4/6 Outer Automorphisms of Free Groups
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AutomorphismFree groupGroup actionInterior (topology)GeometryLocal GroupTheoryMusical ensembleGraph (mathematics)CurveManifoldTheory of relativityNetwork topologyDescriptive statisticsSet theoryFinitismusEquivalence relationDoubling the cubeN-sphereComplex (psychology)SphereMetric systemPhysical systemSimplex algorithmSimplicial complexTerm (mathematics)ThetafunktionSocial classOpen setFilm editingSurfaceGraph (mathematics)HandlebodyDifferent (Kate Ryan album)Multiplication sign2 (number)SpacetimeRight angleGoodness of fitVertex (graph theory)Lecture/Conference
07:35
ManifoldNumerical analysisEinbettung <Mathematik>SpherePhysical systemSimplicial complexComplex (psychology)SummierbarkeitLecture/Conference
09:44
Numerical analysisOrder (biology)N-sphereSphereMereologyPhysical systemPolar coordinate systemFilm editingRight angleOpen setBoundary value problemLecture/Conference
13:01
Graph (mathematics)Numerical analysisSet theoryPhase transitionInfinityComplex (psychology)Complex numberSpherePhysical systemSimplex algorithmTerm (mathematics)Complete metric spaceSquare numberOpen setHomöomorphismusCorrespondence (mathematics)Graph coloringGraph (mathematics)Vertex (graph theory)Boundary value problemSpacetimeRight angleFlow separationMaxima and minimaConnectivity (graph theory)Lecture/Conference
20:20
Graph (mathematics)ManifoldFree groupFundamentalgruppeSpherePhysical systemConnectivity (graph theory)Lecture/Conference
21:13
Graph (mathematics)Einbettung <Mathematik>Free groupFundamentalgruppeLoop (music)Physical systemComplete metric spaceElement (mathematics)Lecture/Conference
22:18
Graph (mathematics)Network topologyFundamentalgruppeFundamental theorem of algebraSphereLoop (music)Identical particlesIsomorphieklasseElement (mathematics)Numerical analysisDuality (mathematics)Free groupPhysical systemSigma-algebraSimplex algorithmDot productGoodness of fitBasis <Mathematik>WeightSummierbarkeitPoint (geometry)HomöomorphismusHandlebodyVertex (graph theory)LengthSpacetimeCoordinate systemRight angleLecture/Conference
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Graph (mathematics)Duality (mathematics)Free groupHomomorphismusSpherePhysical systemHomöomorphismusDirection (geometry)HandlebodyIsomorphieklasseInverse elementRight angleAutomorphismProper mapCorrespondence (mathematics)Lecture/Conference
33:48
Endliche GruppeOrder (biology)Exact sequenceGroup actionHomomorphismusCalculusComplex (psychology)SpherePhysical systemStochastic kernel estimationStudent's t-testQuotient groupGoodness of fitLecture/Conference
36:43
ManifoldShooting methodSpherePhysical systemTube (container)Transverse waveLecture/Conference
37:41
MathematicsProof theoryGroup actionComplex (psychology)SphereMultiplicationPhysical systemStress (mechanics)Tube (container)Quotient groupMultiplication signManifoldLecture/Conference
40:23
Graph (mathematics)CombinatoricsMathematicsNetwork topologyProtein foldingArithmetic meanProof theoryComplex (psychology)Kontraktion <Mathematik>SphereTheoryPhysical systemTerm (mathematics)Morse-TheorieSurgeryParameter (computer programming)Open setInterior (topology)Many-sorted logicSurfaceGraph (mathematics)Identical particlesCanonical ensembleBoundary value problemSpacetimeCategory of beingTheoremCovering spacePoint (geometry)Graph coloringBounded variationLecture/Conference
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Graph (mathematics)HomotopieMathematicsGrothendieck topologyGroup actionComplex (psychology)SpherePhysical systemTerm (mathematics)Connectivity (graph theory)Point (geometry)Social classOpen setGraph (mathematics)IsomorphieklasseFisher informationEquivalence relationDuality (mathematics)Free groupLecture/Conference
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Latent class modelGraph (mathematics)HomotopieManifoldNumerical analysisEntire functionAnalogyHomologieAutomorphismProof theoryDuality (mathematics)Free groupGroup actionKlassengruppeComplex (psychology)Kontraktion <Mathematik>SpherePhysical systemTerm (mathematics)Stochastic kernel estimationSurgeryLinear subspaceConnectivity (graph theory)HomöomorphismusMany-sorted logicGraph coloringSurfaceGraph (mathematics)Characteristic polynomialVertex (graph theory)Multiplication signBoundary value problemSpacetime1 (number)LogicInfinityEquivalence relationMorphismusLoop (music)Quotient spaceStability theoryRight angleLecture/Conference
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Musical ensembleDiagram
Transcript: English(auto-generated)
00:16
Okay, so I cheated while you guys were out drinking coffee, I drew the picture that I was supposed to draw.
00:22
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 guy, the one-dimensional simplices is here. And I forgot to say that what is O n, I have all these disjoint simplices and then I've got these face relations.
01:01
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.
01:21
So that's how I make this into a space. The space of equivalence classes of Markov metric 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.
01:42
But basically I want to, if I can get a graph by collapsing some edges that's still in my space, then 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.
02:12
Picture of a marking, I could just start all over, right?
02:28
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 curve complex definition.
02:54
So remember we've got our three manifold and I kind of have two ways of picturing it.
03:00
Either as this times two or as a three sphere minus two n balls, which I think of as glued together.
03:25
Okay, so a sphere system is a set of embedded two spheres, of disjointly embedded two spheres.
04:08
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, a handle body and doubled it.
04:21
So what is this picture? I cut my double 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.
04:46
And then to get it back together, yeah, yeah. You weren't here yesterday. No, I missed your talk. You missed my talk. We spent a long time on this.
05:01
Okay, okay, okay, so then you can skip my question. Yeah, okay, good. A set of disjointly embedded two spheres, yeah, yeah. Let me just, sorry, let me see. A set of embedded two spheres, which can be isotoped to be disjoint.
05:41
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.
06:05
And Si and Sj aren't isotopic. So you've already seen a definition like this of a curve system on a surface.
06:21
It's a set of disjointly embedded, 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.
06:50
N is the sphere complex, so it's got a vertex for each sphere S.
07:10
And it's got a K-simplex for each system of K plus one spheres.
07:29
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.
08:02
And in the case of this manifold, a simply connected piece is going to be a three ball with some punctures.
08:28
So each is S3 minus a bunch of balls, some number of balls.
08:41
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 the picture and half of them in this picture because I'm going to double them to get spheres.
09:01
Or 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 pieces, so if I cut along these spheres, this is the picture I get.
09:24
This is a three ball with six things missing, six balls missing. So this is S3 minus six B3s.
09:42
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?
10:01
No, I say after I cut, I get, yeah, the cut open guy is each, yeah, the cut open guy. If I add another sphere then I cut this three ball more into two pieces. One is a three ball, a three sphere minus three balls, or in other words a three ball minus two balls, same thing.
10:30
And the other one is a three ball minus five balls. Okay, so right, and if I add more spheres I cut it up into more pieces, et cetera.
10:47
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.
11:10
So, exercise. How many spheres does it take to cut MN into simply connected pieces?
11:23
It takes at least N. Well, it's not an exercise, it's trivial.
11:40
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,
12:00
then any bigger sphere system will also cut it into simply connected pieces. So the define S, yeah, I don't want to do that. I don't want to say it like that.
12:20
So it takes at least N spheres and you can't fit more 2N minus something, three, four.
12:46
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
13:01
without 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, right, so let S infinity of M be the sub-complex spanned by simplices by incomplete systems.
13:58
Right, so if I have an incomplete system, here's an incomplete system,
14:08
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.
14:23
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.
14:40
So that means the simplices corresponding to incomplete systems actually form a sub-complex. Then, definition 3, O n is equal to the sphere complex of M minus S infinity of M.
15:15
Okay, so that's pretty short, too. That's the third definition.
15:29
So in other words, well, maybe I should have written it that way. So that is the union of open simplices corresponding to complete systems.
16:08
Yes? Is there any particular reason why you don't call it S infinity of M n? Because I forgot? Thank you. Yeah.
16:31
No, it's 2n. Exercise. How many spheres can you fit in?
16:41
What is the maximum number? I think that's on the exercise sheet.
17:05
Okay. Severe systems, yeah. So that's nice, but you might take some convincing to believe that this is the same space I defined before.
17:20
So I'm going to try to convince you in terms of graphs. Oh no, this is the one I want to use.
17:43
Okay, so in terms of graphs, I had, yeah. So what's the correspondence between definition number two and definition number three? I showed you the correspondence between definition number one and definition number two,
18:01
but let's do definition two to definition three. So let's start with a sphere system, a complete sphere system, sitting inside of M n.
18:38
So here's my sphere system, do some colors.
18:50
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.
19:01
So there's two really simple looking spheres, right? I'm drawing half of them. But if I take two spheres and connect them by a tube, that's still a sphere, yeah? So let me take a tube.
19:28
There's a picture of another sphere. Okay, it's a sphere. It's either separating or non-separating. So I can change this picture by homeomorphism to make it look like one of my easy spheres, but I won't.
19:46
So for right now, let me just draw easy spheres, okay? So let's take a sphere system. It's got to be complete, so I have to make sure that the complementary pieces are simply connected.
20:12
And now I need to, so that's the sphere system, complete sphere system. Now I need to produce a marked graph. So what I do is just take a vertex in every complementary component.
20:27
There's two complementary components here. And then I'm going to draw an edge for every sphere.
20:42
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.
21:12
Here's MN. That's half of MN anyway. Okay, now that I've got the free group identified with MN, I now know how to identify the fundamental group of my graph with FN.
21:25
Every loop I can see, well, that's a loop in MN. So it's some loop in, it's some word in FN, some element of the fundamental group of FN. So this marks every, the graph.
21:50
I got a complete system, and I'm identifying FN with the fundamental group. I'm letting G of S be the dual graph.
22:06
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? For a marking, I need an isomorphism between FN to G of S.
22:27
If you like, I could pick a maximal tree and draw a picture. I shouldn't have made the spheres green. Pick some maximal tree. Then this loop is some element in the fundamental group of FN, so I call that U.
22:42
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. UV and W is the basis for the free group.
23:02
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?
23:22
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.
23:51
Well, so that's the simplex. The vertices are S1 up to SK.
24:00
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.
24:21
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 this simplex precisely by giving these numbers, W, I,
24:44
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?
25:10
In the simplex corresponding to S is a weighted sphere system.
25:24
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 one.
25:41
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 SI.
26:11
Okay, good point.
26:23
So what was I going to, what comes next? A point is a 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?
26:41
Supposing I have a marked graph, how do I get a sphere system?
27:03
Well, here's my graph. I won't draw the marking just yet. What I'm going to do is put an edge, put a dot in the middle of each edge.
27:32
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.
27:46
And all of these dots will become disks, right? And when I double it, then I'm going to double this handle body,
28:02
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 double handle body. So there's a homeomorphism. Actually, there's lots of homeomorphisms between this and my standard MN.
28:33
Okay, so I can pull back this sphere system.
28:42
H inverse of S is a sphere system in M. And the dual graph here is the graph I started with.
29:05
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.
29:24
Right? 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.
30:00
So now I've pulled back some system.
30:03
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.
30:26
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.
30:40
This is F. And I really wanted G. So what do I do? Let me see. There's a map here so that if I do G followed by G inverse F.
31:05
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. And then I can model this by a homeomorphism of MN.
31:33
So something FG. Let me just call it the same name.
31:44
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 okay. I can just change the marking to any other marking I want by performing another homeomorphism,
32:04
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.
32:22
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.
32:47
So that's, do I need to say anything else about, oh yeah, I do need to say something else about this. Sorry, please keep explaining about H again. So I just chose an arbitrary homeomorphism H.
33:02
I know that this thing is homeomorphic to, I mean, this is the 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, it looks like two sides have a genus.
33:21
Am I just not looking at it properly? Oh, yes. Yes, okay. I drew the picture wrong. This should only have two bumps. Yes, sorry. Yeah, okay.
33:41
This is a handle body of genus 2. When I fatten up that graph, I get a handle body of genus 2. It's homeomorphic, yes, to a handle body of genus, to a doubled handle body of genus 2. Okay. I can't see actually who was asking that question.
34:01
Yeah, okay. 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.
34:25
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.
34:41
That's good. Recall, we have this short exact sequence, pi naught diff, or homeo I guess I called it, of mn maps to out of fn.
35:04
And the kernel, what was I calling it, dt maybe, is a small finite group generated by Dane twists in two spheres.
35:21
Okay. 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.
35:42
So this is the claim that a Dane twist is trivially on s of mn.
36:00
So then this is where 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.
36:24
Maybe I need a bigger picture. Yeah, here, bigger picture.
36:43
I'm doing a Dane twist in this sphere, and I've got some sphere system in my manifold. So I claim I can isotope the sphere to be transverse to the sphere I'm doing the Dane twist in.
37:02
So s equals s1, sk, the sphere system. I can make the si transverse to my sphere, to the twisting sphere.
37:27
So that means that pieces of these spheres here will kind of come shoot through here and intersect basically in tubes.
37:44
And then shoot on and do something else and maybe come back and intersect here in another tube somewhere else, etc. 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?
38:05
Well, it wraps all the way around, all the way around, and comes back the other side. I should have practiced drawing this picture.
38:29
Okay, 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.
38:46
The new tube, so this is a new tube, that's the image of the tube, is isotopic to the original tube.
39:02
So in fact, up to isotopy, the Dane twist didn't do anything at all to this sphere system. So that is the idea of why this, so dt acts trivially on s of m.
39:26
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. If it passes through multiple times, how do you know they don't get knotted together in some hard-to-extract way?
39:46
This is the idea of the proof. Okay, yeah, no, it gets more complicated. So these are basic ideas that go back to Francois Lodenbach.
40:12
Well, he proved lots of nice things about spheres in three manifolds. Yeah, so if you're interested, I recommend you go back and look at his papers.
40:25
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,
41:05
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.
41:26
That would take a while. So there are proofs. So the proofs, the original proof was due to Mark Kuller and myself in terms of graphs and Morse theory,
41:50
the combinatorial version of Morse theory.
42:09
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.
42:27
I should put Scora using trees. And actually, he's used folding paths.
42:46
He never published this proof, but Giraudin-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,
43:04
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 Scora defined sort of a canonical version,
43:23
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 kind of canonical method of finding a folding path,
43:41
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 contractible,
44:06
the nicest argument for that is also due to Hatcher, and it's the same argument. It uses surgery paths.
44:24
Okay, 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 from my course,
44:59
and it does the sphere complex proof.
45:07
Everybody know about Open Math Notes? It's a good thing to know about. 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 written handwritten notes or something like that,
45:22
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. Okay, so that's right.
45:44
Let me see, what have I got? Eleven, where did I start? I've got ten minutes? Okay, so I just want to, at the beginning of the class I mentioned the spine and the boardification.
46:00
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.
46:26
What if I was actually interested in out of FN? Well, in the graph picture I could have assumed, 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,
46:47
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.
47:04
If I take MN and I take out, well a three ball, there's this tiny little three ball, B3.
47:25
Now I make all my sphere systems miss B3.
47:40
Let me call this MN1. So I get a sphere complex.
48:04
In terms of graphs, if I have a sphere system then I've got a dual graph,
48:22
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.
48:43
So if I do this then I get a lot of FN now acts and I get the same,
49:03
so one nice thing about this proof of Hatcher's 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.
49:34
Of course in my outer space I took out the stuff that was at infinity,
49:53
which I might call ONS. So I'm doing exactly the same thing I did over here.
50:01
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.
50:22
I should say that Hatcher's proof number three also shows the entire sphere complex is contractible.
50:45
So I get this contractible sphere complex, this contractible subspace, which works out of FNX on this and it acts properly on ONS.
51:02
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 homotopy of graphs with leaves, marked graphs with leaves with a single leaf.
51:28
There's no reason to stop there. I could also throw out a bunch of balls.
51:44
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.
52:03
And what is it? Yeah, it's pi naught homeos of MNS
52:24
and actually I want it to fix the boundary. Yeah, the kernel by Dane Twists is the group I'm calling ANS.
52:48
So we can't recall this because I'm just defining it. So in terms of graphs, so now I have a whole bunch of balls that I've taken out.
53:12
Let me draw the dual graph in a different color. That was my dual graph. And I have a bunch of balls in there.
53:22
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.
53:44
So dual to a sphere system is a graph with leaves.
54:06
So for graphs it was easier to describe out of FN in terms of homotopy equivalences of a graph than it was in terms of homeomorphisms of this manifold because of this annoying Dane Twist group.
54:20
It's also true that you can describe this group in terms of graphs with leaves much easier. ANS is homotopy equivalences of, well, let's take a graph with leaves.
54:40
Here's a simple graph with leaves, S leaves and N loops. Call this RNS. So I'll look at homotopy equivalences of RNS and I probably want to fix the univalent vertices.
55:08
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.
55:23
So out of FN you might think of it as the mapping class group of a surface analog. A graph is the analog of the surface. But there's also mapping class groups of punctured surfaces, right, where you have S punctures on your surface and the mapping class group is another group,
55:43
which is interesting and closely related to the mapping class group of an unpunctured surface. And so you might think of what I'm doing here thinking of a graph with leaves as the analog of a surface with punctures. And so I'm talking about homotopy equivalences of graphs with leaves
56:05
instead of graphs 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 and at the same time you're thinking about automorphisms of free groups.
56:23
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 cohomology,
56:44
these groups, they show up. When you try to calculate the Euler characteristic of the quotient space, they turn up all over the place. Sometimes I just start with these groups right off the bat, but this time I didn't.
57:06
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.