space both trajectories from PDQ for P and Q critical points this is just a set of greater politeness on the mediation not scanned so let me jobless when this great introductory front he custody minus and so the space is not in general Compaq compact occasion consists of of so much less space of all possibly broken creating trajectories humor plus and minus in the middle may passed through a bunch of other critical points the analog in contact acknowledges the falling over the top stratum commercial space it's going to be the
set of you from punctured Riemann surface of June 0 2 are crosswise on users entitled get plus at the puncture syntactic and minus the punches genus of C 0 and news J. hallmark and the quotient by and moreover I think it's going to go on a good idea to specify entomology class of the current soldiers just like this like this and last but the impact of fire only about the consider multi-storey curves on but now that the notion of what a multistory curve is a little more complicated than that and then usual staircase on because so the the Senate connected components of a look of the of the of the building as but no longer linearly on because multiple negative so the strata of the compact locations are indexed by treaties so as you set of land and the direct entreaties the character actor some such that every vertex has unique incoming edge along along with labels corresponding to the the topics as to the so the badges should be labeled with Rick orbits and the vertices be labeled With a moduli technology classes so far H to wipe it's it's really some homology relative to the other it's a typical tree look something like this for all the edges of the file never again draw the direction on the edges because all just draw the trees such all the edges are oriented now all the edges are labeled with the words and the something internal report this year mn homology classes for breach of the vertices and this corresponds to a 2 whole market curve Crawford building which looks like this so each of these curves is separately equipped with a matter too are Arkansas after the translation each component have to Arkansas but the translation but right analysts said ass as little more structure namely there's no operation entries namely contraction edges so by the tree like this 2nd contract and the interior energy is the only option is Is the censure ,comma given the tree the booklets so you the right yeah yeah so so yes I lied allow these like dangling edges so that they have they don't have any incoming Vittek haven't gone if so we can regard us as a category where officers are contractions of ages so now never any tree an ETA watchlist space of 4 pull off occurs modeled on the predecessor of "quotation mark aggression you represent definition of what the actually is but it took us to see what this is let me actually add 1 more piece of data it's it's really better to say that we choose our base .period 4 were bits on the input and output at its peak so which is based .period forget blast gamble 1 minus going to skin after we don't user-based Clifford police work and the notion of morphism between
trees includes some data had a match at the base now with what building also wanting so on the unity in the treaty you can form a curve on we just his dual graphics so far the curve C said the the fact that so far J. home let me just say so UBS entirely 2 To the chosen rape orbits panicky for edges on In the correct policy classes on page we also want shoes on so at input and output edges please specify asymptotic markers Major mapped by way you to the base .period and at the Interior edges there's every edge which is not an input or output edged on we specify a matching between on between attendance fears that these 2 points in In C so but what that means is just if this were barber's Malta recovered and you have to specify matching between the positive and negative and on so that you know how the on glue the curve to to resolve a you'd had yeah yeah just yet the president at the time of the input of pledges these Rabe orbits are all he could to base points and again if if any of them are mostly covered on their multiple asymptotic markers you might choose which which amounts to the next but the mission license and the technical conditions and and listing all of them specific things although the precise definition for the experts but what I mean we should think of this as just whole market buildings whose dual graphis is that is the treaty so now they have only knows then usually the user's drove away from the whole world where people can get a yes yes yes it I guess I wrote this way for convenience but it's it's really best think of like each each component mapping to a separate copy of Arkansas send them on the map was only well-defined the translation and 1 so now all we can to find the compact survive modular space that embodies the common Union and of the primer for all trees mapping to volume was his view that the yes yes yeah these fees the labels of their 1 to get out of the contract write to these these contractions of of edges if I can contract gamma response to gluing the whole on at this rate and a single a single curve mapping Arkansas television natural runoff apology on this compact find space the it's soft on on every restricted every piece here it's the default you think just uniform convergence come on compact subset but citizens since the day home off a curb equation is is on these curves are all all smooth and they satisfied even operate estimates on smoothness and as a result there's really only 1 reasonable topology Michael on this space you can say it altered Caesar 0 convergence topology of Caesar seemed fairly convergence that they give you the same culture the event topology on the compact space while you do it I said to the neighborhood occurred like this is given by we can take 0 neighborhood of each of the components but you're also allowed to the glue the curves at every interior and on the pyramid of virtual desperate for over was asking vendors and that the space and foreign is and sailor but more about the structure of this space sometimes how the Stratford together the hearing illustration here so what's considered shares a tree it has 2 possible office and Missouri contracted a single edge in the house the company and contracted everything so the way this this is going to work with would the goal of this you have the other player yet none there the incident that the corresponding structure of the compact much less space look like this you think so this corresponds to hold her like this however plane here this Have you have a curve like this the sort of the 2 glowing parameters you can glue here English good here and these are independent operations both take
the glint Prime on the space cooling premieres at any each Ray Bobbitt is the non-degree real numbers 0 being no blowing so then they think if we go here and along this strip or if they get something that looks like that we could everything is including Justice so this is what the compact Fred much less space will look like if it's transfers landed near a curve like this since this from soap at every tree has an index and the dimensions the next 1 this time so there's the what they're the 2 definitions 1 is that the index of the linear iced operator some operator on and by the next year meeting calculator as sort a difference of certain Masovic comes comes under indices of the report itself on the boundary through some topological which which depends on a tree I'm in fact the behind in the case say see 1 equals 0 onward dealing with contractual where were bets and you can just assigned each Ray Warburton number and and then and then the next the trees just as the difference between the top and the sum of about what I I need to actually slightly correct this definition the reason is the following if you have the same looking at like the Maasai space of cylinders and compact fibers 1 thing that could happen that 1 possible limit you could have is a curve with 2 negative there is at the core with 3 negative and other top and then to hold off claims to that reduces the score sponsor this tree in this corresponds to the street now the stratum Of the compact and much less space of curves correspond chemistry has a stratum where we have curves that look like this and then there's no there's no label Of these 2 components but if you look at curves for corresponding to this tree these these 2 planes are very distinguished 1 of them corresponds to 1 Vittek funk response the other so soldiers just want out of office in group of fatigue and that's that's what actually it appears another thing to say about this compact vacationers that a little bit and different from something which is usually considered so this is this is what another compact kitchen for larger and the expectation look like so sometimes he considered this time this problem this or then there's a server Standard as if the compact vacation work you consider this in this curve to be lying in the same copy of our crosswise and as a result there's a notion of what the relative vertical position of of these 2 and so this stratum now becomes convention 1 the system a block of this compact vacation so that this compact fication a but is not so nice from the perspective of proving the master equations describe equals 0 because on this is John here which is could mention 1 and a prairie has contribution and you have to show that you have to argue that it's fun that doesn't contribute where In this compact location words could mention to it's it's clear by definition in the economy contributions and that this is mostly just a technical point that I strongly believe that these 2 compact vacations contain essentially the same information I know this I could these marvelous spaces of a little more structure so S on has but call concatenations which is some sort of vaguely symmetric when little structure :colon lies at the heart of the authority vested from lately model structure so how I'm saying here is that we have a collection of trees and you on given some data have had identified some Paris of input at edges without the edges on you form a trick no confirmed that so this corresponds to so of course abstractly you know that like this if you have a collection of trees TI and the matching data you can get the concatenation by this and on the level of Michael spaces there's a match this map from the product of white spaces the guy who was less space for the concatenation those maps it is
not quite the isomorphism because it each edge like this on where he lived 1 tree to another In this much less space there's a choice of asymptotic market at this reported and on MySpace there's there's no choice of Aston tight markets that a matching of the bands but not the trespasser .period market and so if the Red is mostly covered this map is not 1 to 1 but it is still on under this isomorphism once went you quotient by this ambiguity so for quotient that action by the squad the gamma were gamma ranges over the say junction edges the edges of the were good 2 S has these as as the set of sort of trees and has these 2 structures 1 is it's a category we have a morphisms between trees and the other is this new model structure is a way of concatenating Trieste Italy once and the 2 structures on code just and basically all of the structure but that's here how the much spaces fit together come and the this this this the structural come up out there again later serve fundamental to the proving the master equations of soap opera me now say little about the basic analysis on which the it was the school certainly we all know what on it's a 1 1 basically just corresponds like stacking them on top of each other and the other corresponds to the good 1 is like that I think a treat the 9 1 1 operation just used have have a single treating Dixon subset edges to contract now the operation is still a bunch of trees and you Pixar pairs of edges and you stack them on top of each other without introducing a contraction there they're somewhat similar if you if you on decided that you only care about trees with a single Vertex this would not be an unreasonable thing to decide inside you only care about trees of a single Vertex then these 2 operations sort of becomes just become 1 British operation if you have a way of stacking single vertex trees the contract all the edges he could have a single Street I need you could do you could work on you could work like this to come and that I think it's I it's convenient to separate the 2 operations before their marriage .period 1 so what I wanted to annex adjusters and basic 1st extended analysis for how on hobbies do it which allows 1 to get a handle on what these monthly space actually look so there could be good references and software from proposed for most aspects of Holdorf recursion supported jumped Tuesday reference but predicted for this you can look at McDuff someone for 4 more details they they'll treat the case of technology you the sense that so but 6 "quotation mark that provide surface sea and sun porches and now I wanna the way describing this much less space of Kirk buildings and because type said today that will get some spaces maps so this is the space of of smooth perhaps you from C to crosswise so you should be a class people believed take the locally that his parents it has K derivatives which lie in LP on and moreover it should approach Our trivial cylinders over the rave orbits at and the punctures ,comma exponentially and guided by the of the differences so that should mean is that you invest the wires the balance of speed ,comma of tea this difference the Times eater Walter S. should also be in the UK piece of infinity Kansas 1 sofa so this is some cylindrical coordinates near these punches plus-minus we can also depends on possible so this is a very smooth polished manifold manifold full of all the modular space of solutions of the dog equation lies inside the space at the center of exported maps now "quotation mark and more
precisely can be described as follows the cave Pete Delta In over overdoses this bundle this fiber over given that you is the the UK next 1 Delta the on 0 1 forms valued and pullback of a bundle of Arkansas not every month so this is a list smoothed by public national unions and dimensional space of all maps and infant the bundle over its cage map so that there's a section here namely teach math we can associate from double argued so the fewest if you as a matter bellboy use section of of this bundle overseas was absent due to have argued this is called a section at best so then on the modular space of industry work and busses on top gamma minus some bottom is exactly the percentage 0 0 and the question by this is the diversity of the world they are in the the death of his views on right yeah so long as subsidy is just remember and L is the length all of this report again of every city of Arabe orbit of length down on this Is it too has called the trivial cylinder over gamma so just looks like this Arkansas and it's it's a whole lot in ridicule and the in general to whole market maps which has finally over energy will on the asymptotic 2 To this day attributes older in the customized so a basic property section it's derivative is a friend home thought of me just and review some of them the of the analysis on the theory of which is relevant for much of on curves so a linear map paid from the 2 are free enough about spaces is called for at home the only Colonel is finite installed that the images they closed and the cultural issues vitamins and In this case other 2 things you can define 2 variants of a 1 is the index sounds which is just the mention of her normalized mentioned the Cockrell In find them and in the case that you never find them and so this is simply the dimension of the mice dimension show 1 can think of the index is simply a particular way of using aid to regularize the undefined the difference between the demands that the mice the venture you can also define on a given a In defined orientation line I don't think so because so the orientation wanted a so the determinant of a is the top wedge power of the colonel and so it was tough power of duel an absolute values means take the orientation wanted this is the real vector space of surviving take the orientation of orientation and vector space is homology of the spear this time so this is it's isomorphic Eze and using generators the same things choosing orientation and moreover is this is from the much to agreed and it's it's parity being that the dimension of the so the example we really care about here the following sales if the farm the and that are all bungles
over complex manifold and and Al is you take operator of order and then on this black From sections of the UK Press and the section of on this front inserts it's important here that long if he is not 1 infinity this for at-home statement to be true ,comma in fact could really just for quick equals to 1 and things are slightly easier but and it's taken care of 2 to be 6 Thomas and yes the derivative of of section the squabbling eyes operator and it is a town it is an elliptic so this is where this spread home statement ,comma comes from over a dozen than fall exactly from America here because see is not compact as these cylindrical ends so you need a flakes extension of this result and Peter Walker McGowan come and this the spread home statement on for for the operator with ends suffered events on users crucially the fact that the boundary were periodic orbits on Monday Janet says crucial written off in the final of the index of tea to be the index of this operator the ass the as incident of both of these are invariant under deformations of operator Eddie the final operator the family fretful home operators the nexus constant and and orientation line forms a a local system over this "quotation mark family of operators federal so this if you get a section of a bundle it's differential is only well-defined this on the zeros of otherwise it depends on his connections so upper library is front home the statement only makes sense Over .period large white space and it turns out that there's a it's a nice family connections you can put on the spot while the namely connections which are pulled back from a connection unattended bundle of our crosswise and those connections and Assistant home for all for all of us I'm surfer overall maps you were regardless of of a mother the whole market or not and begin again because they that the operator and this allows you to define on the index for any tree regardless of what bases-empty you're disconnected so this is well-defined even know just looking at it served up prurient might depend on the Marshall Space being some property watchlist but answers to the index and there's also on the orientation line which already like this foreign Batista orientation of the of the section so in In In many cases for example ,comma if the 1st 3 classes 0 and only looking at consider only considering contract award on the index is simply on camera can be expressed as the difference between the index of the began classes and the index of again minuses deliveries various you can you can define index indices of red corpuscles property and you can define orientation lines of the report with its property well actually that Oh this this you can do in federal cases this you can always do In you always do orientation lines for for individual pockets we is on so you are you look at a whole Moffat plane positively asymptotic to rate to the Red Barber and take its orientation line and this is by definition orientation 1 of the real I might worry there lots of choices of planes or maybe gamma isn't contractual so any trace of plant on with used used his abs abstractly so you have the trivial cylinder and that gives you like it you can the least have like deposit and asymptotic 2 together because of all the fine notion of that and you simply can't extend that vector bundle to see and put some some pressure on operator there and take determine lines Meczywor inflation like this look at me I'm so that the only choice there on up to deformations a tracer relative during class mean you can you can argue that independence Dr. Kunkel favor from this definition there's some this from tort action of the mud-covered multiplicity that was talking about earlier just corresponding to the rotating retaining the orbit by
2 poverty right now but the real I they were want interested notice In is what's the structure of the smart-aleck spaces so you have to do that we need but in this event and the in the setting of international selling free of a map tests From Peter F. to smooth lap while also spaces I thought you really smooth is just for some recent case such as Baby 1 come on let's suppose that the differential it's 0 you need to have this rejected then there is a local workers and between the cruelty and reserves to near 0 the so that was the subject of the Beslan operator is is a very important property it we would be very happy if it's always world Help and I'll probably know by embarked here Reg the Lotus Square is screenwriters operators suggested the corollary now of of this implicit function there is that on at least I don't compact if I just look at 1 particular stratum This is a smooth manifold of the mention the index of T minus the number of vertices of each vertex of the gives you our actions the of the Kosovo put another remark which this obvious for the experts but on this there's there's pointing out is that from this main results without defining high-technology on hold it on this was spaces all cut transversely show that the real but the real content is how to more perturbed the much less spaces abstractly so that they become on transfer In in the many cases and somebody geometry if you choose a generic almost complex structure for input data is is generic them with your Montoya spaces are all transfers therefore an implicit function their overall manifolds of expected dimension and this allows you on to define their multifarious simply dimensions are much less bases time music-oriented appropriately do you know this is not a lot workers who good work you should use all of the people who live here this is going on the year ourselves so that you have difference in the ratifications only seen that it's if everything is transfers from the differences only seeing for index to and the problem among you on this 1 because there's so much this year this is just you know all of the last of the these in on let's see so so the high-dimensional ones if you have like smooth manifolds 1 thing you can do is some the fundamental class homology push forward to appoint is trivial but the fundamental fast and smooth borders of prefer to appoint is not trivial or maybe maybe you can friends may be friends for some time and if you get these orders and classes you can construct Buddhism classes you can do that the Clinton Siegel construction together not just technology groups
back actually as a stable home atop a tight which is unfair and or maybe maybe less stable homotopy tank but something in between so now there be 1 interesting thing to do if you have a greater say over the you yeah probed for ideas great I mean created I think so that neither the grading just think corresponds to like them like the grading of home atop groups of a stable home topic was the highest in more than a quarter of the nation's most places where you'll various radio stations on that Cecil not only that I think that the kid other case to keep in mind is like find emotional Mauceri so if you if you so suppose we have more function fundamental manifold and the principles that it but the more structure we endow the much less spaces with under more the more the Wichern recover so free if we simply an album with no structure but we can get better the homology the homology them with the 2 coefficients if we endow them with orientations reading at the homology them with the coefficients if we endow them with framing then we can recover the stable home atop a type that the sum summer liable for fairly elaborate construction amusing the framed words some classes of like the relative friend borders and classes of all of these much less space at the fit together to put to build some From the stable homotopy defense and in the semis fired the missile case the reason that I think they don't meet EU in Uganda ,comma No you don't get 1st to speak up to up the suspension on In the case of conflict Margie I don't know whether 1 expects to be able to get a frightening that that's probably a bit too much on but 1 could still stable home top type is probably not possible or not and I expect to be possible maybe something more than half a mile the groups 4 models over some interest expected In vessel finally let me discuss the extra structure we need to endow the mudslides basis with in order to define a virtual fundamental cycles modular space 0 all transfers of the topological structure is enough to say what the virtual fundamental cycle should be in the general case you're much less space is simply some horrible contact Costa topological space it's and as a couple to space you don't doesn't say anything about them what the virtual fundamental cycle should be about these muscle accounts should be so How will you have to do this figure out what the additional structuring to remember in order to be able to define so other lizards that basic idea is the following so we have some horrible model based on the X and over some given open subset of the much less space yield we embed this menu into some the conversion of as you as a 0 sets of a function to the vitamins vector thanks offer some manifold Sophie do nobody all right the Virgin dimension explaining this this index the expected dimensions of that the index of the 1st bombing of the nearest operator ex-officer mouthful of dimension D plus last mentioned he so diagram like this is called a fine international reduction and and it tells you what the much less space 1st what the virtual fundamental cycle should be locally in the sense that if you perturb the section itself up to be transfers to 0 you get some perturbation of this small open subset of acts on the Net for any it's as if your perturbation office chances to 0 this provision of access a manifold of dimension the and that's it represents the virtual fundamentals like a locally before you and at home to recover them from Global Virtual fundamental cycle of X what we what we have to do is serve glue these local cycles together
given some appropriately compatible and Alice of charts of this fall this the idea of fun defining the virtual fundamental cycle in this way but has has has also that and is very old the around applied any for too many problems in this area on because of militaristic various cremation structures 2 to encode 1 sort of notion of Annapolis of of this type on this common currently there are other ways by the authors the essentially and this is the same thing in 1 in related contexts of prior all of us on the continuing with current issue structures on the also and recent work of winter Hyman on this with the notion of and and also Spivack person well and Joyce and arrived manifolds strike manifold as
Irvin persuasive is a maximal
Atlas of smooth charts easy is to a manifold as an atlas of this type is to a direct manifold the drive manifold as some sort of intrinsic notion of a space equal to stretch like this always saw what was going to do described is on Wednesday for the implicit Alice switches into another way of formalizing a collection of charts like this which are sufficiently compatible so that they allow us to construct a virtual fundamental cycle so before I give you start I think today I won't give you the definition of an English at rather I will on construct 1 on the much less spaces of whole Moffat curves initial somehow motivated by what the accident so maybe I should go for a constructive 1 on non-Muslims physical off occurs when construct on 1 of us much simpler space so let's just take a vector bundle soviet B is a defined image the manifold and others affected by law and take a section not necessarily transfers to 0 scientists in 0 sets clearly it's it's virtual fundamental cycle over what I mean by that is just what you'd get if you perturb asked to be transfers and take that they could use cycle the on he found that this was supposed British given the set as not necessary transfers to 0 on what is what is the Atlas was associated with so so and what's a new set of all Perez the elsewhere the president triples in the office when often where the the offer some opens up to the he also is selected space and land it's a map from the office to the restriction of access to the offer the linear linear and then I'm going to make sulfur you said Paris the next and the offer of sorts of for some half I'm going to make the final for example Putin X offer to be the locus for a little bees and Venus open Sept corresponding to the offer 1 and the sulfur the fastest for us as of we "quotation mark health of the offer 0 and so happens here if I just right if I if I remove this condition set he he offered 3 0 and is giving the definition of acts said in the past from XXL awful Louie Dewey added House over the 1st 3 restricted B 2 lioness opens at localized we added extra degree of freedom of certain elements of this vector space and then we we modify the equation so there's an obvious maps From effects of the office forgetting everything except for the fall in the future 0 sat on this matter on this imbalance inside inside accident in fact it's exactly acts in the sector this open subsidy of just by definition so exactly a chart of this sport and I can also observed that if I pick the offer big enough and when office of nontrivial enough maps 6 rejected re-enter fiber F then it's very easy to make MySpace cut out transversely the linear operators just the usual in Europe but at the secular nearest operator at a point of vaccination the offer is the usual arise operator the 2nd at plus this land of make land office rejected is differently rejected so far this year by year it's a sub yet we used to do so from those stories this year was you know what you got off to a good start "quotation mark and it's SSI
so said so if if I said and I think is yourself as as alphabetical order setting the officals era here Vice ignore this this term goes away so becomes the center be with S equals 0 and again in the office which is exactly this is section you is yes it is just as very section of the cross 0 but so we have a bunch of on charts like this indexed by the set it In fact we have charts index by the collection of the finest subsets of so were generally 4 planet subsets I contained in a supply of the sense of being in the newly opened the offer for fitness it in the set time such that is in the office for all I and this equation satisfied so you have the serve analog of the future and for for these sets as well did I think I'd be the empty set and discovering exactly X the original modulus basic care about Mom and these are these dictate what will hope about U.S. bases are the breakfast they're more likely to be there on Philby transfers in traffic large portions of excerpts of the video trenches flows I will cover acts on and XI has a map to the offer for every often I voted 2 0 said of 1 of that those maps I'm going say X sub I might as a force for opens up top of the charts together also may maybe but let me actually give you a sketch of the definition of Olympus apples on before I the final on much of bases of the content of the soul Christmas price apples on space Texas dollars will be server a simplified version in particular all all it's in the service of isotropic and the show consists ActiveX your compact houses it consists of an index set a cap something so 1st I want these spaces he often are often a reserve by dimensional vector spaces in it 2nd I want spaces Texas I for profanity I want to say just how topological spaces and more work X 1 of the empty set to equal access Anderson that was Granik consists of a bunch of data and the only way that this data actually interacts with the original modular space X is is through this as thirdly maps as sulfur from Mexico to the or Portland I should have naps C R J the Dreiss morphisms from prices for I contained in jail Chrysler could provide just those as flows for Alpha in J and not acting on X J With 0 said that should be identified with an open subset compound
it's musical the thick and much less and I'm also going to specify
open subsets suspects by Reagan so they're a bunch of axioms about hobbies on pieces of data fit together on but they're mostly topological and things like C J. composed of succeeding J should be equal to see if I can think things like this so I defined the exercise in this particular case itself are the tautological projections and it's it's clear 5 take XJ and require some of the office to the 0 I'm just giving open subset of the structures the apparent here Aksaray denotes the locus for excess cut transversely from 1 of the 1 of important axioms is that if I look at it the portion of acts which is covered by the regular locusts in some thickening that should be all the Top operators just opened subset of next and we want to be all that's so just saying that there are enough charts here do as long as the still maybe it's more the current like this so this is a and this operate here is is a subset of that time units it's the causes of the the empty set he said the mission would be about this on the New York City's police were based on most of the war so it could be 1 he yes visited some space in abstract definition is it just some abstract opens at on X in this even tho that was the excise adjust spaces they not don't have enough structure free to tell where were the transfers from where they're not that's that's like quite that's basically why have to include as a piece of extra data human in the definition of Annapolis so in practice we define Atlas you define XI rank to be the locus worse transfers but but it's the EU cannot recover XI rag from the rest of the day the here that's part of what you have to specify I'm in particular its it's OK you can you can always shrink XI right if you want to maker mean there their various transfer saw conditions you might want to apply for instance if you have like the curve with lots of big events the 2 notions of trans resulting might it fun may have a skirt 1 is the linear eyes operator as as a matter from this fixture curve is rejected and 1 is but the linear as operator I'm all Of this is a matter book also allowing the deformations of the complex structure you could say that's rejected without a weaker conditions and you can define acts every giving using either 1 of them just gives you a slightly different commanders at all the rest of the Alps is the same as just you had a more more strict definition of of what's being transferred to me I don't to find 1 of these on not SpaceShipOne technology the results of the country and was the final day of the missiles will be that the index set over the Atlas Soboleva set of quadruples these are all part of that is that it is nominated integer the offer will be home and are invariant call dimension to somehow pulled boundary close to the mansion submanifolds boundary henceforth undiscovered referred to the office of the visor but this is what I mean the
Office of active space and and the Samak linear map from the office 2 Smith functions on the volleyed in the a vertical 0 1 forms so here this is the universal curve and 0 and is the to inform provides space of noble genus stable Noel tumors 0 agreement surfaces with and market .period and sees Iran as the universal curve over that it secretly this is just and 0 and this 1 of the map forgetting the last the sections also should be far in various and you have to lose closed I mean just a close subset 88 could be punted to coast subset the muffle the foundry on invariant some he had 19 and indeed I wanna like use the offer the puncture hold off recursion make effect by by taking the intersections with the offer and but the whole off occurs only to find that the translation so there's no hope of getting anything well-defined unless we make the authority I think it is a compact subset of Y and crossed with art you used by his velocity will work by the plus-minus the Post Meyer said he's like the input and output edges Of the very concept of the interior edges yes you know so I've written here is is a finite set rather than a positive integer but I mean I just mean it's convenient and to look at my less spaces of curves were the points are labeled by some given fine except instead of the images from wanted answers this stuff France economists is the vertical 0 1 forms the fund perfect family this should be supported away from the nodes of the fibers of C however where did you think services that doing for modular space of stable Noel Geiser romance options with more points and this is the universal family seek really comes around .period 1 we can define this thinking much less space although just go and directly defined for any so this is a sad day maps along with the alpha Nicky often In what you To be transfers to the awful with on exactly are often the sections which stabilized the domain the city come from this you get this map classifying seen together rights a sissy this easy equipped already with its marking on by the input edge of tea in the upper edges T-bills market punches and see and we as these are often a section does marked points and those makes the stable and the business-class and since we have this classifying we can now write down the secret all you know this this is very similar to the end results are defined earlier From the analog of the said the offer is this the locus of 6 smooth which are transfers to the awful exactly are often the sections which they policy as before for the natural forgetful maps From M the TI he the offer on and if you take the 0 7 of 1 of them scaling back and the tease about myself besides is defined the non-contact fight version but as just before the the conductor died in Britain was defined as follows so there another bunch of actions some of the surplus would you have to verify for this construction but the vast majority of them are far off I talked to this week ,comma logical given the construction citing breakdown the full definition I probably will write it down next time but 1 thing to keep in mind is that In practice you don't ever have to most of the ax and you'll have to verify there's this sort of universal construction which works to be skinny restore modular space of the whole market curves you can see them as construction in really use anything specific to contact homology respect up just with the devices intersecting the records that the only there is real you have to show is that every curve he ever in the modular space for contact knowledge has some devices near 2nd transfer statistical as once once you verify this property on distressed general procedure for defining gives gives you want the so what what again want 1 thing you have to check here is the regular "quotation mark and and in thick modulus spaces cover the original much space and as I said what this comes down to a showing that every curve In contacted him in this modular space has some devices which intersects transversely once you choose your devices to stabilize the domain it's straightforward to choose he often the obstacle colonel at the about operated at that point basically because I want to stabilize the curve you have these problems give us a absolute coordinates on the curve it's it's embedded economically inside the universal family saved by choosing Land of appropriately you can really see any element in India this space of section the 1 we believe from what do you do when you're for real of the year so so out of I've had simplified things slightly but they often me so that's a bit there's actually this group symmetric upon letters and you're right there's this observed gamma alpha so so what what actual definition of so this creates action and Gammell the vector space and this land also should we cannot offer Ankara and and so now action actually all we have to do is take the offer to be part of the data and then office should be induced by labeling these intersections with 1 Carrara the nicely this action of damn offers on this much less space but by changing the market here and by acting on the offer yeah so this is the leadership all this speculation that was if the bond member now it should be clear that this is somehow not enough I think us so we want because if we just serve to define the virtual Montoya class virtual virtual fundamental class of each much less space individually and it served to find and homology roles of the voluntary and this is enough to give a give of number 4 on the ones which are expected 0 dimensional since they may have none in the blunder no map from this relative H 0 2 6 so instead we somehow need to make some coherent choices a virtual fundamental cycles over the whole system of modular space and to do this we need the Alice's to somehow be compatible on between the different Montoya space so tempted to get such on Monday the State Bar of the be the union a of the crimes over subjects connected 2nd century so for instance is if this is your tree you have sought to some treat the subtree means you pick some subset of vertices such that the resulting and take all the edges on sent to them and required that the resulting sp some tree be connected so some trio Of all with a silver Texas but there's the 3 some trees with 1 vortex there are To some trees with 2 vertices UK analogies these 2 vertices because you don't get connected 3 and there's 1 subtree with With all vertices you're not allowed to choose Nova sees because that's also not connected so I claim that the vote construction gives an assistant index at a bar at the top and bottom now that so what would I do if I if I ever on the thickening day them for for the whole tree on that that's that's this construction here now that the beginning date and just 4 the thickening them as an element of the safety if I think data for some sub treaty Prime he just imposes conditions over the corresponding sub curve he required that some curves the 1st one some treaty Prime be stabilized by the sections with adviser and and then this this element land that when the the office of only makes sense over the some curve you know what you need to find also finally we can say how heavily because that's the compatible so "quotation mark so that music is palaces are compatible calls so for 1 for any dwarfism the prime there's this inclusion of embodies into an NBA to price the abductor quotient thing by about a morphism group activity Prime which Monday for 3 about her enough there's an atlas on MySpace a the at some space a bar of the price and not every subtree of tea prime pulls back to subject he simply taking the inverse image it therefore it every conclusion here so there's Nablus on and party given by just taking a stab at us like responding to this just using these and subservient except moreover you can you can take from this space with the settlers and just to take the restriction of established to the to the substance just by looking at the corresponding stratum in every and much less space and and these are economically it was the same thing and 2nd for any concatenations from this product is much less basis to 1st base of the nation I it was funny there are only 2 of the Constitution earlier this fall it's it's just topological here there's really anything the check unjust after formalizing the compatibility properties that we need and to make the construction work this an isomorphism up to this quotient by the on the Monday from the junction just been Nablus here which is that this trade union of the they borrow to die there's now is here which is a borrower now again any subtree of a given TI which subtree the concatenation of its inclusion yeah and again this is compatible this summer palace of this on "quotation mark coincides with the for a product that was on this product of spaces the myself so now that over time next time I will come and talk about how to give a precise definition of an implicit Alice and all talk about how to what algebraic topology 1 needs to extract a virtual fundamental cycle all the space given an and status a few and home