 Hello, everyone. So the title of my talk is Trisections and Flat Surface Bundles, and my goal is to talk a little bit about trisections, a little bit about surface bundles, and a little bit about how these things connect and what I've been doing with them. So my outline, oh no, okay. So my outline is kind of following some of the results that I have come up with. So first off, if you have a direct product of a genus G surface and a genus H surface, there is a trisection with certain parameters that are computable directly from the genera of the surfaces that we start with, and again, I will talk a bit about what trisections mean and what those parameters mean in a little bit. This trisection turns out to be minimal. So there are no smaller parameters that work for this particular four-manifold, and there's also an algorithm for drawing a trisection diagram for this four-manifold that realizes this minimal bound. Generalizing this a little bit from just the direct product of these two surfaces, if we have a non-trivial sigma H bundle over sigma G, as long as that bundle is flat, then there is a trisection with the same parameters as before, and flatness ends up playing kind of a key part in that construction and making all of the proofs still work nicely. So I'm going to start out by talking about trisections, talking about how to trisect the direct product of closed surfaces, closed-orientable surfaces, and then say a little bit about how that generalizes to the flat non-trivial bundles. So a trisection decomposes a four-manifold into three simple pieces. Those pieces are four-dimensional one-handle bodies, and Gay and Kirby introduced the idea of trisections in 2012 and proved that every four-manifold, assuming smooth, closed, orientable, and connected, has a trisection. My mouse doesn't appear to be working, but on the right here is a schematic of a trisected four-manifold. So the the fold disk here represents the four-manifold. It is, of course, closed, so the black circle that's bounding it isn't really there. But the the three four-dimensional pieces that we get are these x1, x2, and x3. They pairwise intersect along the blue, red, and green arcs there, and their triple intersection is a surface sigma denoted by that black dot in the middle there. A couple other characteristics of this. Each of those four-dimensional pieces, for example, x1, has boundary exactly the the arcs that are there. And those are going to be each colored arc. There is a three-dimensional handle body and taking, say, the green and the red handle bodies together. That's a hayguard splitting of the boundary of x1 and similar for x2 and x3 there. And taking this triple intersection surface in in the center of that schematic and hayguard diagrams for the boundaries of the four-dimensional pieces, we end up with a trisection diagram. So what this diagram represents, again, the the surface here is the triple intersection of the three pieces, and the red curves on the surface bound disks in the red handle body, the intersection of x1 and x2, the blue curves bound disks in the blue handle body, and the green curves bound disks in the green handle body. And if we attach those disks, fill in any spherical components now with three balls, that gives us a three-dimensional spine for the trisection, and it correlates exactly to the union of the blue, red, and green arcs in that schematic. An result of Ladenbach and Ponaru says that that information is all we need to reconstruct the original four-dimensional, the four-manifold and fill in those four-dimensional pieces. So this diagram actually encodes all of the information needed to describe the four-manifold that it represents, and this diagram here is a diagram for s2 cross s2. A little bit of past work on trisections that is relevant to what I do. As I mentioned, again Kirby proved that trisections of closed four-manifolds exist. Everything I'm going to talk about is smooth and orientable, so I'm just mostly not going to say those words. And then a few years later it was, there was a generalization to what's called relative trisections for compact four-manifolds that had boundary. And the diagrams there are a little bit more complicated, a little bit more involved, and there's a bit more going on so that everything works out the way it needs to. We know that low genus trisections, so trisections that that central surface has low genus are standard, which basically means they are a, there's a very small selection of them, and they end up being s2 cross s2, twisted cross s2, cp2, cp2 bar, s1 cross s3, and s4. So there's exactly five four-manifolds that have a trisection of genus two. Some of them have a genus one or a genus zero trisection, but no others can be represented with such a small surface. Some work has been done on trisecting left sheds vibrations, and I work with surface bundles over surfaces. A few years ago Dale Koenig came up with an algorithm for trisection diagrams for three manifold bundles over s1, and Jeff Meyer has done some work with diagrams for spun four-manifolds, and the one important part of those last two things is that not only have they trisected these manifolds, they have come up with a way to generate diagrams for the trisections, because it's hard to study the any trisections, any trisected four-manifold without a diagram to represent the trisection, and there aren't so far a huge number of examples of diagrams to go along with large classes of four-manifolds. So I've kind of added to this by building diagrams for surface bundles over surfaces, particularly when those bundles are just direct products, but more generally when it's a non-trivial bundle that happens to be flat, which just means it's particularly nice. So earlier I said that there is a trisection of the direct product of a genus G surface with a genus H surface that has these parameters here, 2G plus 1 times 2H plus 1 plus 1 and 2G plus 2H, and what those mean is that each of the four-dimensional handlebodies that we've got, the three pieces each have genus 2G plus 2H, that's the second parameter there in the trisection description, and the surface that sits at the as the triple intersection has genus 2G plus 1 times 2H plus 1 plus 1, and it turns out that this is the best we can do for this particular four-manifold, this direct product, and it basically comes down to these parameters, they can correspond to handle decompositions, and the direct product sigma G cross sigma H has, the simplest handle decomposition has 2G plus 2H 1 handles, and that's exactly the genus we got for the four-dimensional piece, and if we tried to reduce that number, that would correspond to a handle decomposition with fewer one handles, which can't happen, so that's kind of a little bit for why this construction is minimal, so what we do for the actual construction is we have our two surfaces, one with genus G, one with genus H, and we take the genus G surface, which if you're familiar with bundle terminology, I'm thinking of this as my base surface, and we decompose it into three disks, these disks are going to be 4G plus 2Gons, again G is the genus of this surface, they pairwise intersect at exactly half of those edges, and triple intersect, the triple intersection of these three disks is all of the vertices, and I will show a picture of what this looks like in just a moment, then we take three disks in the other surface, the genus H surface, and take their product with the disks from the genus G surface, and we're going to end up essentially taking that product, removing it from the preimage of the genus G surface under the projection map from the direct product, and then gluing on one of those other four balls there, and that's what's said here at the bottom of the screen, is that each sector of the trisection, each x sub i will be the preimage of one of the the disks in the base with this small piece cut out, and then the the corresponding piece in the next preimage glued on, and we can actually see directly from this description that this four-dimensional handle body has genus 2G plus 2H, and that's using the right side of this equation here at the bottom of the screen, so the first piece here B sub i cross sigma H with n sub i removed is a disk cross a once punctured genus H surface, that's going to be a genus 2H handle body, and then to that we are attaching a four ball that intersects it at 2G plus one different three balls, and those intersections correspond to the the edges in the pairwise intersections of bi and bi plus one, and that really amounts to attaching 2G one handles, bringing the the genus of that four-dimensional piece up to 2G plus 2H, but that's a lot of notation it's probably a little bit hard to follow, so what does this actually look like in the case where our genus G surface is a torus, so we have G equals one, this picture here is as the the torus are presented as a quotient of a square in the standard way, the three disks we get are in this case hexagons, and they're labeled here as B1, B2, and B3, their edges are highlighted in different colors, I've got B alpha as the intersections of B1 and B2, so that's three different edges there, B beta is the intersections of B2 and B3, and B gamma is the intersection of B3 and B1, and from this decomposition we take the one skeleton and the trisection surface that we're going to get is roughly a neighborhood of that one skeleton, really what we're going to do is we're going to replace the vertices with a three punctured genus H surface, so specifically the genus H surface with those three disks, the n sub i disks that we picked out removed, and then we will replace the edges with edge cross the boundary of one of those disks, and that will get us the surface that we need, so walking through that with an example, this example will be S2 cross T2, so your S2 is the genus G surface, G equals zero, and the torus is our genus H surface, so H equals one, and here the, I have a reminder that with these parameters we should be getting a genus four surface for our trisection surface, so again we start by taking in this case our sphere and dividing it into three disks, I don't have them labeled here, but it's a beach ball essentially, and from this we're going to take the one skeleton and replace the vertices with a copy of the torus with three disks removed and the edges with cylinders corresponding to that edge cross the boundary component that it's connecting there, and so this is kind of what that looks like, and then we identify all the boundary components exactly as you would expect here, and we end up with the surface on the right, so this is going to be our trisection surface in this case, and as I said it is genus four just like we needed, but now we need to start placing curves, and these curves again are going to tell us we're going to have three systems, red, blue, and green, and each curve system will bound disks in a different one of the three-dimensional handle bodies, and that will tell us how to get from the surface to the three-dimensional spine and back to the overall four manifold in the end. The way we're going to get these curves is we are going to build them from arcs in in this case the torus, and really in the the torus with three disks removed, and we are really going to rely on the structure of this trisection surface as the union of we took the the vertices crossed with the torus with three disks removed, we took the alpha edges, which remember were the coming from the intersection of the disks b1 and b2, we're going to cross them with the boundary of the first disk, and the torus will take the beta edges, cross them with the boundary of the second disk, and the gamma edges cross them with the boundary of the third disk, and with this particular surface bundle there's only one each, one alpha edge, one beta edge, one gamma edge, it is a really it's the the simplest example we could possibly have, simplest interesting example, s2 cross s2 is simpler, but it's not really interesting, so all right so what again we're going to be building curves from arcs, and this slide is a little bit crowded, but first if you focus on the picture on the left there these are the arcs we're going to be working with, so at first we have a collection of arcs labeled as omega sub alpha, an indexed by in this case just one and two, and we need these arcs to cut the torus with the disks removed into a pair of pants, and they all need to have endpoints in the boundary of the first disk that was removed, which in this case is the disk on the left, and then we'll take another arc c sub alpha, and this will have endpoints in the boundaries of the first and second disks, so the left and middle disks that were cut out there, and we're going to connect things up, and that's skipping over to the third and fourth pictures there, so the third picture comes from connecting up the corresponding endpoints of the omega arcs, and then the fourth picture comes from connecting up the endpoints of the c alpha arc, just across different copies of the the torus there, and then to complete the curve system, this is a genus four surface, so we need four curves, and we're going to end up adding meridional curves across any any edges that are left here, and in this case there's only one that doesn't have any any red curves crossing it yet, and so we only add a single meridian here, we do something like this for red curves, for blue curves, and for green curves, and we end up with this diagram here, so I'm just going to let this sit here for a minute, you can see there's a lot of symmetry in this diagram, and really that happens with with all of the diagrams that this construction, this algorithm gets us, and part of that symmetry is coming from, I use the same collection of arcs in each case, just with endpoints in the appropriate boundary components there of the of the torus, I think this is probably a good point to stop and ask if there are any questions at this point about the the arcs you choose here, is it do things like lengths matter, or can you take anything sort of in the same momentopy class, or yeah it's really just up to isotope, and I tend to work with arcs that look like this, so that go around, like one arc that goes around the bit of genus there, and one bit one arc that goes through it, and if I'm working with a higher genus surface instead of just the torus, I would have a pair of arcs like this for each torus some end there, but really any any collection of arcs that cuts it into a pair of pants and has endpoints in the correct boundary component is going to work, thank you, yeah, and I will say if you are working with different arcs here, then the curves you get when you connect them up will look slightly different, obviously because they're different arcs, but also the torus pieces that I have here, the one on the top and the one on the bottom, I'm really thinking of them as being reflections of each other, because the tube on the left there is the corresponds to the disc N1, the tube in the middle corresponds to the disc N2, and the tube on the right corresponds to the disc N3, and I want to think of the torus on the bottom as being really the same surface as the torus on the top, but when I flip it over and there's some reflection happening there so that endpoints line up the way that they need to, and that's kind of hidden, but it works out nicely, and will still give us pretty symmetric pictures even with different arcs. Okay, so with non-trivial bundles, this is where the idea of flatness comes in and is going to be really important, so a non-trivial bundle is when there's some amount of twisting incorporated, so a simple example with lower dimensional stuff is an interval bundle over the circle, you can have an annulus, that's just a direct product, there's no twisting going on, or you can have a Mobius band, and you can actually literally see twisting in that. Other examples with surfaces are the torus S1 cross S1 or the Klein bottle, where again there's some twisting and that's a non-trivial bundle, and with those examples, non-trivial means non-orientable in higher dimensions, that isn't the case, you can have non-trivial bundles and still have everything be orientable, which is good because orientability is nice, but really what happens here with this twisting that's going on is some of the arcs are going to be replaced by their image under adipheomorphism of the genus H surface, so going back to this picture here, looking at say the third picture there, if I think of the top half as the starting point, then the arcs on the bottom right now are exactly the same, for a non-trivial bundle I might replace those with their image under adipheomorphism, and so we would lose some of that symmetry that we have, but this won't happen with every pair of arcs, so it would not be the case that all of, that every curve on this surface would change, and in fact I have a way of doing it where only the green curves would change, and kind of isolating the twisting that's happened into one portion of the surface bundle, so that red and blue are left alone and green has changed, and that ends up giving us a different manifold, but for this all to work and work out nicely, we do need the bundle to be flat, and the perspective that I view flat bundles from is that they're really constructed from a disc across the genus H surface by edge identifications in that disc, excuse me, so this might be, the disc might be the fundamental polygon of the genus G surface, and the edge identifications, just the standard edge identifications in constructing that surface, or it could be some other disc, and really what we do though is we identify those edges using adipheomorphism, so we would identify the first copy of edge cross surface with the second copy of the edge, and cross it with the image of the surface under some diffeomorphism fee, and what diffeomorphism that is depends on what the bundle is and what edge we're looking at, and how the things are glued together, but that is the rough idea, and also the edge that's mentioned here is going to be related to which curves are going to be changed in the diagram, it'll tell us where do we actually apply that particular diffeomorphism to a set of arcs, and that will happen when the arcs cross a particular edge from the one skeleton, that's all the material I have, I have a couple other pictures, I ran out of time as I was putting this talk together, but I do have some pictures of some trisection diagrams that I have put together, so this one on the bottom right here is S2 cross T2, it's the same diagram that I showed earlier just rotated, and the one above it is the same manifold, but viewed as T2 cross S2 instead, so here I took the disc decomposition of the torus, you see dividing the torus into the three hexagons that you saw, and then replacing each of the six vertices with a pair of pants, and gluing them together across the the edges, and on the left this is the this is the diagram that's equivalent to the one on the bottom right, just by moving some of the curves around, so if you're familiar with handle slides, those two diagrams are handle side equivalent, and a diffeomorphism of the surface will transform this diagram on the left into the one on the top right, so the intersection patterns there of the curves are the same, if you aren't familiar with handle slides, the idea is working within a single set of curves, so just with the red you can slide one curve over another, and the diagrams will represent the same thing, and the idea there is each curve bounds a disc, and if you have like a small disc and a big disc kind of going over it or around it, you can slide the big disc over, and that's still going to be a disc in the space you're working with, so that's the idea of a handle side, so these are our three different trisection diagrams for the same manifold, and they're all related by some of the equivalences of trisections, which I didn't really talk about, but really it comes down to handle slides and surface diffeomorphisms really trisection diagrams of a manifold, and then here I've got some more interesting bundles, on the left is just the four toys, so T2 cross T2, and you can see some similarities here in the trisection surface with the top right surface here, it has the same general hexagonal structure, but now the vertices are replaced with punctured tori instead of punctured spheres, and on the right it's a little bit hard to see the difference, but this is actually a non-trivial diagram, and I don't know why my mouse isn't working, but on the right edge of the right picture, if you look at the green curves at the top and the bottom, the ones that connect the two bottom holes of the tori, and connect the two top holes of the tori, two of the curves are different, instead of just going directly through each of the holes there, it goes through, and then around and through, and so that's one example of how the curves might change with, in this case a very simple surface diffeomorphism, and a fairly simple non-trivial bundle there, and then this last slide is something I didn't talk about at all, but the construction does also work for bundles where the base is non-orientable, so on the left here is a torus bundle over rp2, top right and also bottom right is a sphere bundle over rp2, and compared to the previous diagrams there's more twisting going on with the curves, although overall it still has roughly the the same structure to the diagrams, but you can see kind of some twisting happening there with the curves, specifically as they cross the tubes that go through the middle of the picture, and in the bottom right is a tri-section diagram for the spin of rp3 done by Jeff Meyer, and it's actually the same as the diagram above it for an s2 bundle over rp2, and it's hard to see why it's the same, but there's a surface diffeomorphism between those two, and I don't think any handle slides between them, it's just a surface diffeomorphism that takes, I don't remember what curves, map where, it's been a while since I've drawn those pictures, but yeah, so that's just a couple examples of some non-trivial diagrams, non-trivial surface bundle tri-section diagrams, and some trivial surface bundle diagrams, thank you for listening. So let's go ahead and thank Marla for her wonderful talk, you guys can feel free to unmute yourself and applause or use the, she says like an applaud sort of options at the bottom, thank you Marla. So Marla, you said that the product tri-sections are minimal, that's not true in general for the other ones or is it true sometimes or? It would be true sometimes and really it's going to come down to when the the rank of the fundamental group is 2g plus 2h, then they'll be minimal. Do you know if there's like some condition you could put on like the monodrome that like if it has this property then it's minimal? It's not something I've thought about a whole lot just because I haven't had time, it's on my list of things I would like to know is when are the non-trivial bundles going to have this tri-section as minimal and also it would be interesting to look at when they're not minimal, how big you could make that difference. Between the parameters and the rank of the fundamental group which would presumably be give a minimal tri-section. Can you remember if you say, can you say a bit why these tri-sections are minimal? Yeah so it the tri-section parameters here correspond to a handle decomposition and really what this is saying with the saying that all each of the four-dimensional pieces has genus 2g plus 2h is saying there's a handle decomposition that has 2g plus 2h1 handles the difference of these two parameters number of two handles and 2g plus 2h3 handles and one each zero and four handles so it's closed and there's no reason going back to this picture that you can't just swap your labels on x1, x2, x3 and that sense of triality also corresponds to a way of kind of switching around the numbers of one, two and three handles. But in this case the number is the same and really it boils down to the fact that you have to have at least 2g plus 2h1 handles and any attempt to make those parameters smaller is going to decrease this 2g plus 2h which says then you have a handle decomposition with fewer one handles or you could generate the the fundamental group with fewer than 2g plus 2h generators. Yeah something I didn't really talk about is if you try and reduce the the surface genus here at all it's also going to reduce this number but this number corresponds to one handles and generators of the fundamental group. Sorry why? For the the flatness condition on these bundles what exactly is involved in that or is it like technical or? So this is something that came up fairly recently and I will I will try to explain it as best I can without having slides to go along with it. So the what I the perspective I'd been using for a while for non-trivial bundles is using the monodromy representation which is a map from the fundamental group of the base into the mapping class group of the fiber. So fundamental group of genus G into mapping class group or group of diffeomorphisms of the genus H surface and that representation is is unique or gives you a unique manifold when the base has genus at least two and otherwise the the representation still exists but it's non-unique but with that representation going into the mapping class group there really wasn't any reason for these maps here to be the same all along an edge they would just have to be isotopic and that complicates the edge gluings a lot. Sorry why? Yeah so it's possible that flatness isn't actually necessary but the the particular way that I have constructed my curves and proved that things work relies on really a single diffeomorphism for each edge in the one skeleton and that comes from flatness.