 OK, so I cheated while you guys were out drinking coffee. I drew the picture that I was supposed to draw. This is a picture of O2. Here's two simplices. Those are these two simplices. Notice they both correspond to theta graphs or cages or whatever you want to call them. And the one-dimensional simplices is here. And I forgot to say that what is ON, I have all these disjoint simplices. And then I've got these face relations. So in other words, this simplex is glued to that one because I can get this graph from that graph by collapsing some edges. So anyway, that's the second definition, space up. So that's how I make this into a space. The space of equivalence classes is a mark made to graphs such that G is finite, blah, blah, blah, blah, blah, blah. Yeah, so that's good. And this is a description of the topology where I'm being somewhat imprecise here. But basically, I want to, if I can get a graph by collapsing some edges that's still in my space, I think that's a face of the simplex and I glue it where it's supposed to go. Third definition, same space. Still defining the same space three different ways. Picture of a marking. I could just start all over, right? Third definition, guess what? It's going to be in terms of this three manifold and spheres in the three manifold. You should like this definition because everybody likes the curve complex. The definition looks a lot like a curve complex definition. So remember, we've got our three manifold and I kind of have two ways of picturing it, either as this times two or as a three sphere minus two end balls, which I think it was glued together. So a sphere system is a set of embedded two spheres. Yeah, disjointly embedded two spheres. Sorry, you didn't get your picture on the right. Oh, this is the picture. So I do two pictures of this manifold for you. One was I took this guy and doubled it. I handled the body and doubled it. And yeah, so what is this picture? I cut my handle body, open along three spheres, A, B, and C, A1, A2, and A3. And when I cut it open, I got a ball with six patches on it. Then to get it back together, you weren't here yesterday. I mean, no, you missed my talk. We spent a long time on this. OK, OK, OK, so then you can get my question. Yeah, OK, good. A set of disjointly embedded two sphere. Yeah, let me just, sorry, let me see. Is a set of embedded two spheres, which can be isotoped to be disjoint. So set S equals S1 up to SK of two spheres. It can be isotoped to be disjoint. And SI doesn't bound a ball. And SI and SJ aren't isotopic. So you've already seen a definition like this of a curve system on a surface. It's a set of disjointly embedded, it's a set of simple closed curves that can be isotoped to be disjoint, no two are parallel, and none bounds a disk. So that's a sphere system. And I'm going to do the same thing you do with curve systems. I'm going to make a simplicial complex, S of M. N is the sphere complex. So it's got a vertex for each sphere, S. And it's got a K-symplex for each system of K plus 1 spheres. Simple definition. So that's a simplicial complex called the sphere complex. And I want to say definition, a sphere system is complete. If it cuts the manifold into simply connected pieces, in the case of this manifold, a simply connected piece is going to be a three ball with some punctures. So each is S3 minus a bunch of balls, some number of balls. So for example, well, we already did this. Yeah. Here's a sphere system, A1, A2, A3. Thanks for the question, Indira. Disjointly embedded. I've only drawn half of them in this picture, because I'm going to double them to get spheres. Well, here they are in this picture. There's three spheres there. Looks like there's six, but there's only three, because they're glued together. And I wanted to say something. Oh, yeah, what piece is. So if I cut along these spheres, this is the picture I get. This is a three ball with six things missing. Six balls missing. So this is S3 minus 6B3s. If I was to add another sphere, say I wanted to add this one. In this picture, that's this one. Yeah? You got six B3s or three B3s? I got six of them right there. I see the six of them. Aren't you identifying? No. I say, after I cut, I get the cut open guy is each. Yeah, the cut open guy. If I add another sphere, then I cut this three ball into two pieces. One is a three sphere minus three balls, or in other words, a three ball minus two balls. Same thing. And the other one is a three ball minus five balls. Right, and if I add more spheres, I cut it up into more pieces, et cetera. So we're doing sphere systems. So that's an easy definition. Yeah, a sphere system is complete if it cuts it into simply connected pieces. Right. So exercise. How many spheres does it take to cut Mn into simply connected pieces? It takes at least n. Well, it's not an exercise. It's trivial. Anyway, so I've got this sphere complex, right. So some of the simplices cut it into simply connected pieces and some of them don't. If I have a sphere system that cuts it into simply connected pieces, then any bigger sphere system will also cut it into simply connected pieces. So define S. Yeah, I don't want to do that. I don't want to say it like that. So it takes at least n spheres. You can't fit more 2n minus something, 3, 4. Part of the exercise is to put the right number there. So after you've cut it into two-punctured balls, in other words, three-punctured spheres, you can't cut it up anymore without making having some of them being parallel to the boundary pieces. You can't stick another sphere in this piece. Any sphere you tried to stick in there would be parallel to one of the boundary spheres. So the client let S infinity of M be the subcomplex spanned by simplices by incomplete systems. So if I have an incomplete system, here's an incomplete system, that one, that one, and that one, say, that's a perfectly nice sphere system. It's got three spheres in it, but this piece is not simply connected. So that's an incomplete system. If that's incomplete, then throwing out a sphere will still be incomplete. So in terms of the sphere complex, what does that mean? If I have a simplex corresponding to an incomplete system, all of its faces are also incomplete. So that means the simplices corresponding to incomplete systems actually form a subcomplex. Then, definition 3, ON is equal to the sphere complex of M minus S infinity of M. So that's pretty short, too. That's the third definition. So in other words, maybe I should have written it down the way is the union of open simplices corresponding to complete systems. Yes. Is there any particular reason why you don't call it S infinity of Mn? Because I forgot. Thank you. Yeah. I'm doing the exercises. So you should do that in like 3M, like 3 minus 3. No, it's 2N. Exercise. How many spheres can you fit in? What is the maximum number? I think that's on the exercise sheet. OK. Sphere systems, yeah. So that's nice. But you might take some convincing to believe that this is the same space I defined before. So I'm going to try to convince you in terms of graphs. Oh no, this is the one I want to use. I'll get it right. OK. So in terms of graphs, I had, yeah. So what's the correspondence between definition number 2 and definition number 3? I showed you the correspondence between definition number 1 and definition number 2. But let's do definition 2 to definition 3. So let's start with a complete sphere system sitting inside of Mn. So here's my sphere system, and some colors. Let's make something simple. Incidentally, I keep drawing these really simple looking spheres. I would like to point out that spheres can be pretty complicated looking. So there's two really simple looking spheres. I'm drawing half of them. But if I take two spheres and connect them by a tube, that's still a sphere. So let me take a tube. And there's a picture of another sphere. OK, it's a sphere. It's either separating or non-separating. So I can change this picture by a homeomorphism to make it look like one of my easy spheres, but I won't. So for right now, let me just draw easy spheres, OK? So let's take a sphere system. It's got to be complete, so I have to cut every. I have to make sure that the complementary pieces are simply connected. And now I need to, so that's a complete sphere system. Now I need to produce a marked graph. So what I do is just take a vertex in every complementary component. There's two complementary components here. And then I'm going to draw an edge for every sphere. So that gives me a graph. If my sphere system wound all around my manifold, then my graph would wind all around my manifold. But I can't draw those. Those are too hard to draw. So there's a sphere system. There's a graph. What about the marking? Well, I'm going to identify, once and for all, the free group with the fundamental group of this manifold, mn. Here's mn. That's half of mn, anyway. Now that I've got a free group identified with mn, I now know how to identify the fundamental group of my graph with fn. Every loop I can see, well, that's a loop in mn. So it's some loop, some word in fn, some element of the fundamental group of fn. So this marks every the graph. I've got a complete system. And I'm identifying fn with the fundamental group. I'm letting g of s be the dual graph. So I've got g of s is embedded inside mn. And the embedding marks g of s. So what do I need for a marking? I need an isomorphism between fn to g of s. If you like, I could pick a maximal tree, draw a picture. I shouldn't have made the sphere screen. Picks a maximal tree. Then this loop is some element in the fundamental group of fn. So I call that u. This loop is some element in the fundamental group of fn. I call that v. And this loop is some element in the fundamental group of fn. I call that w. And now I have a marked graph the way I've drawn it. u, v, and w is a basis for the free group. So that's how to get a marked graph given a sphere system. What about the other way around? Supposing I have a marked graph. How do I get a sphere system? Yeah? What about the metric? What about the metric? Good question. I forgot to say that. So this is a complete system. So the simplex, very good point. Simplex corresponding to s. Well, so that's the simplex. The vertices are s1 up to sk. So I think of that as I think of a point in that simplex. It has barycentric coordinates. A point in the simplex. So I think of this is sitting inside of r to the k. The same picture. That's the simplex. And this point has some coordinates, s1, s2, s3, the sum of w i, s i. So I can locate this point in the simplex precisely by giving these numbers, w i, by giving the sphere system and certain weights to the spheres. So w i is weights on the spheres. A point in what do I call it, sigma of s. Did I give it a name? In the simplex corresponding to s is a weighted sphere system. So if I want to think of this green system as a point in the space instead of the whole simplex, then I've got some weights, w1, w2, w3, w4. And the sum of those weights is 1. So I think of those. I also have an edge for every sphere. So I put the dual graph. Edge lengths are given by weights on the s i. Good point. So what comes next? Point is weighted sphere system. Oh, yeah, right. So I've shown you how to, given a sphere system, how to get a marked graph. And what about the other way? Supposing I have a marked graph, how do I get a sphere system? Well, here's my graph. I won't draw the marking just yet. What I'm going to do is put a dot in the middle of each edge. So here's x1, x2, x3. Now I'm going to take this graph and make it fat. So I'm just going to thicken it up to make a handle body. And all of these dots will become disks. And when I double it, then I'm going to double this handle body. And I'll get something that's homeomorphic to m. I've got a handle body. It's got genus n. And when I double it, I'll get something that's homeomorphic to mn. It's a doubled handle body. So there's a homeomorphism. Actually, there's lots of homeomorphisms between this and my standard mn. So I can pull back this sphere system. h inverse of s is a sphere system in m. And it's got the dual graph here is the graph I started with. So when I pull it back, the dual graph over here will be the graph I started with. Unfortunately, this graph might not have the right marking. I mean, this sphere system might not be the one with the correct marking. I started with a particular marking. I want that marking. So the dual graph g, which is conveniently drawn on the left-hand side here. So now I've pulled back some system. I've got a dual graph that's isomorphic to the graph that I started with. But as I said, that might not be the right marking. But it does have some marking. So h inverse of s corresponds to gf. f is some isomorphism between the free group and pi 1 of g. It's not the right one. Well, I'll just change it until it's the right one. This is f. And I really wanted g. So what do I do? I, let me see. There's a map here so that if I do g followed by g inverse f, if I put f inverse g here, then the composition is f. And I know that I can model this. This is an automorphism of fn. I can model this by a homeomorphism of mn. So something fg. Let me just call it the same name. So I started with this sphere system over here. I pulled it back over here and got some horribly messed up sphere system. I wasn't happy with that sphere system because it had the wrong marking. But that's OK. I can just change the marking to any other marking I want by performing another homeomorphism, which will mess it up even further. But it will still give me a sphere system whose dual graph is homeomorphic to g. And it will give me the right marking. So I take my original system. I pull it back here. This was h inverse of s. And then I push it forward down here, or pull it back, I guess, in this case, to fg inverse h inverse of s. And now I have a sphere system in mn that gives me the right marking. So it's a little more complicated than the other direction. But it all works. OK. So that's, do I need to say anything else about? Oh, yeah. I do need to say something else about this. All right. Please give me an explanation about h again. So I just chose an arbitrary homeomorphism h. I know that this thing is homeomorphic to, I mean, this is a handle body. And when I double it, I get something homeomorphic to my mn, my standard fixed mn. So I choose a homeomorphism. Maybe I can't see properly. From here, it looks like one side has the list of the two sides of genus. So it might just not work here properly. Oh, yes. Yes. OK. I do the picture wrong. This should only have two bumps. Yes. Sorry. Yeah. OK. This is a handle body of genus 2. When I fatten up that graphic at a handle body of genus 2, it's homeomorphic. Yes, to a handle body of genus 2, a doubled handle body of genus 2. OK. I can't see actually who is asking that question. Yeah. OK. No, this is good. So when I teach calculus courses, I tell my students that all the typos I make on the board are intentional. It's for pedagogical purposes. But yeah, I'm not sure they believe me. Calculus students are very demanding. OK. Sphere systems. Oh, yeah. Yeah, yeah. So what's the action of out of Fn? That's a little bit tricky. I still have pictures of Dane twists up there. That's good. Recall, we have this short exact sequence, pi not diff, or homeo, I guess I called it, of Mn, maps to out of Fn. And the kernel, what was I calling it, dt maybe, is a small finite group generated by Dane twists in two spheres. OK. So it's obvious that this group acts on the sphere complex. You have a sphere system. You apply a homeomorphism. You get another sphere system. In order to get an action of this quotient group on the sphere complex, I have to show that the kernel acts trivially. So this is the claim that a Dane twist acts trivially on S of Mn. So then this is where, if you think, if you thought about this picture, it's one reason I wanted to use this picture. Here's the idea. Supposing you're doing a Dane twist in some sphere. Let's make it. There's my Dane twist sphere. Maybe I need a bigger picture. Yeah, here, bigger picture. I'm doing a Dane twist in this sphere. And I've got some sphere system in my manifold. So I claim I can isotop the sphere to be transverse to the sphere I'm doing the Dane twist in. So S equals S1, SK, a sphere system. I can make the SI transverse to my sphere, to the twisting sphere. So that means that pieces of these spheres here will kind of come, shoot through here, and intersect basically in tubes. And then shoot on and do something else and maybe come back and intersect here in another tube, somewhere else, et cetera. So what happens when I do a Dane twist? Let me just simplify this by doing one picture. I have this tube coming through from the outside to the inside. And when I do a Dane twist, what happens to it? Well, it wraps all the way around and comes back to the other side. Should have practiced drawing this picture. That's what happens to that tube. Well, nothing really happened to that tube up to isotopy, because I can just take this loop and lift it up over the inside sphere, back down. I claim that that's isotopic, the new tube. So this is a new tube. That's the image of the tube. It is isotopic to the original tube. So in fact, up to isotopy, this Dane twist didn't do anything at all to this sphere system. So that is the idea of why this DT acts trivially on S of M. So I do get an action of the quotient group out of FN on the sphere complex. I can see if the tube only passes through once, why that's true. It passes through multiple times. How do you know they don't get together in some structure abstract way? This is the idea of the proof. OK, no, it gets more complicated. So these are basic ideas that go back to Francois Bodenbach. And yeah, well, he proved lots of nice things about spheres in three manifolds. So if you're interested, I recommend you go back and look at his papers. Right, so anyway, so I think we now have three definitions of this space. Definition one, two, and three. The theorem, of course, is that any version of the space you care to use is contractible. And out of FN acts properly. In the exercises, you'll check that. And the fact that it's proper, you won't check that it's contractible. That would take a while. So there are proofs. So the proofs, the original proof was due to Mark Culler and myself in terms of graphs and Morse theory, the combinatorial version of Morse theory. I don't say combinatorial Morse theory, because that has a meaning that is a slightly variation on that. Using graphs, the second version using trees. I should put scorer, using trees. And actually, he's used folding paths. He never published this proof. But Giroudon-Levit generalized it vastly, and they have a published proof. So the idea is basically what I showed you when I gave you that little sketch that the graph version was connected using Staling's folds. Except if you lift these folds to the universal cover, you're folding in the tree in infinitely many spots at once. And scorer defined a canonical version. But the problem, as I pointed out, is there's lots of paths between two points. There's lots of ways to fold to get from your graph to the identity. So he gave a canonical method of finding a folding path, and then you can just retract the space along the folding paths. And the third proof, using sphere systems, is due to hatcher. And if you're familiar with the proof that the arc complex on a surface with boundary is contractable, the nicest argument for that is also due to hatcher, and it's the same argument. It uses surgery paths. OK, so I'm not going to present any of these proofs. However, I should mention that I've actually given basic courses about outer space in the last few years. And there are notes in open math notes, notes from my course. And it does the sphere complex proof. Everybody know about open math notes? Everybody know about open math notes? It's a good thing to know about. This is the AMS has a site, which people post notes that they're not quite ready to publish as books or something, but they've given a course and handwritten notes or something like that, or informal notes. They post them on open math notes before they make them into something publishable, or they may never make them into something publishable. OK, so that's right. Let me see. What have I got? 11? Where did I start? I've got 10 minutes. OK, so I just want to, at the beginning of the class, I mentioned the spine and the boardification. I'm not going to obviously get to that today. But I do want to mention, especially the sphere complex point of view gives you a very nice way of thinking a little bit more generally about these groups. Supposing we take, so I've been talking all day today about out of FN. What if I was actually interested in out of FN? Well, in the graph picture, I could have insisted that all of my graphs had base points. My markings, I don't want to think of them now as maps of the isomorphisms with the free group. I want to think of them as homotopy equivalences of a rose that send the base point to the base point. I want everything to preserve base points. Yeah, but there's an easy way to fit this all in the sphere complex picture. If I take MN and I take out, well, a three ball. Here's this tiny little three ball, B3. Now I make all my sphere systems miss B3. MN might, let me call this MN1, MN1. So I get a sphere complex. In terms of graphs, if I have a sphere system, then I've got a dual graph. But this little component, this little ball, lives in one of the components. And I can think of that as telling me where to put the base point. I just draw a little edge, or I could think of putting the base point there at the end of a leaf, which is the best thing to do actually. So if I do this, then I get a lot of FN, now acts. And I get the same. So one nice thing about this proof of hatchers is that his proof, using these surgery paths, works perfectly well if you have a puncture. The same proof shows S of MN1 is contractible. Of course, in my outer space, I took out the stuff that was infinity at infinity, which I might call ONS. So I'm doing exactly the same thing I did over here. I'm taking the sphere complex except manifold instead of just being a double-handle body is a double-handle body minus a ball. And I'm throwing out sphere systems whose complement is not simply connected. And so I get a subspace of this sphere complex, which I called outer space. I should say that hatchers proof number three also shows the entire sphere complex is contractible. So I get this contractible sphere complex that's contractible subspace, which works out of FNX on this and it acts properly on ONS. So I get a proper action on a contractible space, which is what I wanted. And then, so that was nice, I threw out a ball and I got the automorphism group. And I could think of that as a space, if I liked, of a homeotopy of grafts with leaves, marked grafts with leaves with a single leaf. There's no reason to stop there. I could also throw out a bunch of balls. If I throw out S3 balls, I get something called MNS. I can look at sphere systems in MNS. I get a sphere complex. And what's the group? Well, it's not out of FN anymore. It's some group, which we call ANS. And what is it? Yeah, it's pi not homeos of MNS. And actually, I want it to fix the boundary. And the kernel by Dane Twist is the group I'm calling ANS. So we can't recall this because I'm just defining it. So in terms of grafts, so now I have a whole bunch of balls that I've taken out. Let me draw the dual graft in a different color than the, that was my dual graft. And I have a bunch of balls in there. I wanted them all to be in possibly different components of my manifold so I can draw a little leaf to each one. This one's also in that component. So dual to a sphere system is a graft with leaves. So for grafts, it was easier to describe out of FN in terms of homotopy equivalences of a graft than it was in terms of homeomorphisms of this manifold because of this annoying Dane Twist group. It's also true that you can describe this group in terms of grafts with leaves much easier. ANS is homotopy equivalences of, well, let's take a graft with leaves. Here's a simple graft with leaves. S leaves and n loops call this RNS. So I look at homotopy equivalences of R and S and I probably want to fix the univalent vertices. So I'll call this the boundary. So what I'm doing here should look very familiar. If you're familiar with mapping class groups, so out of FN you might think of as the mapping class group of the surface analog. Graph is the analog of the surface. But there's also mapping class groups of punctured surfaces where you have S punctures on your surface and the mapping class group is another group which is interesting and closely related to the mapping class group of an unpunctured surface. And so what you might think of what I'm doing here is thinking of a graft with leaves as the analog of a surface with punctures. And so I'm talking about homotopy equivalences of grafts with leaves instead of grafts without leaves. And I've got S leaves. So it turns out just in the case of mapping class groups that it's very convenient to think about these groups at the same time you're thinking about automorphisms of free groups. When you try to prove things about out of FN, even if you're only interested in out of FN, it turns out that these groups just keep inserting themselves. So when you try to prove things like homological stability they show up. When you try to prove that there's some sort of a duality between homology and co-homology these groups they show up. When you try to calculate the Euler characteristic of the quotient space they turn up all over the place. So sometimes I just start with these groups right off the bat but this time I didn't. So I think that's all for this time. Next time what I'm going to do is tell you what the spine is. It's very easy now. That will literally take five minutes and then show you how to use the spine to do stuff.