SPEAKER: So we should welcome Tom Mrowka from MIT is going to tell us about instanton Floer homology for trivalent graphs.
TOM MROWKA: It's a pleasure--
I take it that applause is for me actually showing up to a talk.
Anyway, so this talk, actually it's a little bit different than the title. I thought it's really sort of two half hour talks. The first half hour is about how the part of mathematics that I work on, how it has sort of been influenced by some of Thurston's ideas, and where they have led. And then the second part is going to be more about the instanton Floer homology for trivalent graphs.
So the place that I want to start this is with the observation that one of the main features of low dimensional topology is that it's hard to embed low genus surfaces in three and four dimensional manifolds. Of course, in three dimensional manifolds, it's in some sense rather obvious. Non-trivial knots don't bound disks, et cetera. In dimension 4, that became understood from work of Kervaire and Milnor, where they found an obstruction to embedding two spheres.
And the part of the story that Bill's work comes into is in the late '70s, he introduced the Thurston norm, which kind of organized the possibility of embedding low genus surfaces. So that the Thurston norm of a homology class is the minimum minus the Euler characteristic of an embedded oriented surface that has two components and represents the given homology class. The definition makes good sense for an irreducible 3-manifold.
And he also proved that if that 3-manifold carries a taut foliation-- so that's a foliation for which there's a closed curve that meets all the leaves-- then the inequality holds. I think Eliashberg likes to call this "the inequality" and tells it the Euler characteristic of an embedded surface is bound by the bounded by the Euler class of the taut foliation evaluated on the surface or keeping track of the signs. This is telling that the Thurston norm of a homology class is bounded below by the evaluation of the Euler class of a taut foliation on the surface. And it's one of Dave Gabai's early theorems that in effect that characterizes the Thurston norm, in fact. And then it also led to the work on sutured manifolds et cetera, which is now fundamental in understanding 3-manifolds contact structures, foliations, et cetera.
The beginning part of the talk is going to emphasize the fact that the inequality has a four dimensional generalization, which is called the adjunction inequality. And that this is a story for 4-manifolds now. And what it tells you is that-- this is proved by Kronheimer and myself originally. And it tells you that if you have a 4-manifold, then you can a find certain distinguished set of cohomology classes, so you have this inequality. I forgot the sign, made it-- there's a sign error there. I should have written it like this.
This is valid for surfaces that have-- again, S is supposed to be embedded surface with now a non-negative self-intersection number. And you notice, of course, when your 4-manifold is a 3-manifold cross-S 1, and, your surface really sits inside the 3-manifold-- and this is just exactly the D inequality. S dot S is 0 in this case.
So one thing that's still kind of annoying, even many years after Peter and I proved that such classes exist, is that they're still extremely mysterious. Somehow, a taut foliation is something you can imagine, think about, get your hands on, whereas these cohomology classes that 4-manifolds carry are much more mysterious. They arise from understanding some of the-- in any of the invariants of 4-manifolds from Donaldson, that was where we originally started. Seibert-Witten invariants or the Ozsvath and Szabo's Heegaard Floe homology invariants.
So it would be great, you know, you more geometrically-oriented folks, to find a geometric structure on a 4-manifold for which this inequality holds.
AUDIENCE: But that inequality is just true for each individual Ki?
TOM MROWKA: Yeah, that's right. So the way the first part of the talk and the second part of the talk tie together is that the work that Peter and I did in the early '90s, that gave us this inequality, we've revisited it. And it's now become the story that relates some instanton stuff to Khovanov homology. And we'll see. So this original stuff is, that's just what I said, great.
Some recent theorems that we can prove is that Khovanov homology in its reduced form detects the unknot. And we can also prove-- so Jake Rasmussen has an invariant that comes out of Khovanov homology, which is a lower bound for the slice genus of a knot. So you'll remember, if I-- there's S-3 bounding a four ball. And if I have a knot K, then the aversion of the minimal genus problem is I could look for surfaces inside the four ball that bound the knot, that's a four-ball genus. And this S invariant is a combinatorially defined invariant defined by Jake.
It comes out of the combinatorics of Khovanov homology. And in Jake's story, he was able to prove that it gives you this bound for the four-ball genus. And using the fact that we could relate Khovanov homology to gauge theoretic invariants, we could prove that in fact this bound persists, if you allow yourself to imagine that instead of just the standard four ball, this was just a homotopy four ball.
So exactly the same bound holds. That term's of some interest because there was some work of Friedman, Walker, Gampf and Scott--
TOM MROWKA: Morrison, yes, thanks. Where they were hoping to use this SN variant to test if some potential constructions were counterexamples to the Poincare conjecture-- four dimension Poincare conjecture-- and by finding hopefully a knot where it bounded something of two small genus in the homotopy four ball. But the SN variant was the obstruction. So that's not going to work.
By the way, where's Ron? Hi, Ron. So to make-- you know, sometimes in order to make life interesting, you have to make bets. So Ron and I have now made a bet, which involves one of us paying for dinner for the other somewhere, involving the Poincare conjecture, four dimensional smooth Poincare conjecture. So I'm betting that more or less within the next 10 years, it will be solved. And Ron's betting against.
AUDIENCE: Neither is betting on how it's going to go?
TOM MROWKA: No, I'm betting that it will be proved in the next 10 years.
AUDIENCE: Proved true?
TOM MROWKA: Yeah. And Ron is betting that nothing will happen in the next 10 years. OK, not for any good reason, but dinner is a good bet. Nobody loses.
I want to explain a little bit of the mathematics around where the four dimensional version of the inequality comes up. And this is the story the Donaldson invariants. We're going to do this in a little more generality.
The Donaldson invariants are invariants that come from studying anti-self-dual connections in a principal bundle, principle SUN bundle or more generally a PUN bundle on a 4-manifold. And we look at this modulated space of anti-self-dual connections that's a subset of-- AA is the space of connections, in the bundle, G is the gauge group. And this is a distinguished sub-manifold depending on the metric inside the space. The space of connections mod gauge is pretty close to this mapping space, the space of maps from x to the classifying space of BSUN pulling back the bundle-- the component that pulls back the bundle P. So in this form, it's quite standard to understand it's algebraic topology, at least rationally.
And after a lot of work, what you can prove is that there's a well-defined integration map over this cycle. And it defines a linear function on the cohomology of this mapping space. And by the way, to understand this cohomology, the standard trick is to look at the evaluation map.
And then what you can show is that the cohomology, say with Q coefficients, is generated as a ring by the Kunneth components of the pullbacks of the Chern classes under the evaluation map. And a nice way to provide some notation for that, each of the Chern classes gives you a map from the homology of your manifold to the cohomology of this mapping space by pulling back the Chern class and then slanting against the homology class. It behaves on degrees, as indicated.
And in the case of SU2 bundle, what you find is that if you sum over all SU2 bundles, the evaluation-- so here, this side is sort of a formal expression at the moment. If you take that exponential of-- a mu map of a two dimensional homology class, that gives you a two dimensional cohomology class. And I take its exponential, which is an infinite sum in the symmetric algebra, and I evaluate it on all modulized spaces--d and there's some extra little factor here-- then this turns out to be remarkably simple. It's just a function of the self-intersection number of the surface, and then there's some distinguished cohomology classes. Oops, there's some coefficients missing there.
So it's just the sum of exponentials of evaluations of these distinguished cohomology classes. These are the ones that come up in the adjunction inequality. And this nice formula holds when the 4-manifold has what's called simple type. So simple type means that the Donaldson map, this map, annihilates the ideal generated by what happens if you take on the manifold, a zero dimensional homology class, you get a four dimensional cohomology class. You square it, and then-- anyway, this particular, if you look at any element of the cohomology that's a multiple of this class, if it's killed, then the manifold has simple type.
All of the four manifolds that we know have simple type. For example, symplectic manifolds always have simple type. It's very easy to check if 4-manifolds have simple type. We still don't know if all four manifolds have simple type.
And Ron and I have another bet. We drank a little bit too much last night. But in fact, this one has money for the solver involved. My inclination is that all manifolds have simple type and Ron's is not. So if somebody proves either finds a counter example-- a 4-manifold that doesn't have simple type-- or proves that they all do, then one of us will give you some money.
AUDIENCE: Is the sum on the left finite or infinite?
TOM MROWKA: This is an infinite sum.
AUDIENCE: So in particular, you have convergence of that series.
TOM MROWKA: We don't have-- well, yeah, so that's right. This thing converges, that's correct. That's part of the content of the theorem.
AUDIENCE: Is that saying that those mu2 of Ss are getting very negative or something?
TOM MROWKA: No, this side is-- each of these modulized spaces is finite dimensional. So when you evaluate each of them, you get a number. And then it's a fact that there are infinitely many of these guys. There's a convergence issue, but this side, they're not getting-- there's an exponential, so there's a factorial that's killing off growth.
So these things-- it's saying something about how quickly they grow. It's not saying that it goes to zero.
AUDIENCE: Is this an equality between numbers?
TOM MROWKA: It's an equality between numbers, yes.
AUDIENCE: So the series on the left does converge?
TOM MROWKA: Yes.
AUDIENCE: What makes it converge?
TOM MROWKA: Magic.
AUDIENCE: Why do the terms get small?
TOM MROWKA: Well, there are factorials in here. Reciprocals are factorials in the definition of the exponential. So it's saying-- if I look at this evaluation, it's just saying that as K gets big-- so you have to evaluate these guys in different modulized space-- but as Ks gets big, these just don't grow so fast. Because you have to divide them by K factorial in the sum. I mean, just like e to the 10, that makes sense, right?
AUDIENCE: That's a big number.
TOM MROWKA: Yeah, it's a big number, that's OK. That's a big number over here, too. Anyway, so there's the balance in the variance. Then there's instanton Floer homology.
So instanton Floer homology or cohomology is the Morse homology or cohomology of the Chern-Simons function on a space of connections, on a PUN bundle over a 3-manifold. So in the case that the bundle is trivialized, this is the definition of the Chern-Simons function. And the beautiful fact about the Chern-Simons function in dimension 3 is that its first variation has a very nice, simple expression.
The space of connections is an affine space. If I pick a connection, and then I move out along a line through a connection-- move out along this line here. And then watch how the Chern-Simons function changes. Its infinitesimal change is the integral over y of pairing of the curvature of the base point with the tangent vector that you're moving along in.
And in particular, this is telling you that the critical points of the Chern-Simons function are the flat connections. And so the gauge equivalence classes of critical points are more gauge equivalence classes of flat connections. Gauge equivalence classes of flat connections are-- you can identify with representations of the fundamental group up to conjugacy. So this Floer homology is sensitive to the existence of representations of the fundamental group.
And furthermore, playing along with this formal analogy with a fine dimensional story, if you pick a Romanian metric on y, then you can think of the space of connections is becoming a Romanian manifold finding the inner product to just be the L2 inner product on the algebra valued 1 forms. And then the downward gradient flow for Chern-Simons becomes this PDE. And this PDE, if I think of this a as-- a solution would be a one parameter family of connections. And if I think of that as a connection on the 4-manifold r times y, then this is equivalent to the anti-self-dual Yang Mills equations.
So it's the analysis of the anti-self-dual Yang Mills equations, thanks to Ulanbek, Taub, Donaldson, et cetera, that allows you to construct the Morse complex. But in any case, there is a Morse complex. You have to be careful, as I'll explain later.
It's not defined for any 3-manifold, it's not defined for all three manifolds and all principle bundles. There's a restriction that you need to keep in mind. But when it works, this is what it is. And it's an invariant of the 3-manifold and the principle bundle on it.
Now this Floer homology is related to the Donaldson invariance in that it's the right thing when you try and take the topological quantum field theory point of view. In fact, this is one of the original motivations for simply formulating the notion of the topological quantum field theory. That is to say when you have a 4-manifold with boundary and a principle bundle on this 4-manifold, it restricts to the given one on its boundary, then you can extend this map that we had before. So for a close manifold integration of the modulized space gave you a map to the real numbers. Here, it gives you a map to the instanton Floer homology instead.
Now tautologically, if you look at the Morse homology construction, when you change the orientation of the 3-manifold, that changes the sign of the Chern-Simons function. So it changes homology to cohomology and vice versa. And so what you can often prove is that if you have a closed 4-manifold and it's decomposed along a suitable 3-manifold, you can often obtain the invariants of the closed manifold by a chronic repairing between homology and cohomology for the relative invariants.
And an instance of this map, if we take our 4-manifold to be just an interval times the 3-manifold, then we can think of-- the map that we had before, remember the cohomology of the configuration space that is generated by a map from the homology of the manifold to the configuration space. And then the cohomology of the configuration space, that's this mu 2 second Chern class. So then you would get a map from the homology of the 3-manifold to that endomorphisms of the instanton Floer homology.
So in the case that x is an interval time is y, in general, if you have a manifold with boundary, simplifying slightly what I said before, you get a map from the two dimensional homology of this manifold into endomorphism, so Floer homology. So for this particular case, we get a map of this sort. And in fact, these guys are acting by commuting endomorphisms.
And what you can show further is actually that the spectrum-- if I represent geometrically my two dimensional homology class pi-oriented surface, then the spectrum of the resulting operator on the Floer homology, it's eigenvalues are even integers that lie between the Euler characteristic and minus the Euler characteristic of the surface. And in this way, you actually can recover the Thurston norm. So in effect, the Thurston norm is determined by knowing the spectra of these operators.
And in particular, if you're careful about when Floer homology is defined, then one thing that this story eventually proves is that if you have a 3-manifold with B1 positive that's irreducible, then there exists a representation of the fundamental group of with 3-manifold into SO3, which doesn't lift to SU2, in particular it's a non-trivial-- it's an interesting representation. And I think it's a nice conjecture, which maybe is one that there's more hope for proving, actually with the exception of the three sphere, every 3-manifold admits a non-trivial representation into SU2. So at least now, thanks to all the great new work, at least hyperbolic manifolds up the finite covers do so.
Thurston comes back into-- it just doesn't get away from our story. So I told you the story about how Donaldson invariants in Floer homology, just for ordinary 4-manifolds and 3-manifolds, it turns out to be useful-- as Thurston taught us-- to think about orbifolds and then generalize this story to the context of orbifolds. So what we're going to do, we're going to do this in the simplest possible way, first starting off with a 4-manifold. So we want a 4-manifold containing a surface, so an ordinary, smooth 4-manifold containing a smoothly embedded surface. But we're going to reinterpret that data as defining for us an orbifold. And it's an orbifold that just has cone-angle pi along that surface.
So the simplest possible thing by requiring the cone-angle to be pi is that there's not really any extra choice in the orbifold structure. And what we're going to do is do gauge theory on these orbifold SUN bundles. The interesting extra bit of data that you get when you're thinking about the bundle is that above this orbifold locus, above this surface, in order to construct the orbifold structure on the bundle, you need an endomorphism of order 2-- so Z2 action on the lift of the bundle, which is going to be non-trivial along this. That should be along the surface.
And in fact, although it took Peter and I a long time to do this, once you've swallowed the orbifold pill, then there's no particular reason to require the single locus to be a smooth manifold. It's useful to generalize, to allow S not just to be a smooth sub-manifold but to be a kind of singular surface, which has become in the physics literature and not invariant literature, it's been called a foam. We'll come back to that later.
We're just going to look at these simple, orbifolds. And then we can-- as with most analysis, once you decide to work on orbifolds, the only thing that's complicated is just keeping track of what an orbifold means. But the analysis itself is you just repeat all the analysis that you did and just make it invariant under some group action. So there's not much extra work to do to work in this context.
But now the interesting thing that happens here is that the Floer homology, what does it look at? Well, it's now representations of the orbifold fundamental group. But a little bit more concretely, it's representations of the not complement but where meridians of the knot in this story will go to some element of order, too. So they're not representations naturally of the whole fundamental group, but they're slightly damaged along the knot, the [INAUDIBLE] element of order two.
So I want to think about this in a the simplest possible example first. Let's look at representations of the fundamental group of just a knot in the three sphere complement into SU2 and where the meridians go to the two sphere of elements inside SU2, whose square is minus 1.
So remember, I'm doing a couple of things, which I hope are not too confusing at the same time. For the Z2 orbifold business, we want to look at PUN bundles, so PU2 bundles in this case. So PU2 is SL3. And we need something of order 2 in PU2.
Now I can think of that a little more concretely as looking at it as a map to SU2. But rather than it squaring to 1, an element of order 2, it's something whose square is in the center of SU2. Now there are two choices whether or not it's plus or minus 1. Minus 1 is the interesting one, so that's what we're going to look at.
So we're looking at conjugates of this matrix inside SU2. That's a 2 sphere inside SU2. If you think of SU2 3 sphere, that's the equator where plus and minus are the North Pole and South Pole of the 3 sphere.
Now what is the space of representations that we get? Well, for the unknot, of course it's the 2 sphere. Here we're looking at knot representations at the conjugacy, just representations. So for the unknot, we get the 2 sphere. Or if you have an unlink of k-components, then you just get a product of two spheres. That's easy.
A little more interestingly, if you look at the trefoil, then the representation space has two components. There's a component which is a 2 sphere, which is just factoring through the map from the fundamental group or the complement to the homology of the complement, just reproducing this guy, really. And then there's another component of the space of representations, which is a SO3 or RP3.
What's interesting about these two examples is that the homology of these representation spaces actually reproduced the Khovanov homology. So the Khovanov homology, as I am evidently assuming you all know, is some combinatorial invariant of knots. It's quite a pretty and interesting story. But if you compute it, then for the unknot of k-components, it's just Z squared tensored together K times. And for the trefoil, the Khovanov homology is four copies of Z and a Z2. So it's interesting that even there's torsion there, and this picture sees that torsion.
Now just as an aside, it's kind of fun to think about what these representation spaces look like. And there's a pretty picture to try and get to know them, which is-- I mean, in general it's hard to compute what representation spaces look like. But there's a little trick that sometimes helps you here.
So for not complement, we have the usual Wirtinger presentation. So the blue guys are the two arcs of the knot crossing each other. This guy is going over this guy. And then there are generators for the fundamental group that is xi, xj, and xk, which come from taking some base point out there and then running that curve to here and back. And of course, you get this relation.
Now the neat little fact here is that conjugating-- so now we're looking at a representation where all the meridians-- these are all meridians of the knot, these xs. So they all land on this 2 sphere in this SU2 case. And what you find is that this relation, when you combine it with the fact that these guys are landing on this 2 sphere, conjugating by xi when xi is in the 2 sphere is actually the geodesic reflection. You know, relative xi is a point on the 2 sphere. And 2 sphere is a symmetric space. It has a map, which has that as its unique fixed point, and x pi minus 1 on the tangent space.
And conjugating by rho of xi acts by that symmetric space involution. And so what you see is that whenever you have one of these Wirtinger relations, then what that's telling you is that the guy that's conjugating is the midpoint along the geodesic, between the other two points. And so once you know that, it's easy to figure out what the representation for this space for the trefoil is, for example. Of course in general, it's impossible, but this helps you in some examples.
Anyway, so we can extend this game to SUN. And we can consider elements in SUN. Again, we're late doing PUN, but we're going to think of them by lifting them up to SUN and requiring that their square is some central element of SUN. So this is the picture.
But now in the SUN case, then this automorphism has two eigenvalues. And now there are more conjugacy classes than in the SU2 case, depending on the dimension of this eigen space. We pick one of the square roots of that guy as our favorite. And then they're different conjugacy classes, according to the dimension of that eigen space.
So let's suppose it's fixed at some K dimensional-- that eigen space is K dimensional. So these guys now, generalizing the previous story, form a copy of a Grassmanian sitting inside SUN. So that conjugacy class is a Grassmanian inside SUN.
And we're going to make things simple. Look at the case of k equals 1. So that means there's one dimensional space with this eigenvalue and n minus 1 eigen space with the opposite sign. That's the CPN minus 1 case.
And in that case, what you find is that what's the representation space then-- for the k equals 1 case-- it's just copies of CPN minus 1-- for the unknot, it's CPN minus 1. I'm just telling you that that generator lands somewhere on CPN minus 1. Or it's a product of CPN minus 1 for a knot of K components.
And if you look at the symmetric space game, having said symmetric space, these are all symmetric spaces, and that's symmetric space game for computing things keeps going, then now what you find is that, say for the trefoil, the representation variety is a copy of CPN minus 1 and a copy of the unit tangent bundle of CPN minus 1, generalizing the case of the 2 sphere CP1 and the unit tangent bundle of the 2 sphere RP3. In the case that n equals 2.
And again, these guys now reproduce the Khovanov-Rozansky homology of these examples. And again, that's kind of-- this guy has some torsion in that its homology which is the same as the torsion in the homology of Khovanov-Rozansky. So it smells like there's a pretty story there.
Now I have to try and be a little bit clearer about when these Floer homology groups make sense. So I was kind of brushing under the rug some technicalities, and now we have to pick up the rug a little bit and try and be a little bit more honest. So first of all, we're dealing with PUN bundles, so there's an extra characteristic class. And there's a picture.
So when you're dealing with PUN bundles, so there's a characteristic class. They're still orbifold bundles, so the bundle is only defined on the complement of the surface. So this is the four dimensional line, so that there's a characteristic class w in H2 of x minus S with ZN coefficients. And we're going to keep track of that guy-- sorry, I didn't-- sorry. I forgot the first sentence here.
We're looking at PUN connections. And in general, PUN connections can have more complicated stabilizers than SUN connections. And there's a trick to avoid that, which is to look at PUN connections modulo gauge transformations that lift to SUN. Then the story becomes simpler.
So if you have an SUN connection, its stabilizers is always just a subgroup that leaves invariant some block decomposition of CN. But if you have PUN, then the stabilizer can become something more complicated. But anyway, there's this little technical trick of only using gauge transformations that lift SUN. That makes it a little bit harder to keep track of things functorially, when you try and define maps in Floer homology.
So that's dealt with by-- again, these PUN bundles have this characteristic class. And we're going to keep track of that characteristic class-- so here's this pair, 4-manifold with surface. It's bundle defined on the complement of this guy with some characteristic class. And we're going to keep track of actually a geometric representative of it. So that's what this blue stuff is, it's an auxiliary surface inside the 4-manifold, which is keeping track of W2. That surface can have boundary on the S that we have. So just to try and be honest. And sometimes a surface could be not orientable.
Now over here, here's how we're going to think about the critical points of Chern-Simons mod gauge, where this means SUN gauge estimations that lift SUN. There are conjugacy of representations of the fundamental group now of what? Well, in the 3-manifold case there's still-- we're doing this orbifold business along the knot K. But we have this PUN bundle, so there's this extra characteristic class, for which we pick a representative, which is this blue curve in this picture.
And now what we're going to look at is the critical points of Chern-Simons correspond to representations of the fundamental group of-- actually the complement of both the knot and this representative of the characteristic class w. But now they have the property that if-- sorry, that should be omega. So there's w, the cohomology class, and there's omega, it's a representative of it.
And what we want is that if you take the holonomy of the connection around w for a curve that links it once, it's some central element of SUN. And it's inside this Grassmanian, if the loop links the knot once. So this first condition, if I pass to a PUN bundle rather than an SUN bundle, it means the bundle extends. But it will extend as a non-trivial bundle.
That's a little bit technical. But just so nobody gives me counter examples to what I say, I feel I need to say it. Again, like I said before, an orbifold bundle doesn't necessarily give you a bundle on the 4-manifold. It only gives you a bundle on the complement.
Actually, this isn't really important for this talk. But this is-- I'm not going to say, never mind that.
So the problem when you're trying to define instanton Floer homology is the fact that the quotient space connections mod gauge-- this isn't usually a manifold. It's not a manifold because this doesn't act freely. And we're trying to do the Morse homology really on the quotient space.
It turns out it's not actually so important for the Morse homology to work that the quotient space isn't a manifold. What's really more important is that none of the critical points lie in the singular set. None of the critical points have non-trivial stabilizer.
So we want to put ourselves in a position where we can avoid reducible representations of the fundamental group. And there's a simple trick to avoid that. If you look at just the 2-torus, there it is. Well of course, if I look at the representation space-- representations into SUN of the fundamental group of the torus, that's just a pair of commuting matrices. That's a complicated gadget or a pair of commuting matrices up to conjugacy. That's a complicated gadget.
However, if I look at representations where-- I look at representations where-- the representations of the kind that we're looking at, representations into PUN, where the bundle that carries them is non-trivial, as long as the bundle is non-trivial, it turns out that there's a unique [? opticonjugacy ?] representation. It's easy to write it down.
So that said, if I look at the Hopf link H, and I give myself omega, a curve joining the two components, then of course a complement of the Hopf link is where it tracks onto a 2-torus. And this curve hits the 2-torus once.
So what I can do to make sure things work is I start off with my knot in my 3-manifold. And I prescribe a little bit of extra data, so I can canonically stick in a Hopf link. So this extra bit of data is really a plane in the tangent's place plus a vector in that plane. Anyway, I can canonically stick in a Hopf link and make sure that my bundle on that Hopf link is the one that has only irreducible representations.
Then for this gadget, the instanton Floer homology is always well-defined. So you can canonically get a knot invariant in this way, so we call this construction on the level of the knot, going from K to K natural and then define this version of the instanton Floer homology to be the ordinary instanton Floer homology, just not with a Hopf link added to it.
Now the nice thing that this does-- so remember, the critical points of Chern-Simons are actually their conjugacy classes are representations. And that beautiful story that I was telling you relating to Khovanov homology, I was insisting that we were looking at not conjugacy classes or representations but just representations, period. And adding this Hopf link actually fixes that story.
The representation on the Hopf link, when I restrict to the neighborhood of the Hopf link, there's a unique, irreducible representation. So the way the representation space of this guy is related to the representation space of this guy is that actually representations knot up the conjugacy of this 3-manifold correspond to representations opticonjugacy of this 3-manifold and knot. So by doing this trick, we've put ourselves in a good position.
Now as I mentioned before, the ordinary instanton Floer homology, it's functorial under cobordisms. There's a map from-- I've generalized it from the way I've said it. But if I've got a 4-manifold with two boundary components, then I can think of-- integration over the modulized space on this manifold gives you a map between the Floer homologies.
And that story persists to this story, where now we have a story about cobordism of pairs. We start off with a 3-manifold with a knot in it. There was this extra data keeping track of the bundle, that's the omega. Then if we have a cobordism to another 3-manifold with knot and bundle data, as long as that cobordism has to be compatible with the bundle data, then we get a map.
So what can we do with that map? Well, we're studying knots, it's natural to consider skein exact sequence and see-- I mean, skein, knots that are really related by a skein relation. So we have imagined three knots in the 3 sphere, which differ on the inside of a 3 ball in the usual way, like this. There's another picture of these three guys, which show you that this has a natural Z3 symmetry. If I think of these guys as running along the opposite edges of a tetrahedron, there are only three different ways to put them in.
So we get three different knots in a natural way. And these knots are cobordinate between natural surfaces-- using natural surfaces. There's this sort of twisted rectangle, which is sitting inside the-- so that's the rectangle that goes between these guys, K2 and K1. So this guy sits inside his tetrahedron. And then I can put it into the tetrahedron times i by thinking of it so that these vertices become just intervals.
And this picture over here is the level sets of a function that I'm going to use on this twisted rectangle to stick it into S3 times i. So I want these two edges to be at the 0 end of S3 times i, these two edges to be at the 1 end of S3 times i, and then this function interpolates between them. Of course, there's not a function at these vertices, but those vertices just get stretched out. So there's a natural cobordism between these guys.
So now there's a little tricky point that I didn't emphasize yet. So these cobordisms-- that we're constructing-- typically are not in general orientable. If K2 and K1 both have the same number of components, then that cobordism is necessarily a Mobius band. And it's not orientable. And you need to be very careful when you have non-orientable surfaces because if I'm thinking about the holonomy around that surface, it matters which way I go typically, I mean whether I'm going in a positive sense or negative sense.
In the case where N equals 2, this matrix in SU2 is conjugate to this matrix. So the holonomy going around one way is conjugate to the holonomy going around the other way. That eventually allows you to define maps for non-orientable surfaces in the case that N equals 2 or more generally, if N is an even number and K is half that number, then there's still a map for non-orientable surfaces.
So for N equals 1, cobordisms actually do induce maps. So we get this trio of Floer homologies and trio of maps. And in fact, this is an exact triangle. So just as Khovanov homology satisfies the skein relation, there's an exact sequence, exact triangle, the same thing is true for our instanton Floer homology.
And the proof that this is a exact triangle is relatively straightforward by just understanding the topology of the composite cobordisms, which is completely elementary. And then the interesting technical part of it is using those composite cobordisms to construct suitable chain maps between the mapping cone of one of these maps and the third opposite the complex with the opposite vertex.
And so eventually, iterating the skein's exact sequences is what leads to the spectral sequence that computes the instanton knot invariant and whose E1 page turns out to be the Khovanov complex. And that spectral sequence is what's used to prove the theorems I mentioned earlier.
Now I'll just go on for a moment. And bigger than 2, you need to extend this story in an interesting way. Because actually the definition of Khovanov-Rozansky is much harder than the definition of Khovanov homology. It doesn't-- Khovanov homology somehow-- if you just work in the world of knots, you can define it. For Khovanov-Rozansky homology, you need to at least work with knotted trivalent graphs. And the story there's algebraically much more complicated, using the notion of matrix factorizations et cetera.
And maybe I'll just end up with this guy. Actually, I'll end up with this one. Yes.
So we've already committed ourselves to using orbifold. So why not use orbifolds where the singular locus is no longer a knot, but we could imagine a trivalent graph, for example, as being the singular locus, where the orbifold group at that point is Z2 times Z2. If we do that, then we can still do the analogous story, but now there's an interesting relation. Namely, if I look at the holonomy of the connection around this guy and around this guy, it has to multiply to the holonomy around this guy.
So for example, if these guys both have one dimensional eigen spaces, sort of K equals 1 here, then this guy will have to have K equals 2. And so you can now extend the story to these knotted trivalent graphs, where each leg is labeled by a vertex with a kind of conservation law at each of the vertices. And you can check that for very simple trivalent graphs like this guy, that the representation space in this sense-- you can check it's a 2-step flag manifold. Its homology agrees with Khovanov-Rozansky. And actually this also gets rid of the orientation problem.
You can still prove in the case that when you're only dealing with 1s and 2s that this sequence leads to a a-- so now the instanton Floer homology of this guy is well-defined, and these three knots are trivalent graphs fit into a exact sequence. And anyway, that's sort of where we are. And we're hoping that eventually we'll be able to relate this instanton Floer homology for trivalent graphs actually to the Khovanov-Rozansky story. That's still quite unknown, but anyway, OK, I'll stop there.
AUDIENCE: So you're saying that maybe this [INAUDIBLE], is it possible Khovanov-Rozansky homology is just a homology that's basic representation by leading examples or--
TOM MROWKA: No, it's not. And that wasn't true for just the instanton Floer homologies. For simple knots, they turn out to be the same. But I think 3, 4-torus mark for example. I think it's the smallest example where the instanton Floer homology group has smaller rank than Khovanov homology.
SPEAKER: Any questions? Let's thank Tom again.
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Tom Mrowka of MIT surveys recent developments emerging from a reinvestigation of various versions of instanton Floer homology for knots or links in three manifolds at the 50th annual Cornell Topology Festival, May 6, 2012.
The festival was organized by the Cornell University Department of Mathematics with support from the National Science Foundation.