 Welcome to all of you. This is a special activity today. We have a special colloquium The speaker today is Ian Aigle from the University of Berkeley He's giving a talk talk and colloquium on the virtual fiber and conjecture So before he starts I would like to say some words about Ian He has been in the University of California since 2007. His main research areas are in the topology and geometry of three-dimensional manifolds And he was educated if I understand it correctly in San Diego, the University of San Diego And then before that in Caltech And he's the winner of a number of grants and awards Including the Oswald Weblen Prize in Geometry, Mitter Professor in the fall of 2012 Senior Berwick Prize at the London Mathematical Society, Simon's Apatical Fellowship, Clay Research Award, etc Well, very recently Professor Aigle is the recipient of the 2016 breakthrough prize in mathematics, which as you know is the most generous prize in fundamental science in mathematics and physics and As a great gesture, which I think fits very well with the ICTP mission Professor Aigle has donated Percentage of that prize to promote Mathematicians from developing countries so and the thing that's a good gesture that we all appreciate from from Developing countries perspective. So so let's welcome Professor Aigle. Yeah, thanks a lot for the introduction for the Invitation speaker. It's been a great conference this week so Today I want to talk about this virtual fiber and conjecture, but my I wasn't quite sure the audience's background so most of the talk I'm going to spend sort of on definitions and background and topology and Hopefully some examples to give you an idea of what's out there So the one of my postdoctoral mentors William Thurston in 1982 he had a An article in the Bolton of the American Mathematical Society that sort of gave an outline of his Viewpoint on three-dimensional manifolds and topology and sort of revolutionary ideas and At the at the end of the paper he had a list of 24 questions Which have ended up being quite influential in the in the field over the past 30 years or so so the the goal of talk is to focus on one of the one of the questions from this list and Question number 18 so He says does every hyperbolic three-manifold have a fine sheet of cover which fibers of the circle? This dubious sounding question seems to have a definite chance for a positive answer so Obviously as a question over time it's been Sort of upgraded to a conjecture called the Thurston's virtual fibrin conjecture But in any case you never actually as far as I know conjectured it so the The goal of talk then is to just explain what what is this question mean because I'm assuming that I'm basically assuming people in the audience have like a undergraduate Math background there maybe some some classes of mathematics, but Not much more than that. So I wanted to describe some aspects of manifolds and Especially the three-dimensional ones and then Describe how you how you manufacture three-dimensional manifolds different ways you can make them out of lower dimensional spaces and various other operations So in particular out of curves and surfaces to make a three-manifold So basically by products and certain kind of twisted products called vibrations and then That indicates some of the limitations of this and then and then Describe Thurston's questions for a particular class of the three-dimensional spaces and then all at the Then I'll indicate some of the some of the tools that go into the the resolution of this of this question That are outside of the area of three-dimensional topology actually Okay, so that's the that's the goal for the for the talk today then So what is a manifold well, it's a space and so by that I mean a Metrizable topological space so in in topology with though we're not concerned with the metric but There's there can always be an underlying metric on the on the on this spaces We're considering today and which locally looks like some n-dimensional Euclidean space and that's called an n-dimensional manifold So here's a sort of standard description So you have a topological space and you cover with different patches that look like that have coordinates Euclidean coordinates So on different patches you'll you'll get these coordinates, but there might not be a global coordinates over the entire space, so But over on the overlaps between these coordinates you see Identifications nice what are called diffe morphisms between the subsets of Euclidean space the sort of coordinates might not match up, but But the the derivatives and everything is sort of non-degenerate derivatives are all non-degenerate, etc So You can also so that's sort of an abstract way of thinking about manifolds The original way actually they were thought about is as subsets of higher-dimensionally Euclidean spaces So if you have a n-dimensional manifold it turns out there's an embedding thermal Whitney It says you can always embed it into a 2m dimensionally Euclidean space So it's something you can think of it sitting like this picture inside of a higher-dimensionally Euclidean space and it locally Looks like a lower-dimensionally Euclidean space up to some kind of wiggle So that's sort of a quick Description of a manifold if there's any questions feel free to to pipe up if something's not clear okay, so Okay, so it's a space then Covered by Coordinate charts which have these which gives a smooth correspondence with the Euclidean space and so I'll be mostly interested in Today in one two and three dimensional manifolds so so a A two-dimensional manifold then you have You know coordinates like this a standard x y-axis And you can think of that as kind of a product It's a product of a line with the line so you need two coordinates each coordinate gives you one dimension and then the the pair coordinates gives you a product space And in three dimensions then we have three dimensional coordinates locally again There's you can think of as a product either of two dimensions with one dimension or of three different dimensions and So in some sense Dimensional manifolds are made by this kind of products of lower dimensional manifolds And so the kind of goal I guess in some sense for the talk today is to see well How how can that product structure? Can you extend it sort of globally over the manifold in some sense? And you can't always do that that's sort of the upshot So Okay, so that's that's one way of viewing these these coordinate charts Okay, so two-dimensional manifolds. Let's let's look at those. These are also called surfaces and Under certain technical hypotheses bounded connected and orientable then they're determined just by the genus so we have Two-dimensional sphere you can think of these as sitting inside of three space Taurus and then connect some so you can remove discs from two of the manifolds and and glue them together Connect some operation and make as many handles as you like So that's essentially the classification of surfaces So I guess I I skipped over the one-dimensional case so they only they only closed bounded one-dimensional manifold is a circle and There's the Taurus is obtained by taking a product of two circles So You can see it here Here's sort of global. There's you can put global coordinates on a Taurus Where one coordinate is one circle one coordinate is another circle sort of like what latitude and longitude Except the coordinates of latitude and longitude on the earth. They don't they're not global in the sense that at the north and south poles they They're The longitude is not uniquely determined So there's not some kind of global coordinate system whereas on the Taurus there is and none of these other Surfaces higher genus surfaces. Can you get some kind of a global coordinate system? So So now How do we construct three-dimensional spaces? well You can construct them from one and two-dimensional spaces in a variety of ways So for example, you can make the analog of this Taurus here you can make a three-dimensional Taurus just Product with one more circle now you've got a three-dimensional space. I didn't have a picture of that, but anyways, hopefully you can imagine that so So that's that's one way so that's a product construction that gives you a three-dimensional manifold Now there's a generalization of products, which are called bundles. So These are these are spaces these are spaces where The you don't have a sort of global coordinates But in a certain direction when you go from one coordinate chart to the another there's a certain coordinate direction that always lines up Either a one-dimensional coordinate or maybe a two-dimensional coordinate if we're in three-dimensions So that's sort of rough description of what a bundle is. So a A classic example is the unit tangent bundle to a surface Then this gives you a three-dimensional manifold. So I here's a picture of the two sphere and At each point I can draw a little circle and And that gives me a third coordinate so I can go to any point and then I can there's some angle a third coordinate Direction and this this gives you a three-dimensional top logical space So if you if we did this on this torus that I had in the previous page Let's see. I guess if I do little circles on there Then I would actually just get the three-dimensional torus. It turns out that that's that unit tangent bundle is actually just a product of the circle But here it turns out that this this data this extra coordinates direction so it doesn't have a global Consistency like zero that no way saying is there's no there's no section of the unit tangent bundle on the sphere global section so So this is but yet in any given little coordinate patch You can sort of make sense Of the third coordinate the third coordinates sort of line up consistently So that's an example of a bundle and there's it It's also a bundle in the sense that there's a map from this three-dimensional space to the two-dimensional space You just forget the the angle of the circle so So that's a that's a rough description of a of a bundle and This is also that we can also think of the circles is giving what's called a vibration of the manifold so at every every point the manifold there's a certain direction that is specified by the by the circles of those of this unit This unit tangent circles Anyone know which manifold this is so I guess this is called RP3 the three-dimensional productive space Okay, here's another example of a fibered three manifold this yellow Curve here is the trefoil knot if you think of this inside the three sphere and you can extend to a a Fibration or a foliation by these circles of all of the three-dimensional sphere or the complement of the of the trefoil knot so that's another example of a kind of Folliation or vibration and if you identify You take you can take a quotient where you just identify all of these curves to a single point And it turns out then you get a map from the three sphere to the two sphere And now it gets to the one on the previous page So so that's another example of one of these vibrations of constructing a three-dimensional space from a kind of twisted product between a two-dimensional and one-dimensional spaces That's actually The terminology for this is it's called a cyphert vibration. So these kinds of these were introduced by cyphert and trefoil in the 30s Okay, so what Let me go back a step and to say a little bit about topology. So Topology the goal is to classify topological spaces up to homeomorphism. This is so one of the broad underlying goals and What does it mean to be homeomorphic? So they're sort of topologically indistinguishable. So as I said At the beginning we're considering manifolds as having a metric on them But in topology you forget about the underlying metric and you're you only care about So to close by points go to close by points So if you can make a a one-to-one bijection between two spaces, which is not stretching or tearing Or gluing anything together then it's a homeomorphism sort of intuitively so you can make that precise by saying you have One-to-one bijections between the manifolds which are continuous maps going both ways From say x to y and y to x so that the if I compose these maps, then I just get the identity map so that's the the technical description of a homeomorphism But what this means is that there's a one-to-one correspondence between these two spaces that sends Nearby points in one space in say in some metric to nearby points in another space And a continuous map is one that takes the pre-image of balls will be a union of balls in the in the in the pre-image so Okay, so so that's that's what we mean by homeomorphism And so we're interested in in classifying in particular three-dimensional spaces up to homeomorphism So some classical invariance that don't that only depend on the topology Our orientability that's basically saying that if you if you're living say in a three-dimensional manifold And you walk around and you're right-handed you don't come back to you and become left-handed So that would be a change of orientation There's dimension just the if it's a manifold then the dimension of these Euclidean Coordinate charts that was just describing That's a topological invariance. So you can't have a surface that's homeomorphic to the three-dimensional manifold for example And then there's things called Betty numbers in the again in the surface case The Betty numbers essentially telling you how many how many holes you have or what the genus of the of the surfaces in some sense And then there's a refinement of that called homology And then the fundamental group which I'll talk a little bit more about so the fundamental group is is sort of the One of the most important topological invariance in three dimensions. It turns out in terms of the class the in terms of the homeomorphism classification so Now now we know that Compact so it's closed and bound at three-dimensional spaces. They're completely described up to homeomorphism by a Conjecture of William Thurston made in the 70s called the geologization conjecture and that was solved in 2003 by Perlman this So you might have heard about this that it implies the point-create conjecture and But it has it has much many more ramifications for the study of three-dimensional spaces Now I'm not going to get into the description of the geometrization theorem today, but because we're going to focus in a bit on hyperbolic spaces So So now I described a little bit about Fibrations of manifolds by one-dimensional fibers sort of tested product construction But there's you can also look at Fibrations of manifolds with two-dimensional fibers, which are called also called foliations So a three-manifold Fibers over the circle if there's a map from the three-manifold to the circle where the pre-majorary point is a is a surface That's which is called the fiber Now if If M is closed then in fibers of a circle then The the fiber will be some genus G surface We already know the classification of those and you take that surface and you thicken it up You just cross it with interval. So it's a product except so there's nice sort of Global coordinates here in some sense But now we glue the top to the bottom by a homeomorphism of that surface And you call you get what's called the mapping tour. So another way that topologists make Spaces topological spaces is via gluing construction. So there's a natural way to to metter eyes or to apologize this when you When you glue up so you can imagine walking through the space and as you go through the bottom You magically appear on the top kind of like Pac-Man going off the side of the screen and appearing on the other side if you ever played that video again But Pac-Man actually lives on a two-dimensional tourists, I think anyways the So you can imagine Making these these sorts of constructions to get three-dimensional manifolds and it's a kind of twisted product because You're gluing up here by some homeomorphism and it turns out that The homeomorphism the surfaces are very rich and complicated and interesting so You get a lot of interesting manifolds this way. So here's another way to think of a of a foliation so you can think of The every point in the surface in the three manifold. There's a surface going through it So this is a foliation of a complement of some Not inside of the three tourists that's drawn by by Ken Baker. I guess I Think I have a picture here. So that's You can see dynamically the foliation Is moving the sheets are moving around each of these not so this is the three-dimensional space here is you it has a Three-dimensional translation symmetry here, so this is And you remove all these rods from the space and it's the complement of that has this for foliation. That's That's in the picture there so that's An example of a vibration So They're called foliations because I think it evokes you know the analogous term in Geology where you have these these sort of two-dimensional strata that the three-dimensional space is made up of so that's One way to think about it Well, so as I was just saying surfaces have a very rich family of homeomorphisms and That that gives you a broad class of three-dimensional spaces this way Which are in some sense made up, you know out of a kind of twisted product out of one and two-dimensional manifolds If you took the identity homeomorphism, then you just get the surface crossed with a circle But from three three manifold apologies. That's not usually the most interesting manifold But but by taking it sort of more non-trivial homeomorphisms you get lots of free manifolds unfortunately, most closed three manifolds are not fibered by by circles or surfaces, so This is not a very general way of constructing three-dimensional manifolds So if you fiber over the circle, then it turns out that you're you have positive first bending number which is a way of saying well essentially what it's saying is that this map to the circle is topologically non-trivial there's no way to sort of Contract it off of the Continuously there's no way to what's called homotope it to a point whereas if you took a three-sphere map to the circle You could you could contract that down to a point. So anyways the Most three manifolds in some imprecise sense though Had first bay number zero and so for that reason we don't expect we expect most three manifolds and in some sense to To not be fibered of the circle And most three manifolds are also usually not foliated by circles like the examples I showed before so this the the fundamental groups of these circles turned out to a generic circle turned out to give Something in the fundamental group that commutes with every other element slides in the center Which gives an obstruction to them being foliated by circles whereas most three manifolds again in some sense Don't have an element in the fundamental group like that. So if you're familiar with the fundamental group and hopefully that made sense Anyways, so so these these constructions are nice, but they're not they're not a very general way of obtaining three-dimensional spaces Okay, so what I want to talk about then is now transition to Describing hyperbolic manifolds and then we'll come back to this question about about fibering. So This is another way actually of constructing three manifolds that's involved that's underlying the geometrization conductor So what's hyperbolic space? Well, you can imagine it having a table with a chunk of glass in it so that The the the index of refraction of light is is proportional to the to the height above the table Then then it turns out that that light will follow a geodesic path which So in a certain metric on this space Where so if you take you can take a distance between two points It's just the time it takes for light to travel between them and then light rays will either go vertically or they'll go along semicircles that are perpendicular to the tabletop and As you approach the table Light will slow down and it'll get slower and slower and never actually reach the tabletop So the tabletop is sort of infinitely far away in this in this in this notion of distance so If you could actually make some some glass with varying index of refraction in principle You could look in there and get an idea of what it looks like in hyperbolic space Which would be kind of cool So this gives a kind of physical model for Upper half space model of hyperbolic three space now this has a lot of Isometry so an isometry of the space is one which takes If it takes two points to two other points not only is it continuous But it preserves the distance between them so the time that light takes a travel So it might not be so obvious, but I can one way so you can think of this is going infinitely far up and Left and right and back and forth and you can rescale that Just by some factor and it turns out that that's an isometry So any of these if I take this geodesic here and I rescale it It turns out that light takes the same amount of time to get between two points You've scaled it so that the points are higher up, but they can move faster up here because light light moves faster The higher you go, so it ends up being takes the same amount of time to get between points You have translations reflections those are So they're clearly preserved the distance But once not so obvious is an inversion through a sphere that's orthogonal to the tabletop So maybe I'll go to the next page So if I take take a sphere orthogonal to tabletop and take rays going through there and I send a point At distance r to distance 1 over r outside the sphere That's called an inversion and it turns out that that also preserves the distance the speed of light in this in this medium so So the the upper half space model hyperbolic space you can you can parameterize it by the complex plane So if you think that the tabletop as two parameters But as a complex number and then the vertical direction for positive r is a third parameter then then that's upper half space and it turns out that the The isometries of the space are given by so-called Mobius transformations, which PSL 2c This is 2 by 2 matrices whose complex matrices is determinant is 1 and the pH just means that You you might out by plus or minus 1 So the action on hyperbolic three spaces induced by the action on the complex Projective plane CP1 by Mobius transformations of this form. So if I have a 2 by 2 matrix a bcd, then Z goes to az plus b or a cz plus d gives you a Transformation of this hyperbolic plane here it might send infinity to somewhere And that extends over hyperbolic three spaces a number of ways of doing that using like for example the quaternions Anyway, so that's a Description of the the isometries of this space. So it's a very rich group Okay, so now we say that a hyperbolic a Manifold of it's a hyperbolic structure or hyperbolic metric if there's a Discrete group so there's there's a subgroup of PSL 2c So it's just a collection of matrices closed undertaking products and inverses such that M is homomorphic to h3 mod gamma. So what does that mean? Well h3 mod gamma is just you look at a point in hyperbolic space and you look at all its orbits under this group So there'll be a set of points sprinkled throughout space and it's just the the space of all those points So if I if I move this around Then you get a three-dimensional manifold. So you so I identify all those points if you like It's a it's a another kind of quotient construction Um, discreteness means that these points these orbits don't accumulate The the elements of the group don't accumulate at the identity element. So these points stay sort of separated which Guarantees that the quotient space will have a will admit a metric as well So an example then is I Can take these two mobius transformations as z goes to z plus one that's just a translation of this Hyperbolic space and then there's an inversion here One over either to pi over three z plus one it turns out so you take these two transformations and you you multiply them together in all possible ways and it turns out that you'll you'll get a discrete set of Subset of psl2c You question by that and it turns out you get the complement of the figure eight not so you can think of this is lying inside of The three-dimensional sphere for a topologist when I say not we usually mean a closed loop of string and So there's there's no ends and you can also think of it as taking a ball and Having a worm Eat a hole through it in the in the shape of the figure eight knots. So so this thing here is the figure eight knot complement But just as a caveat. This is not a compact manifold There's no boundary here actually so that's sort of off and infinity in in this hyperbolic Metric and there's a way to put a distance the distance upstairs a lot time light travels upstairs descends downstairs and gives you a distance down there Is there any questions so far all right, so Some other examples of hyperbolic manifolds classic examples the cypher Weber dodecahedron space so I can take a a Dodecahedron sitting inside a hyperbolic space and I can glue opposite sides as it says here with a little twist And all in pairs and it turns out that you You can take a dodecahedron hyperbolic space with the dihedral angles are 2 pi over 5 and you end up getting a closed manifold this way This is a compact hyperbolic manifold so this is one way you can get manifolds is by this kind of gluing construction and there's an associated group here that has a Fairly explicit Expression, but it's a little harder to write down As I said the figure eight not compliment is a new example and then there's the white head link compliment So this is again if I think of this as sitting in the three-stereotickets compliment it emits a complete hyperbolic metric Okay, so So now I've discussed Several different ways of constructing three-dimensional spaces from lower-dimensional spaces in some sense And all these are encompassed under the notion of a vibration or a fibering So there are vibrations where the fibers are at zero one or two dimensional. So I Why is why is this let's go back to this construction? So why is this a five a vibration? Well There's hyperbolic space has a map to this figure eight not compliment and the pre-image of points will be the orbits of this group I act in a hyperbolic space. So the pre-image of points will be a Discrete set of points in hyperbolic space. So there's a there's a there's a covering space in other words Where you can get The so the hyperbolic space is sort of being wrapped up onto this onto this figure eight not compliment It's you can it's easier to see this in one dimension where you can imagine the real line wrapping around the circle And it's a local homeomorphism. So if I take in the little patch coordinate patch here It'll lift to coordinate patches up here now if if the The cover if so if the covering space is contractible It is Simply connected I mean so every loop is contractible in that covering space Then it's what's called simply connected and it's connected then the fibers of this vibration Comprises the fundamental group. So this is what we saw in the example of the figure eight not compliment the The fundamental group that I gave acts on hyperbolic space the orbits are in one-to-one correspondence with that group That's a subgroup of matrices of PSL to see And that's what that's what you see in general with covering spaces. So So this is a very general way then to make three-dimensional three-dimensional manifolds And it turns out that hyperbolic three-dimensional manifolds at least ones with Say compact ones. Let's say They have what's called residually finite fundamental group means they have lots and lots of finite cheated covering spaces And now again the geometrization theorem conjecture theorem Says that in some sense most three-dimensional manifolds tend to be hyperbolic again, that's not a very precise statement, but that's sort of It's it's it's turns out to be a very generic way of constructing three-dimensional spaces Okay, so now let's say what it means I think I can say what means to be virtually five word finally and describe what Thurston's question was so like I said the figure eight knot and Whitehead link complements they've made a hyperbolic metric, but in fact they're a fiber of the circle the fiber is actually a compact surface with boundaries so or you can think of it as a Surface like in the figure eight knot case. It's the surface of genus one with But it's been punctured or it has boundary depending on when you look at a manifold The open one of the one with boundary Similarly the Whitehead link has that property so we say that manifold is virtually fibered if there's a finite you to cover And told of them so that's a fine-to-one local homomorphism when these covering spaces such that I'm told of fibers over the circle So virtually means it's it's true up to some finite index Virtual is one of those overused terms, but that's what's come to mean in in topology. So now we Thurston's question now we've Can describe what it means so he asked whatever every hyperbolic three manifold is virtually fibered Does that have a fine sheet of covering space which fibers over the circle? so we know that we can't get every three manifold as One of these fiber bundles over the circle So it's a twisted product, but this question is saying well, maybe we get every hyperbolic three manifold by taking these Mapping to our these Fiber bundles over the circle and then cautioning by a finite group of Symmetries find a group of isometries and that that finite group of isometries though will not it'll it'll move around these surfaces It's not going to preserve the foliation. So the foliation won't descend downstairs and give us another vibration. So that's the The caveat I guess Okay, so now I want to transition to discussing some of the geometry needed to To answer this question Is there any any questions again? All right, so So we're going to move to some higher-dimensional spaces now. So a cube or hypercube Is just a product of intervals. So we take K intervals so you can think of this again is there's coordinates here x1 through xk where all the coordinates lie between 0 and 1 Here's a picture of a four-dimensional cube at Caltech or at least the the one-dimensional skeleton of it Now You can another way of as I said of forming topological spaces by gluing constructions So you can take a bunch of these cubes of different dimensions and you can glue them together along faces or facets of the cubes So Here's a bunch of cubes that have been glued together in various ways So here's a three-dimensional cube glued along an edge to a two-dimensional cube and glued to a bunch of three-dimensional cubes, etc And so this is what's called a cube complex if everything's one-dimensional then you'll just get a graph Now inside a cube complex. There's some special code I mentioned one well in general immersed cube complexes where every Every cube there's a there's various hyperplanes. So if I set one of these XI coordinates equal to say one half Then I get One lower-dimensional cube and I can do that in k different ways if it's a k-dimensional cube And then you can you can glue all these together to get these lower-dimensional cube complexes and they might cross each other Because the hyperplanes will cross so So these are this is called the hyperplane complex and it these things sort of separate the space into two pieces, which is where they Where the importance or relevance in the come in so Gromov Singled out a special class of cube complexes called Katzio cube complexes And I'll give the combinatorial characterization that the the links of the vertices are called flag simple complexes So again, here's some cubes with their hyperplanes, so if it's a k-dimensional cube you'll have k hyperplanes and as I Indicated the hyperplanes might not be embedded. So here's one that comes back and self intersects so that That'll come up in just a bit so a Simplice complex so a simple complex is a complex made out of tetra heat or simplices and Has the property that in the in the one skeleton the each n plus one complete graph Is the one skeleton of an n-simplex so you you can get this from a one-dimensional graph By just gluing on simplices wherever you can so Here's here's an example of a Part of a cube complex, so I grew three squares together. These are two-dimensional cubes And now I chop off the corners around the vertex I just chop off the corners of the cube and I get a bunch of intervals. This is a graph and here I see a K3 graph, so it's a complete graph on three vertices and that's the one skeleton of a two simplex So there better be a two simplex filling that in there But this is actually sitting inside of a cube complex So that that two simplex it has to be the corner of a three-dimensional cube So we better see a three-dimensional cube sitting there And the point behind this that I don't really have time to go into though is There's a kind of positive curvature here if you think of these euclidean cubes with the standard product metric Then there's a kind of concentrated positive curvature at this point and putting a three three dimensional cube there Sort of keeps it non-positively curved in a certain sense, which again, I won't have time to to describe precisely Now There's a theorem of Micah Segui who's who was speaking here last week That associates these capture cube complexes To any set of co-dimension one data space with wall So here's an example of a space with wall So I have this is the two-dimensional hyperbolic space and I have a family of lines here Which is invariant under the group PSL2Z that's the modular group and each of these lines divides the hyperbolic plane into two pieces and So there's a canonical way that Segui developed to associate a cube complex to that So where the the walls here will correspond to the hyperplanes in this complex so each hyperplane here will be associated to a one of these these lines and we see triples of lines crossing and According in over here. We'll see triples of these hyperplanes crossing So we got a three-dimensional cube whenever we have three lines crossing here or these pairs of lines that cross each other they'll give the squares here, so That's hopefully gives you an idea of how this construction might go So you from a lower-dimensional thing in these space of walls you get a very high-dimensional cube complex But the nice thing about it is that it's much more rigid and it and it satisfies these this flag condition that Cat's your condition I described It's a not not positive. It has a certain kind of non-positive curvature to it. So Here's an example of a cube complex of us. That's a surface. So Here's some walls on a surface Really, I should do this in the universal cover the hyperbolic plane and then project down, but When you apply his construction to the surface, then you get this green So each these this green graph on the surface separates the surface into a bunch of squares And so this gives you a two-dimensional cube complex that's homeomorphic to the surface Okay, so that's an example and here notice the each each of these walls here is embedded So and this this cube complex has certain properties that make it which I'm not going to go into but it's The key property is essentially that the the walls here are embedded that there's some other conditions Which make it a what's called a special cube complex? And I'll come back I'll come back to that point in a little bit so So in 2009 Conor Markovich proved that any Hyperbolic three-manifold compact closed hyperbolic three-manifold contains an essential surface, which is nearly geodesic So what they proved is that inside a three-manifold and I'll show a picture in a couple slides of an example of this but you have a surface that maps in it's it has self intersections, but it closes up and In the it lifts in the in the in hyperbolic free space to a properly embedded plane. That's nearly geodesic And Then from this shortly after bears run and wise proved then that any three-dimensional man of hyperbolic manifold Close compact one is the is the fundamental group. It's it's fundamental group So this this group gamma that I described before It's a fundamental group of a cat zero cube complex in a sense. It's most likely going to be very high-dimensional space. So somehow So you apply the Gibbs construction to surfaces, so here's a picture of one of these surf circles This is not one that's nearly geodesic but an example of a kind of circle that at infinity of a hyperbolic surface inside of hyperbolic three space that's that descends so to a closed surface in the manifold and There's a condition here that if if you have the surfaces that separate every pair of points of geodesics Then you can get this this cube complex that from sageeves construction So you get these half spaces from surfaces in hyperbolic three space and then apply sageeves construction to get this cube complex And the cube complex kind of organizes these these surfaces that are intersecting in a very rigid and geometric way And they're far from being embedded in this in this construction And that that implies then that these these cube complex So you start with a three-dimensional space and you've made it much more complicated having a very high-dimensional space But the nice thing is that this high-dimensional space sort of organizes the information of all these immersed surfaces now That's so here's a picture from the cover of Bill Thurston's book Where you can see the kind of geodesic planes inside of hyperbolic space? so this is the There's a right angle dodecahedron here in hyperbolic three space that's Maybe the fundamental domain of a three-dimensional manifold And this is actually what it would look like to live inside a hyperbolic space these These curved geodesics you saw in the upper half space model They actually look straight to your eye because there is a plane that goes through them Are these a thickened up a little bit so they look a little bit curved but the but they're actually would looks the thin line geodesic would look straight to your eye and We have you can see these sort of two-dimensional surfaces And if you take if you put a point each odecahedron and then take a cube around each vertex Then you end up getting a cube complex. That's homeomorphic to hyperbolic three space And that one that will be a cube complex as the same dimension but in general this Construction will give very high-dimensional cube complexes so then in 2008 and then based on some newer work of why Donnie wise if you have a three manifold whose fundamental group is the fundamental give you a cube complex where all these hyperplanes are embedded I Think I called them walls here, but I meant hyperplanes Then the manifold is virtually fibered it has a fine sheet of cover that fibers or the circles so Morally what's happening here is all these immersed surfaces in some covering space They get organized and cut and paste to give you a nice fiber of a vibration So that's Anyway, so I Have time to go into how that how that correspondence works, but that's That's the that's the relation now between this high-dimensional geometry and the the five the vibration theorem And it turned out around that the time that this theorem was proved Haglund and wise had Had a method of producing lots of these cube complexes with these embedded walls And it the proof makes use of some other three three dimensional manifold theory called suture manifold theory that Now I Talked about hyperbolic three space and hyperbolic three space you has this this metrics distance structure But that that distance has a very some very nice properties namely it's Called a Delta hyperbolic metric space so the notion of Originated with in hyperbolic geometry so a metric spaces is geodesic if between any two points there's a line connecting them which is Isometrically embedded interval so you can get between two spaces two points in the space by a lot a line that's shortest shortest distance line Then We say that a metric space is grown up hyperbolic. So this is a space with a distance function if It satisfies what's called rips triangle conditions. So if we have ABC points in the space We connect those three points by geodesics then The geodesic between B and C which I write with this interval notation here lies in a neighborhood of Distance Delta for a uniform Delta about the union of the other two geodesics So here's a picture So Euclidean space is not Delta hyperbolic. So if I take it and say equilateral triangle for any Delta you choose You can take a very large Triangle where the union of these two sides the Delta neighborhood does not contain the third side So in other words, I can get from B to C without having to stay near the other two sides of this triangle Whereas if I take a metric space, that's a tree so like one-dimensional cube complex cats are a cube complex then The geodesic connecting A and B goes along here. So I'm restricted to lie along the subspace A to C goes here then be the geodesic going from B to C lies in the neighborhood of these two So it's actually zero hyperbolic, and that's actually essentially a characterization of trees in Hyperbolic space a triangle they have they are Delta hyperbolic So if I take two edges here, and then the third edge, you know The rail light going between here and here is going to travel within Delta of this Right light ray, and then it's going to move over to this one and travel along there So in fact hyperbolic space is log of 1 plus root 2 hyperbolic in this sense in this sense okay, so now It turns out that these cube complexes constructed by bears run and wise They have fundamental group equal to the three manifold group there They're a universal cover this simply connected covering space Will be Delta hyperbolic for some Delta so So now I can state This conjecture that the wise formulated in 2011 Which implies the virtual fibering conjecture so what he conjectured is that if you have a compact cat to a cube complex Whose universal cover has a Delta hyperbolic metric for some Delta? Then there's a finite sheet of cover of it with embedded hyperplanes now We've gone from three-dimensional hyperbolic manifolds to these high-dimensional Delta hyperbolic spaces And it turns out that this sort of generalization is is crucial to the proof that there's a lot more Structure to these this broader category that they can be taken advantage of in improving this sort of thing And then the fact that it's Delta hyperbolic here is crucial So there it turns out that there's there are cat's square complexes in fact two-dimensional complexes Which don't have any finite sheet of covering spaces constructed by Berger and Moses so This this Delta hyperbolicity is crucial here and it comes from the kind of hyperbolic structure that's the three manifold has and so So then the corollary of this is that if you have a three-manifold Compact three manifold like this cypher where we did a key build space then it'll have a fine sheet of cover that fibers of the circle So the resolving that question Now the proof makes use of a lot of this tech the technology that Downey wise developed and in particular the mountain of special quotient theorem and work that he did in collaboration with Haglund's and Shoe and also a Big part of the work dependent on some of my joint work with Daniel Groves and Jason Manning who's here this week And their technique we use a technique. They call relatively hyperbolic dain filling which again I don't have time to go into but it's so there's a huge amount of machinery that that I made use of and I Don't really have time to describe the proof But the rough idea is that you take this cube complex and you take all the hyperplanes in there And you chop the cube complex along these hyperplanes You take a number of copies of the cube complex you chop them up into these little pieces And then you reassemble them in a non-trivial way inductive way to get this fine sheet of covering space that we wanted which had all embedded hyperplanes But it's a very delicate thing to do and so I don't really have time to go into the description of that But anyways, that's just to give you a flavor of what's going of how this works so now I'll just mention one open question that's That it's active area research now that's some of the tools developed here are useful for so The question is if you have two hyperbolic free manifolds. You have their fundamental groups these two by two matrix groups Because their matrix groups they actually have lots and lots of finite quotients and you get them from this this theorem as well now These three manifolds it turns out their homomorphism type is determined by the the fundamental group so Then you can ask if two of these three manifolds if they had all the same finite quotients of their fundamental group Then are they actually homomorphic to each other? This is kind of a classical way Classical and variant in topology, so Topologists distinguish in various knots by looking at so-called knot colorings Which is a way of describing a certain kind of dihedral representation of a knot group And if you have a knot that has a coloring and none that doesn't then they can't be the same knot for example They're you really talking about the homomorphism type of their complements. So that's So the classical way of distinguishing free dimensional manifolds and so there's been some progress on this so Brideson Reed and Wilton and Bois-Loh and Boyer show that the figure eight knot complement fundamental group Is determined among three manifold groups by its finite quotients and then actually they proved this for many other classes of manifolds of a similar flavor certain Manifolds that fiber puncture torus and the the vibration plays an important part here and the Consequences of these these kind of theorems. I was describing so I'll stop there Thank you very much again for nice talk questions Yeah So what if you have the three manifold fiber to a recircle? Is there some measure of complexity of the three manifold? For instance, I mean high genus sort of fibers means maybe more complex and can you sort of list them by in order of increasing complexity Yeah, that's a great question. So There's an invariant of free manifolds that's The the volume hyperbolic free manifolds have a volume. It's actually a topological invariant again because this metric turns out is essentially canonical up to Isometry so So that's that's one complexity But it turns out that there's certain three manifolds like even the whitehead link that I put up earlier that they fiber But they fiber in infinitely many ways and the fibers can be arbitrarily large genus so So you could have like a complexity. That's like the minimal genus of a vibration. That's There's other kinds of invariance so-called dilatation and various things you can be extract from from these fibrinx Yeah, that's There's a variety of ways that that question can be answered questions Yeah To higher dimensional hyperbolic manifolds also have this vibration structure or is there something special about a three dimensional hyperbolic manifold? Yeah, that's a that's a great question. So It's it turns out that Higher-dimensional hyperbolic manifolds they They they can't like I say a four-dimensional hyperbolic manifold. It can't fiber In a way where the where the fibers are nice and high play say hyperbolic three manifolds There's there's weaker versions of vibration that so actually recently Donnie wise and some collaborators. I think one of the collaborators is here I think they showed that Certain four-dimensional hyperbolic manifolds they have a the fundamental group has a map to Z where the kernel is finally generated Which is what you have in this vibration case the there's a map induced by the map to the circle which on the fundamental group maps to Z and the kernel is the fundamental group of the Surface which is finally generated. So that's maybe the only weakest analog We have so far, but I would say most likely not or anyway, so So that's Three dimensions is very special in that way Because the surfaces have a rich group of homeomorphisms where it turns out then three manifolds have much more restricted homeomorphism classes of homeomorphisms in particular hyperbolic three manifolds that The different morphisms modular isotope is only finite group of the set of isometry. So you can't really get a Hyperbolic four-manifold out of a vibration, fortunately More questions probably on this side of the Yeah, Michelle Say again, I guess that even dimensional hyperbolic manifold cannot fiber over the circle because the error characteristic is non-zero You have this chair veil. Yeah. Yeah, that's a good point. Yes, that's this open for all the dimensional I guess Right. So yeah, that's that's a better answer to the question. That's a question by Gromov in fact, right? So if you if you fiber of the circle then the only characteristic is zero whereas Yeah, chair chairing gas beneath theorem says that the hyperbolic volume of the four manifolds proportional to the other characteristic So it has to be non-zero. Yeah, that's One questions Other people think about the question that I asked you at some point you say that the most three-dimensional manifolds have a First Betty number equal to zero. How can you? Yeah, okay. Yeah, so that's only a heuristic statement. There's You can make that precise in certain models. So three manifolds have a description in terms of a Higard splitting you can decompose them into two handlebodies and One consequence of that is that they have a presentation for the fundamental group with the same number of generators as relators So you take a free group K generators and then you you you kill K different elements and take the smallest largest quotient group of that And so generically you expect that those K Relators will will kill the homology of that free group And so that's one one way in which that can be made precise But anyways There's there's also censuses of these three manifolds you take tetrahedra equal and get together in all ways And you see which hyperbolic manifolds you get for example or other manifolds and again most than just experimentally seem to have first Bay number equal to zero So that's why you have to pass to a finite cover before you get hopes of have a fiber bundle Is there anything else you can learn about the three manifold from looking at this cube complex you've associated to it? Yeah, any other thing you can learn about the three manifold. Yeah, there's a lot more. So there's what's called local extended residual fineness or separate separability so the way that That Well, what what that says is that these surfaces for example these calm Markovitch surfaces There's always a fine sheet of cover where they can lift to an embedding So that's one consequence of this cube complex or rather the special cube complex structures. They have this Nice nice property of subgroups There's yeah, there's other other consequences so-called goodness of the fundamental group that the The co-emology is sort of detected of the three manifold group is sort of detected by It's finite finite quotients of the fundamental group and that that plays into I mean, that's not right directly That's sort of indirectly related to cube complex section more related to the fibrations The question okay, so before we finish. Let me just remind you that I was as useful for colloquia. Oh There's a question there. Oh Yeah, hi, I don't know much about that things but I have just a question. Is there any physical implementation of this question number 18 or not? They get the physical implementation of what to the physical world to my world. I don't know means some physics. Oh, yeah, okay, so So one I guess okay, so examples of three manifolds In physics is that what basically we're asking so, you know, we live in a three-dimensional space together with time So you take a Cross-section our universe and it's a three-dimensional space and we don't know if it goes on forever Or if it might close back in itself like the surface of the earth It was a you know hypothesis. I guess maybe in the in the 90s that maybe our universe is is finite and Now we know how to list all those finite manifolds But now we look at WMAP data and it looks looks like there's no indication of any kind of topology there that we can signature that we can see yet So but that would be that would be a killer application, you know, if we knew that the three man that What the three manifold homeomorphism type of our universe was that would be that would be interesting there's other applications in in study of knots DNA bacterial DNA conform closed loops and It turns out that So studying knots is a is a sort of relative version of topology where you have a you're allowed to deform the knot that it can't pass through itself but it turns out that the The homeomorphism type of the knot complement is a nearly complete invariant of that knot So there's a lot of these things. There's like effective algorithms. So you have To you get two knotted strands of DNA and you can plug them into this program called snappy And it can tell you are they the same or not for example, so there's I Don't know how much these things have been applied but a lot of techniques are applied in some theoretical sciences in various ways it's Let me finish my my sentence before so it's a Every colloquia we have the refreshments outside. So everybody's welcome to join us to have some refreshments except for the diploma students who will come here and then ask questions to To the speaker for the next few minutes and with the promise that they can have some food up there That's all this thing