 Okay, so I can still hear myself in a double loop. Okay, so back to some examples of, is there no loop? I can hear it, but okay. So it's confusing to me because it's running, coming back to me. Let me try and ignore it. Okay, I am muted. You're muted. Yeah, but it says microphone on. But I don't know why. I started with microphone off. And then I don't know what it is. Okay, maybe now it will work. Well, we don't hear you, right? Maybe. Oh, because you hit that, you think? Because who knows, right? Okay, I'm sorry. So I want to talk about the, the contact structures, and here were some examples and now we're, I'm sorry, I got a little distracted. So just basic facts about contact structures is that our Buddha says that all contact structures are locally homogenous and all locally look like the standard contact structure that we had here. On the other hand, Benekin in 1980s, early 80s showed that there's contact structures on R3 that are not standard contact structures. So if a contact structure is like contains a disk, like the one we see here in the picture, then this disk is called an over twisted disk. The contact structure is called over twisted. And if a contact structure has no disk that looks like that, then it's called tight. So this dichotomy between tight and over twisted was introduced by Eli Arzbeck and he proved that there is a huge difference between over twisted and tight contact structures. Over twisted contact structures are classified up to isotope by their homotopy classes, plane fields and in every homotopy class of plane fields, there is a contact structure. On the other hand, tight contact structures do not exist in every homotopy plane, in every homotopy class of plane fields. And in some homotopy class of plane fields, there are tight contact structures that are homotopic as plane fields, but are not isotopic as contact structure and not sometimes even isomorphic as contact structures. So the early examples are due to Giroux and Eliashberg and Polterovich and they were on T3 and Paul Aliska who is giving the next talk and I back in probably 1995, but it got published in 97. Publications always took a bit of time. Young people now know it takes even longer, but we had examples of homotopic structures on homology spheres that are homotopic as plane fields, but not isomorphic. So anyway, this is just a short history to introduce basic questions which are to construct, recognize and classify tight contact structures. In the realm of constructions, Eliashberg introduced this theorem where he recognized the contact structures that we call Stein fillable, which live on boundaries of Stein domains which are certain kinds of complex symplectic manifolds, but the way to think about them by this construction of Eliashberg is to think about them as obtained as adding two handles along Legendrian knots in S3 or after adding some one handles to S3. So Eliashberg and Gromov proved that all the fillable contact structures are tight, and here I talked about Stein fillable, but there are several kinds of fillability and here's some differences that were established throughout history. So tight contact structures contain weakly-symplectically fillable, which contain strongly-symplectically fillable, and this difference is about how the contact structures and the symplectic structures interact near the boundary, and then the smallest family is the Stein fillable contact structures, and there are these differences established by Etnair-Honda between tight and weakly-symplectically fillable, Eliashberg between weakly and strongly-symplectically fillable, and here's in red, I'm talking about the contact invariant in 2005, Gigini used the contact invariant developed by Osvat and Sabo to show that there are strongly-symplectically fillable manifolds that are not Stein fillables. So I want to talk about the contact invariant. So in 1997, Kronheim and Rovka published a paper, Monopoles and Contact Structures, that introduced an invariant of contact structures in Monopole floor homology. And in probably early 2000s, but paper published in 2005, Osvat and Sabo defined an analog of this invariant in their Higart floor homology, in the paper Higart floor homology and contact structures. So the invariant I want to talk about today is the refinement of this Higart floor homology contact invariant. So in order to do a definition of it, I have to say something about the standard classical contact invariant. So first of all, Osvat and Sabo, of course, defined Higart floor homology for a three-manifold, and they defined various flavors of it, and for a contact manifold, M3xi, in the Higart floor homology with various flavors, they defined a contact element. So I'm today going to talk about the contact invariant in the HET floor homology and its refinement. So you can just think about HET everywhere where you see HF if it's not written. So the properties of this invariant are if xi is over twisted, then xi of xi is zero. And the other property really comes from one of the main theorems in that paper, except for the theorem that the definition of this invariant doesn't depend on various choices made, is that if M prime xi prime is obtained by a Legendian surgery, so adding a two-handle with a particular framing along a Legendian knot in M xi, then there is a co-boardism that's induced by this surgery between M and M prime, and this co-boardism induces a map on Higart floor homologies, and this map takes the contact structure invariant of xi prime to C of xi. So remember that Eliasberg said that all steinfillable contact structures can be obtained by a Legendian surgery on a link in a three-sphere. Then we know that if M xi is steinfillable, then there is this co-boardism from S3 to M, and the contact structure of xi is mapped onto the contact structure of S3, and from easy, you know, super easy calculation is that the contact structure of the standard contact invariant of the standard contact structure is non-zero. So here is a theorem of Osat and Sabo that C of xi is zero for an over-twisted contact structure, and C of xi is not zero for a steinfillable contact structure. So clearly there was a, you know, a natural couple of questions to ask is that if contact structure is zero, is xi then over-twisted? And the other question is, the answer is no, and there are these examples that I kind of alluded to in the beginning called contact structures with Jiru torsion, which keep the contact structure tight, but have zero contact invariant. On the other hand, if C of xi is not zero, does it conclude that xi is fillable? And the answer again is no, and there are various examples, a lot of people, many of them here, methods of proof, often accorded boardisms are built to examples that are known to be not steinfillable, and so sort of ad hoc and clever, various clever ways to prove, still usually and often using in some way the contact invariant, but proving that xi is not fillable. So here's a little bit more straightforward way to prove sometimes that a contact manifold is not steinfillable. So we define an invariant O of xi, which is either a non-negative integer or infinity that we call spectral order, such that this O of a tight contact structure is zero, O of a steinfillable contact structure is infinity, and this is a little confusing statement, but O of M can be detected from a given open-book decomposition. So we don't have to worry about open-book decompositions with different pages, but I'm gonna talk about it in a second. So why is this theorem useful? Calculating this invariant O is very difficult, unless we can show it is zero or we can show it is infinity. Calculating lower bounds is extremely difficult, but finding upper bounds is not that hard sometimes, and so what can we do with just upper bounds? Well, if we have an upper bound and it is not infinity, then we can conclude that the contact structure is not steinfillable, even when the contact invariant is actually not zero. So I'll give you some examples of that. I promised the definition and because O is a refinement of Hiegert-Fleur contact invariant, this is somewhat of a long story that has to mention the definition of the contact invariant, and so the contact invariant is defined as follows. So C of psi is an element in Hiegert-Fleur homology of the three manifold with opposite orientation, but we'll just ignore that, and there is a theorem of Giroud that says that every contact structure corresponds to certain family of open book decompositions on the manifold, and these open book decompositions that are used then to construct the Hiegert decomposition and read off this contact invariant. Quickly, what's an open book decomposition? Starting from a different direction, I'm going to say, first of all, if we're given a pair of a surface with boundary and the difhumorphism of that surface to itself, which fixes the boundary, then we can look at S cross zero one and mod out by an equivalent relation where we send point x1 and identify with point H of x0. We identify all the points with different T and T primes when x is on the boundary, and so when we collapse this boundary, we get what we call a binding, and in the neighborhood of the binding, we have this open book picture that's where the open book name comes from, and you can see from here that we can cut along two opposite planes in these open pages in this open book decomposition and get Hiegert decomposition. So if we cut our S cross zero one in half, because S has a boundary, we can get handle bodies, and in each of these handle bodies, we find compressing disc that gives us a Hiegert diagram. So when we're coming to this Hiegert decomposition from the open book decomposition, this Hiegert diagram has a particular form. So the red handle body has this nice set of arcs that come from the curved alpha along which we cut, and the beta handle body has a form where these blue arcs are H of the red arcs. So it's hard to really be precise about this in a short period of time, but if we have... So here is an example. If we have... So this example is where the surface is the genus one surface with two boundary components, and the H is the composition of dame twists around curves A, B, and C. And so if we did the picture like that, we would have to cut this surface into a disc and then look at where our red curves are mapped to and get this right-hand side to just see what the contacting variant is. Where is the contacting variant here? It's this special point. It's the intersection of alpha and beta curves. And in the picture where we just concentrate on the page and the picture on the page, those points appear in pairs at the ends of this alpha arcs. So anyway, once we have this picture of the Higard diagram, then Osat and Sabo had defined the Higard flow homology by looking at this chain complex with chain groups freely generated by tuples of intersection points of alphas and beta curves and with the boundary map counting certain pseudo-holomorphic curves that realize certain domains in the surface where the domains have some number of boundary components on the red curves and some number of boundary components on the blue curves and they might have genus, but that's sort of the picture when we do this count and we count all the domains from different y's that start at x and go to y's by this formula. So this is the definition of the boundary map on this chain complex and Osat and Sabo proved that the homology of the chain complex is independent of choices and so on. That's the Higard flow homology but when we have a contact structure and we construct the Higard diagram as up here, then there is a contact point, there is a special point which is the intersection of alpha and beta curves on the portion where the picture is standard that defines the contact invariant. So now with this out of the way, I want to talk about the definition of O of the spectral order of the contact structure. So what we do is we introduce a filtration on this boundary map and this filtration is considered by, comes from considering a certain complexity on these domains A. And the complexity of domain A involves this following definition. We're going to define the complexity j plus of A by saying that we're going to add the mass-low index mu of A minus 2E of A plus the difference of the order of x minus or the size of x minus size of y where by the size we count the number of cycles in the permutation of the element where x is a tuple x1 through xd of intersection points of alpha and beta curves and if we think of xi living on alpha i then we look at what beta curves they live on that beta curve that defines for us a certain permutation associated to x and we count the number of cycles in that permutation. So in particular if we think about the contact invariant xxi what is its order? So because the contact invariant is defined to be the homology class of this special point where I intersect the alpha curves with its associated beta curves what we see is that the permutation is just the identity and therefore this is just the number of curves in the set of arcs to cut this down. So there is a reformulation of this in terms of a Lipschitz formula and we get for mass love index one domains j plus of a in more local and combinatorial form where this nx of a and ny of b this is supposed to be a are in terms of some sort of local multiplicities of the points x and y at the corners of x and y of the domain a at the corners of x and y so for example if I have some xi that's an intersection of alpha i and beta sigma of i and the domain a comes in like this then this nx would be one quarter on the other hand if the domain came in like this then this would be three quarters and so on so it's very very combinatorial so why am I talking about this j plus because it turns out that the j plus of a is an integer and not just an integer an even integer and so what we can do is we can split the boundary map of hf by taking each one of these guys to be the sum like we had for the boundary map in standard here floor homology but sum just over the things that have j plus equals l so the boundary of any x splits into these boundaries and there what we get is more precisely the following so we are going to introduce a more complicated complex by tensoring our standard complex by f of tt inverse and then defining the boundary map so splitting the boundary map into d0, d1, d2 components shifting to the left by l and so when we have a situation like this we can do filtration we can introduce the filtration on this complex and by taking fp summing everything from p to the left and what we see is that we get a filtered complex and we can talk about the spectral sequence for this complex and we can define the order of this complex to be the smallest k such that the contact element x psi is dyes in the kth page of the spectral sequence so it would be nice if this was all we needed to do because then we could, yes sorry the filtration doesn't involve the contact structure but the filtration is like not it depends on everything and too much and it doesn't if you just do the homology it's just like not really invariant we don't really know how to make that filtration invariant under anything okay yeah right, yeah so it's even worse it's even when you just restrict to the contact element things still depend on choices so for example if you have an over twisted manifold then it is known that you can choose an open book that has an arc that's taken to the left and so that in the picture for that over twisted manifold there is going to be an arc and then there is going to be a domain simple domain with O equals 0 that kills the contact element so it's going to get killed on the first page so O is 0 okay for that choice unfortunately not every open book for an over twisted contact structure has such an arc so but you may still hope that you still get O equals 0 but no it's not true so here's an example this is an open book with the three times punctured disc and with the twists as noted here so I'm just giving an example which we ran into you can carefully study the whole picture for the Hieger's diagram for this H and you can find that there is actually a J plus equals 2 domain that's the only thing that kills the contact invariant so we know the contact invariant is killed so we know in some level it's going to get killed and it actually gets killed with J plus 2 and so from this the O is 1 so we don't have independence of the open book okay so well how do you fix you don't have independence well you just define it away so so we define O of M of Xi to be the minimum of all open books of O of S H A J and now it's an invariant so well it's not a very calculable invariant and we don't you know it might just be nothing on the so what we do prove is there is the independence of the almost complex structure like for the for the Hieger's Fleur there is also independence of S of H and A so first of all then the thing I said is that we have independence of the open book we have an independence of the open book as long as we allow ourselves to take more than just the basis of arcs so we're going to have to have A be not just cutting to a disc but maybe to a whole bunch of discs how does that solve that question that problem I had before on that on this example so if we add a parallel arc here as I did this green arc on the right-hand side you can see this immersed disc and this immersed disc is a domain plus equals zero that kills the contact invariant and so once I allow myself to have extra arcs then I don't have to change the open book SH that's nice in some sense but it's not nice because we don't know how many arcs we have to add to achieve the actual minimum so there is a problem with calculations is that we don't have an upper bound so now we have an invariant because you have to take an arbitrary number of extra arcs to take a minimum over it's hard to find the lower bound so if you're found O equals zero then that's the lower bound so that's a situation where we can do calculations we can also deal with Stein-Fillable Manifolds we can show that like for the standard contact invariant the O for Stein-Fillable Manifold is special and it is as for the standard contact invariant the Legendian surgery in some sense honors the contact situation in the class but what we have is that if M psi is obtained from psi by Legendian surgery then we have that the O does not decrease so in particular we're calculating that the O of the standard contact structure on S3 is infinity and knowing that the Stein-Fillable Manifold can be obtained by a sequence of Legendian surgeries what we get is this result that O of a Stein-Fillable Manifold is infinity so if we have an example where we find that O is finite well maybe we don't know how much it is but if we find an upper bound then we know it's finite then we know the contact manifold is not Stein-Fillable so here's a couple of examples so the example that I sort of plopped before maybe I can talk about it here a little bit more so I'm looking at a genus 2 surface a genus 1 surface with two boundary components and we're going to cut it up into a disk by putting an arc A1 here and then putting an arc maybe so I'm going to think about this here handle made by identifying those two guys think about cutting this surface along this green thing here so I have the blue arc A2 would be like that I have this green arc that goes that's long like this ok so we cut our genus 1 surface into a disk and with this identification I can draw this genus 1 surface picture like this and so now I have my twists my open book is the diffeomorphism is obtained by twisting to the right along this boundary component this is A and along this boundary component this is B and then it's twisting to the left along C which is like a curve maybe here so when you do these twists and you come to these arcs then you get this picture where the arcs original arcs are dotted and their images are the full arcs and then we're looking for domains that whose boundaries alternate between straight and curly arcs and we want those domains that hit the contact invariant the contact invariant is this pair of points we should really think about one point X1 and then there is this pair of points this pair of points is X2 and then there is this pair of points or X3 ok and so then we're looking for the domains that might hit the contact invariant and so very careful analysis of situation after very careful analysis we can conclude the following here is this point Y which is a triple X1, Y2, 1, 3 that I did in green here and from X1, Y2, Y3 there are two domains and only two domains that leave it so when we calculate the boundary of X1, Y2, Y3 we can see that this gray domain sends it to X1, X2, X3 and there is this one other domain that sends it to X1, W2, W3 so in short the the boundary head map of Y is the sum of these two domains each one has only one representative because that can also be proved and so what we see is that the boundary zero of Y is the contact invariant so we see that X1 is zero on the other hand it's a fact and it falls from some work of Conway and some work of Heden and Plamenevskaya that C of Xi here is not zero okay and it's not zero because this element Y bar doesn't get mapped by the standard boundary head to the contact class so that's one of the higher complexity terms in order for this to happen so there is some more examples here there is a whole family of examples on genus 2 surface with two boundary components and with this here monodromy map with a triple twist around one twist around here one left twist along C and any number of twists P along this D so the fact about this manifold this is negative one over P surgery on a horizontal curve in the circle bundle with $4 on the genus 2 surface so here the Xi zero is a unique virtually over twisted contact structure on Y zero which was shown to be not fillable by Liscan's tip sheets again the same theorems I mentioned before give that this contact class is non-zero and we prove that O for all of these manifolds and contact for all of these contact manifolds is zero by a careful argument that shows that this domain is the only that gives us is domain with J plus equals zero that gives us the O equals to zero so so I want to say that J plus equals zero domains are pretty simple they're immersed they're immersed polygons with what we would call naked corners so so here is one of them so here is Y1, Y2, Y3, Y4 and there's an immersed polygon going down to the contact invariant which is X1, X2, X3, X4 so that's it the thing is that when that it you can prove that if you minimize overall A for a given open book you will have achieved the minimum you don't have to do minima over open books that was the statement that I made that you can figure out the invariant from any given open book as long as you increase the number of As that's our theorem and that was what that example was that I was saying to change the open book for this over twisted manifold to get O equals 0 I just need to add this is sort of a throwback to end this sort of collapsing domains so yes I would actually I just realized I forgot to say our idea for this J plus came from a similar thing by Mike Hutchins who was doing something it's an appendix to a paper of Lachov and Wendell on the you know my brain just stopped like in contact homology right so there is torsion in contact homology and there is a paper there where he does something for ECH that's supposed to be a parallel to their thing and so we were motivated by that to actually do this and I think the thing that we have here is some sort of calculability with the hands on we thought we had them and then our proof fell apart because at some point we thought we can add do we have an upper bound we need to add and as I said it's extremely hard to prove the lower bound if you can keep adding infinitely many things so we don't know we suspect there is but we can prove it because we can't figure out how to impose lower bounds effectively so in principle it's possible that it only really takes zero and infinity values so maybe just one other thing Cass and Kang looked for O in sort of like a region that would describe Ziru torsion and they found that there is upper bound if all the definitions work out like because they are not all worked out yet then you could see that O is finite so that would be another way to see that anything that contains Ziru torsion is not fillable is not fillable