 So, I'm going to give the first half will be a survey, a bit of a survey, but then after that I'll be talking about joint work with Adam Clay. Okay, so for conventions throughout, well I'm going to be talking about three manifolds, and when I use W that's going to be a closed, connected orientable irreducible three manifold. And M will be compact, and with all these adjectives, and the boundary of M will be a union of, okay, and then towards the end of the talk I'll need N, which is just going to be, it's a twisted eye bundle over the client bottle, so it's an eye bundle over the client bottle, but it's twisted, so this is an orientable manifold over the client bottle. Okay, so sort of the three key players here are different properties, three different properties that you can put on a three manifold. They're quite different on the face of it, so let's define them. So I'm going to say, Josh Green gave a mini course last week on these things, so we're going to say W is DTF if it admits a co-oriented taut foliation. I don't want to go into complete details in terms of this definition, but for us this means it's a co-dimension one foliation on the three manifold, and what that is is a partition of W into subsets, which locally, it looks like a deck of cards stacked up. So locally the subsets are pieces of surfaces which stack up one top of the other. That's foliation. To say your co-oriented meanset at every point of the manifold, if you look at the bit of the partition, the leaf it's called going through there, I have a well defined, well I choose a normal direction, and you just want that those normal directions vary coherently around the manifold, they vary continuously. And being taut, that's co-oriented, and then taut means that through any leaf, that's one of these partition things, I can find a loop, goes through it, and it's everywhere transverse to the foliation. It's a fact that in Josh's sketch this last week every three manifold W, this form has a foliation, but it's not true that it's necessarily that you can find one which is co-oriented and taut. So that's a definite restriction on the three manifold, I mean for instance a fundamental group of a three manifold which admits co-oriented taut foliation has to be infinite. So in particular you couldn't find one on S3. I'll say that W is NLS if it is not a Higard-Fleur space. This is going to be, the Higard-Fleur is going to be a black box today, it is what it is. I'll say some words which may or may not mean anything, but so there's, it's a Lagrangian Fleur theory which means you're in some sort of symplectic setup, you have a pair of Lagrangians, and it's a homology theory generated by the intersection points between those two Lagrangians. Now in the context of three manifolds you build a symplectic manifold associated to the three manifold and the Lagrangians are associated to a Higard-Splitting. So that's why it's called Higard-Fleur. So Higard-Fleur L space is our three manifolds which have the absolute minimum and the smallest possible Higard-Fleur homology. And finally LO if pi 1W left orderable. So I'll just call that LO. And here I will say something more, it's much easier to describe what this means. So a group is LO, if 1, it's non-trivial. So that's a convention we take which is convenient, particular for three manifolds and 2, there exists a total order on G such that it's left invariant, in other words multiplication by on the left preserves the order. So it is clear on the face of it these things don't appear to have very much to do with each other. Okay, now I want to say make a few remarks on LO. So maybe just some obvious examples as said, well the reels are left orderable just with respect to the usual order in addition. And so any non-trivial subgroup of the reels inherits a left order. But of course not everything's LO. Inflate G is torsion free. And this is actually a very simple exercise. For instance, if we start out with something which is bigger than one, well then I just left multiply by G on, see G squared equals G times G bigger than, G times 1 equals G. So if G is bigger than 1, then G squared is bigger than G, N, etc. You know G to the N bigger than G to the N minus 1. You just proceed like that. So there's no way you can't eventually have G to the N equal 1. So they're torsion free. So finite groups are not left orderable. And for when we're looking at fundamental groups of three manifolds like M. So it's a consequence of various results in three manifolds that for these manifolds of the sort W that we're looking at here. If they have torsion in their fundamental group that they're finite. So either the group is finite or it has no torsion. Okay, so a key example of a left orderable group. It's sort of a universal example in some sense for countable groups. It's only a plus R and it's not at all hard to see why this is left orderable. You see this embeds into R to the N. So here we have, let's list Q, this and then we can associate to every homeomorphism its values on the rationals. And since the rationals are dense, that's clearly an injective map here. And I can give R to the N the lexicographical order. So if I take two elements of R to the N, which are different? I look at the first coordinate where they differ and then just look at which coordinate is bigger for one of them than the other. And that's the bigger element. So for homeomorphisms, I take two F and G. I look at the two list, FR1, FR2, etc. GR1, GR2, etc. I look where they first differ, say FRN is different than GRN. And then I just ask which one is bigger? Say FRN is bigger than GRN. Well, then you declare F bigger than G. And it's a simple exercise to see that that gives a left order on here. It's left invariant. Okay, so that's a key example. And the fact, the basic fact is that, that's a three, four. If I have a non-trivial group G, which is countable, is left LO, if and only if there exists an injective homomorphism. Okay, and well, I don't want to talk about the proof very much. But it's not that difficult. There's a bit of a, there is a subtle point in it, but quite very, very roughly, you embed. G is countable and it's not hard to see that you can map G to the reels, perhaps as a dense subset, so that the order is preserved. And then, so we have this subset of the reels, which can be identified with G. And you have left multiplication, one G, G times G1, et cetera. Left multiplication by G, an element G of G, gives a, well, determines an automorphism of the subset of the reels. And then if you're really careful, this is the subtle point. You have to be careful how you embed G into the reels. But if you take good care, then you can see that this action by left translation elements of G on the subset of R extends to an embedding of G itself in homeoplasar, which you verify as a homomorphism, in fact, an injective homomorphism. Okay, so in that sense, homeoplasar is sort of universal for accountable left-orderable groups. You can actually, there's a canonical choice of this representation. This is where you make the embedding of G into R as tight as possible. And you end up with something called the dynamic realization of G. It's the fundamental invariant of a left-orderable group. This dynamical realization of the left order. This representation is well defined up to conjugacy in homeoplasar. Now for three manifolds, the three manifold groups are special in many ways. And in this context, it's a result, pi1w is the LO if and only if there exists a homomorphism, say, psi from pi1w onto an LO group. Okay, so you don't actually have to show the group itself, pi1w is LO, you just have to show it surjects onto a left-orderable group. This is a straightforward consequence of the Scott-Shayland core theorem. And the Burns-Hale theorem from theory of left-order groups. And then going back to here, this is if and only if, there exists an on-trivial homomorphism of pi1w into homeoplasar. So in fact, understanding if you're given a three manifold like w, and you want to know whether it's LO or not, well, it's equivalent to finding homomorphism from pi1w to homeoplasar with non-trivial image. Okay, so like I said, the three properties, LO, CTF, NLS are quite different, but remarkably, there's no known example where they differ. So for instance, if the first bedding number of w is strictly positive. W is Corian to Todd-Fell-Leach, and this is Dave Gabbai's thesis done in 1983. W is LO. Well, this is a consequence of, you see, if the first bedding number is positive, then there is a surjection of pi1w to h1w and then on to z. But z is a left-order group, so if I use this criterion, for pi1 of w is LO if and only if I can find a surjective homomorphism on to a left-order group, well, this is automatically satisfied if the first bedding number of w is bigger than zero. So that was a paper of mine with Dale Roffson and Bert Feist. And w is NLS. Again, as I mentioned, the Heygart-Fleur stuff is going to be a bit of a black box, so I'll just say, the thing is, to be a Heygart-Fleur L space, you have to be a rational homology, three-spher. You have to have the first bedding number zero. So I'll just say by def. If you're an L space, then just by definition, the first bedding number is zero. So this will hold. So from now on, I'm going to consider rational homology three-spheres. W is finite, so a rational homology sphere. Okay, so this is, if you want to investigate these various properties, CT, FLO and NLS, we're really reduced by stuff that had been done earlier to the case where you're in a rational homology three-sphere. Okay, so in this case, there's more space. There are some known relations between the various concepts. So if you have a Corian-Tetat foliation, then you're not an L space. Now, this is, well, if the Corian-Tetat foliation is C2 smooth, then it's a result of Asfath-Zabo. If it's not, if it's just a topological Corian-Tetat foliation, then you have to use recent work of Wilkises and Rachel Roberts and independently Jonathan Bowden to do this. So these two papers appeared last year. And so CTF implies NLS. If you are CTF as well, this implies that the commutator subgroup of W is LO. So remember, W is a rational homology three-sphere. Its first homology is finite. That means the commutator subgroup is a finite index in here. So if W has Corian-Tetat foliation, then there is a finite cover, which has a left, W tilde, which has a left variable fundamental group. In fact, this was the, so back in 2003, both Caligari Dunfield and Rachel Roberts, Melanie Stein and John Chereshin used this, this type of argument to show that there were three manifolds without Corian-Tetat foliation. I mean, one would say hyperbolic ones. This particular statement, this is due to Caligari Dunfield. Caligari Dunfield, here, if you're LO, then an argument of Sam Llewellyn and Adam Levine shows that you're not a strong L space, which is the special type of L space. Okay, so these are direct quasi-relations. That's definitive. This is up to taking a finite cover, and this is true up to looking at a very special type of L space. Nevertheless, after a lot of examples had been looked at, so two conjectures came out. So conjecture one, due to myself with Cameron Gordon, and Liam Watson is at W, oh, if and only if, W, CTF, sorry, is NLS, and conjecture two. Well, it's a bit, I'm not exactly sure who to attribute this to, but I'm going to put down some names. So Oshav Ashabo and Yuhash, what's a conjecture? W is CTF, if and only if W is NLS. So in fact, what the reality is, is that when Oshav Ashabo showed CTF implies NLS, they ask if the opposite is true. And the first time I saw this written down as a conjecture was in a survey article of Yuhash, but I really don't, I don't know if he was claiming credit for the conjecture or not. At any rate, here it is, the conjecture. So conjecturally, then each of these extremely, quite different, quite disparate concepts, properties for a three-manifold are equivalent. Okay, and there's been a lot of, I'll talk about some evidence now. And there's been a lot of efforts to try to generate examples. There's been a lot of effort to find counter-examples, but they're still standing, the conjectures, as we speak. So first of all, it follows from the work of a bunch of people, and I hope I'm not forgetting anybody. So this is Eisenbacher-Schneumann. I've already mentioned the paper with Adele Ralston at Burt Feist. This is Stipschitz, and then the paper with Gordon and Watson, that the conjectures are true. W is a non-hyperbolic geometric three-manifold. And roughly, exactly speaking, what this means is the two conjectures can be verified to hold for all manifolds which are either cypherd-fibre or admiro-sol geometry. There has been a fair amount of work looking at branch covers of S3, of Oz Vassabo, combined with the work of myself with Cameron Gordon and Liam Watson, shows that conjectures hold the two-fold branch cover of a non-split alternating knot. So one interesting point here is that these are generically hyperbolic. So none of these manifolds here were hyperbolic, but these are generically hyperbolic. Nathan Dunfield has done, well, I'll come back to him in a few minutes, but he's done a lot of experimentation and quite literally gone through hundreds of examples looking at knots, say, up to 15 crossings, looking into two-fold branch cover. And so there are hundreds of thousands of those, and trying to determine by machine calculation whether you CTF or lower NLS. Now, another paper Gordon and Lydman study for sigma NK, the infold branch cyclic cover K, where K is varying over certain families of knots. So that was two, three Dain surgery. Well, there are lots of examples for Dain surgery, and actually it's a good point because typically what happens, you see, for a rational multi-sphere to determine whether it's CTF LO or NLS, it's really hard. General, if someone just throws it out at you, it's very, very difficult. And it turns out that for certain families it's easy, you know, maybe you can show it's NLS for certain families or CTF, but that then drops certain problems in. If you know, for instance, that they're CTF, well, then you ask, do these things, like typically non-trivial Dain surgeries on an alternating knot is CTF. And so, well, you can ask, are they LO or the NLS? So interesting problems arise here, and let me just mention one. So here's a theorem who's exact, I don't know exactly who it's to be credited to, but it certainly goes back to Oshawa Sabu, who set up the basic material for doing this. So if K, a non-trivial knot in S3, K, R, and L space. So what I mean by this, this is R-framed surgery, well, the R-dain surgery on the knot K, L space, some R bigger than zero, R is NLS, if and only if R is less than two times the genus of the knot minus one. So it's a wonderful characterization of what happens on knots. But none of the Dain fillings given L space, or the ones which given L space are exactly the ones where the surgery coefficient is bigger than or equal to 2G, two times the genus of the knot minus one. So immediately you come up with a question. Can you prove something comparable? Yeah, so R-zero. Yeah, thank you. Some R-zero bigger than zero. That's why I chose R-zero bigger than zero. If I said R-zero less than zero, then I'd have to flip everything around. Okay, so does something comparable hold an L as replaced by CTF? So this is definitive, right? This tells you exactly what's happening for Dain surgery on knots in S3 for giving L spaces. In fact, the recent work of Hanselman, Rasmussen, Rasmussen, Watson, as a group and a subset have extended this to a much more general situation than knots in S3. But there are partial things known. So for instance, example K, a non-special alternating knot. So alternating knots come into flavors special and non-special. It's non-special, the generic ones. These special ones are things like torus knots and whatever. So then Ajah Sabo, K, R is knot and L space for all rational surgeries are. So for every surgery except the meridional surgery, Rachel Roberts, K, R is CTF for all R and Q, but it's unknown. Yeah, the fact is very little is known about this. Actually Nathan Dunfield's talk on Friday will address ways of showing that many Dain fillings, Dain surgeries on knots have left orderable fundamental groups. And then finally in terms of evidence, let me mention Nathan. Just bring back Nathan's machine calculation. So I got to say that Nathan has taken the attitude at the beginning that the conjectures are false. And so he's been working really hard for about four or five years to find counter examples. And as such, he's amassed a lot of experimental data and to date none of it has shown, well, the conjecture is still standing. So Dunfield has examined literally hundreds of thousands of, I guess, more than a half million. Fair enough to say, arising from either, see branched covers of knots are from the senses, from the tables. And of course, no counter examples. So one thing I should point out about these, these are all, these aren't generic examples. For example, the ones in the senses, these are relatively few. If you look at their geometric decomposition into tetrahedra, for instance, it's relatively few, say up to eight or something like that. And for these, well, these manifolds all have strong evolutions. And as such, they're not going to be generic. Nevertheless, I think this is positive. Okay, so now, yeah, well, it's a machine calculation so it doesn't always, he can't always tell. So for instance, to tell something's not left over, well, that is algorithmic. You can write a program and it starts running. And hopefully after a certain amount of time, it tells you no, not left over well. And so typically, he found that work very efficiently. Here, so somehow you start out looking, it's for something in the, you take a particular presentation, you look in the Cayley graph, you know, you're looking at the ball of radius one in there, and the ball of radius two, radius three, radius four. And at each time you're looking at a new ball, you're looking for something. And if that, you find something then that will obstruct being left over well. And he said typically you only have to go out to the ball of radius five to find it. Even, you know, these are knots up to 15 crossings. There are reasonably good algorithms for calculating Hagar flow homology, so that can be calculated. He's developed a method for, you know, finding out whether there's a coriant type of radiation. You know, he builds a certain type of branch surface and it's very special and very effective way of doing this. Now if you want to show it is left order, well, so these are things like showing not a left order, if you want to show it is left order, well that's tough, right? You have to find a left order. And so for us that means we have to get a representation to homeo plus r. Well that's a bit of a mess, right, to try to get people just to know how to, in general at least I don't know how to see like, you know, you have a chance, if you want to look at to find representations into SL2 tilde or something, you kind of understand the geometry, the topology of that group and somehow you might be able to see your way to finding a representation. And in fact what he, that's what he does is he asks, he looks for representations in SL2 r, which he, you know, that is something he can study algorithmically. And once he's got it, then he asks, you try to lift that representation to SL2 r to SL2 tilde. And there's an Euler class obstruction, which can be reasonably straightforwardly calculated. So that's what he does. So yeah, so at the end of the day, what he'll tell you is okay, I looked at 500,000 examples, I can tell these ones are left orderable, these ones are not left orderable, these ones are L spaces, these are not L spaces, et cetera, et cetera. But I think you can find on his website, he's got, he has an article where he describes this experimental work. Okay, so to, a theorem I want to talk about is, well today the only sort of, one general result I stated was for non-geometric, non-hyperbolic geometric manifolds that conjectures hold and in particular it holds for ciphered manifolds. But what I want to talk about for the rest of the time is that if W is a graph manifold rational homology sphere, W is LO, if and only if W is CTF, and that's if and only if, but let me do it like this because there are various groups of people working on this W and LS. Okay, so this bit is due to myself and Adam Clay. And then, well, the filiations we construct here are topological. So at the time we were doing this we couldn't apply the Oshawa Sabo argument to deduce this. But nevertheless the filiations we constructed are fairly nice and so we could show, for these examples, these filiations, that this implies it's NLS. Of course, since we did this Wilkises and Rachel Roberts produced their paper, which is just said in general, if you have a CTF manifold you're not an L space. So I can put them there. Or Jonathan Bowden did this. But this other direction, so as I'll explain it, the proofs of these things and the equivalence here, the equivalence here, all have the same flavor and ultimately you need a gluing theorem and certainly the gluing theorems for Hagar flow homology is what you need here and certainly Adam and I had no idea what to do, but quite recently in joint work and in separate work, so with Jonathan Hanselman and William Watson, Jake Rasmussen and Sarah Rasmussen and in a joint paper, they proved this. Okay, it's really, really, really nice recent work. Okay, so in the time remaining I want to talk about this. I'm going to first though state a, just look at a special case and state a couple of problems that are completely wide open. Any information that could be obtained on it would be very interesting. So let's look at, rather than rational homology three-spheres, we're going to look at zetomology three-spheres, graph manifold, well I'm going to start with zetomology three-spheres, graph manifold, zetomology three-spheres. That just means there are graph manifolds which have the same homology as the three-spheres. There's a theorem, and me that if w a graph manifold set Hs then w is CTF if and only if w is not S3 or sigma 235. That's the point query homology sphere. So I have to say here that what you're seeing is the really other than these two's example, everything is CTF. And hence everything is LO, everything is NLS. But that's not the case. The minute you go out of the realm of zetomology three-spheres, rational homology three-spheres, there's no generic behavior. Roughly half of them have the properties, the other half don't. So this is a special case of a more general conjecture of Ajfa Sabo, which is I think quite interesting and worth pointing out. So it has been called the Hegard-Fleur Poincare Conjecture. Ajfa Sabo, EFW and Irreducible set Hs, zetomology sphere then w is NLS and EFW that's not equal. So you see it's amongst the world of Irreducible tetanomology three-spheres, Hegard-Fleur homology, the fact whether being in L space or not is determining the three-spheres, the modulo of sigma 235. Okay, so there's a conjecture, but this raises a problem, like I said. If you believe in the conjectures the conjectures are those conditions are strongly correlated then you'll wonder whether show w is CTF respectively LO w does not equal. So these are wide open questions and like I said, take your favorite family of zetomology three-spheres and see if you can say something about CTF LO or NLS for that. For instance, this is known to hold for zetomology three-spheres obtained by surgery on a knot. So you can get infinitely many different zetomology three-spheres by one over in surgery on a knot and again that I'm not exactly sure who that's due to the first place I saw it was in a paper by Matt Haddon and Liam Watson. So, and various things are known. So recent work of Taolin and Rachel Roberts show that for all but finitely many Dain surgeries of the form one over N on a knot in S3 your CTF and actually consequently your LO using something called the Thurston's universal circle construction. Okay, so there's a family of zetomology three-spheres which arise naturally those which arise from surgery on knots in S3 where at least for a given knot up to removing finitely many surgeries all these conditions hold your NLS CTF LO. Okay, so how do you is based on something called slope detection? So I'm going to simplify things. I'm going to think of M now only look remember M is a compact connected orientable iridescal three manifold whose boundary is union of tori but let's just assume the boundary of M is connected. I do what I'm going to talk about more generally but it just gets too complicated to be much interest in this talk. Okay, so S of M, these are the slopes on boundary M equal to H1 sorry it's a projective space and so the picture you should have so once I choose a basis say we choose a basis in H1T so this contains that I said 2 so once I choose a basis for H1 instead 2 then I can just map it out here where these are the the first homology of the torus is just the lattice here and then on the other hand the first homology with the real coefficients is the background plane and so is there any color? You're looking at lines so an element of this projective space is just a line which goes through the origin and you call it rational if alpha can be taken to be in the first homology with just the coefficients. Okay, those are the ones which go through a lattice point. Okay, so what I'm going to do I'll tell you in very broad terms how the proof works and then whatever times remaining will discuss some of the points in the proof. Okay, so there are more but there are four ways that interest us but interest me in four ways to describe subsets of distinguished slopes distinguished in fact we're going to call them detected slopes. You can use foliations, representations left orders and del spaces. Say we take foliations we're going to define something called default, set of detected slopes detected by foliations this would be D-rep and so the proof of the theorem on the graph manifolds well there are two steps, you have to prove two results you have to prove a detection theorem FM Cyphert all these methods give the same family slopes in order of belonging D so all these four methods of detecting slopes actually give the same family slopes on the boundary say M we'll call that just D of M and then there's a gluing result W M1 on a torus M2 or M1 M2 Cyphert and W ATF respectively LO respectively NLS if and only if D so the set of detected slopes coming from M1 here's W there's this M1 M2 and there's a torus T so there's a set of slopes on T determined by both M1 and by M2 the set of detected slopes and we want those two sets to have a non-trivial intersection okay so now you can see how at least in the special case of the theorem of a special case of graph manifold which you see I want to know is it CTF well it's if and only if DM1 intersect DM2 is non-trivial but the same thing for LO or NLS and well that's all there is to it so the condition for being CTF is equivalent to the condition for being LO or conditioned for being NLS so the three properties are the same so this gluing result I guess Adam Clay and I did it for CTF and LO and for the NLS gluing result that was the work of Hanselman of the two Rasmussen's and Liam Watson okay so I don't have a lot of time so I think what I'll do is I'll just describe what very briefly what foliation detection is what order detection is and what NLS detection is by the way all these properties are ways of detecting are very natural so you take again boundary M is T and I'm going to assume that alpha is rational just to simplify things I mean you could have an irrational slope being detected but let's do this so alpha foliation detected if there existed a coriant to taut foliation such that F is transverse to the boundary of M that's T and if I look at F intersect boundary of M some what's F intersect boundary of M since it's F is transverse to boundary of M that's a co-dimension when foliation of the boundary so it's just a foliation of this boundary torus by lines and circles and to say that alpha is foliation detected well alpha if it's rational recall here that alpha then I take it to be in the first tomology and what I want is that there's at least one circle there exists at least one circle I want at least one and if so you say that that slope alpha is detected by that foliation F but more generally that the slope is foliation detected there's a equally simple notion of order detection but first let O be a left order on pi 1 M then you look at the restricted restriction of O to pi 1 M is an order on Z2 so the fundamental group of the boundary of M is Z2 and this is an exercise which I don't know if you did last week or not the students here but given a left order on Z2 so what is that doing the Z2 which I'm thinking of now in this plane the plane R2 some elements are positive some elements are negative but what you can show is that there's a line there's a unique line determined by the order such that on one side of the line everything is bigger than zero and on the other side everything is less than zero that's a fairly simple exercise so the order that line is uniquely determined by the order and so that's giving me a slope so every left order we say that alpha is order detected if I can find a left order on M which one I restricted restricted to the boundary I get that line okay and then finally I'll finish off in the last minute with NLS detection which appears quite ad hoc but because I suppose in a way it is I mean the Haegard-Fleur people presumably can take what I'm about to say and then convert it into Haegard-Fleur language where it comes out more naturally but but it's the right notion for for us and you say take a rational slope and what do you do you take a I want to glue together M with N, remember N is the twisted eye bundle over the client bottle and F is a homeomorphism from boundary M to boundary N and I wanted to take the slope alpha to the what's called the rational longitude of there's a well-defined slope on the boundary this is called the rational longitude and I want the gluey map to do this and you say let's call this well I won't call it anything say that alpha is NLS detected okay so this you know all seems rather rather ad hoc and unconnected to various forms of detection but it turns out that the work I alluded to earlier about Cyphert manifold the three conjectures hold well that work at least in a relative version of it shows you that there are very simple connections and straight you know direct connections between these various ways of defining things I don't have time to do it but if anybody's interested I can explain after so I'll stop here