 We can take co-limit over subspaces of an infinite dimensional vector space. So the left-hand side was then called just cd, topologically in rest subcategory. And the right-hand side, well, it is what it is. There's a name this often is given. Loops infinity minus 1, mtod, but it's just defined to be that thing. Just a couple words about the definition or the notation. This sequence of pointed spaces, td, v, forms what's called a spectrum. And that's what this notation, mtod, means. If you pre-compose with direct summing with r, that's the suspension of that spectrum. And then there's operation taking v for loop space and take co-limit. It's called loops infinity, and this minus 1 is also known as suspension. Anyway, the object I want to investigate in this lecture is the morphism spaces of this category. So I mentioned lecture two, I think, that in this co-limit these morphism spaces become something fairly geometric. They have a bunch of components. One for each diffeomorphism class of co-bordism from there to there. And each component is a classifying space for the diffeomorphism group real boundary of that co-bordism. So that's something you might be interested in, whether or not you're interested in co-bordism categories of field theories. It's a classifying space for manifold bundles. Co-homology of it will be characteristic classes of manifold bundles, so on. It means that w has some boundary, and it's supposed to be the symbol for boundary. And it means diffeomorphisms with point-wise fixed on the boundary. Slightly better to say near the boundary, but up to homotopy is the same. So that's the object I'm interested in. And I want to read this weak equivalence as kind of calculating this side in terms of this side. So a priori, but the individual morphism spaces, maybe I'm even interested in path component of a morphism space. So each diffeomorph w is a path component of a particular morphism space CD. Extracting information about a path component in a morphism space is a slightly awkward thing to do from knowing something about the classifying space. So in general, if I have a topologically enriched category, the best you can say is that there is a map from hom x, y, that's a space, to the space of paths starting at x and ending at y. So x, y are in the object space of c, which is n0 of c, which sets inside bc. Omega is like loop space, but it's instead a path starting at x and ending at y. This is again just by construction of bc. A morphism from x to y gives you by definition a one simplex. So this is a continuous map. Okay, so that's kind of a universal thing we always have. So this may or may not be an interesting map. The case where, okay, let's call it something, alpha. Maybe it's a weak equivalent. It's a kind of standard fact that this is a weak equivalent for all x and y, if and only if hc is a groupoid. If this topological groupoid has this property, then we can opt to homotopy reconstruct the morphism spaces from knowing the classifying space. If hc is not a groupoid, we shouldn't expect any particularly good relation. So, okay, so otherwise it's kind of awkward to reconstruct anything about these spaces out of knowing the homotopy type of the classifying space. Nevertheless, we still have the map. So for the cobaltism category, d-dimensional cobaltism category, if I pick two closed d-1 manifolds inside this harm space, I have a component that has the homotopy type of classifying space for smooth bundles. And I have this map and instance of this alpha. And this theorem, which is proof I haven't talked about, tells me the homotopy type of this classifying space. So this becomes the co-limit. Well, okay, either this is empty. Maybe these are not bordant. Maybe I'll set that way. This is either empty if m0 are not bordant m1. And otherwise, it's the loop space of what I wrote over there. So that's called omega infinity, m-t-o-d. Which is the co-limit, omega v, t-d, comma b. The reason this is a potentially interesting thing to do is that this right-hand side is a priori a lot easier to understand. So this is kind of a... I'll take the point of view that this is understandable. For example, the rational cohomology. Okay, it might not be connected, but the rational cohomology of a component is a polynomial ring. It's the free symmetric algebra on cohomology of BOD, I think. Something like that. And if we have cohomology classes here, as long as we have a map, we can at least pull them back. Yes, the only... So if you have a loop space, all the path components are homotopy equivalent. So that's probably a little bit surprising. So the only dependence... The homotopy type of this is either empty if you're taking paths between two things or taking a path between them. Otherwise, they're all homotopy equivalent, independent of M0 and M1. Yes. And therefore this answer is independent of that. So I get cohomology classes in cohomology of BDEV boundary of W, just by pulling back. Okay, maybe I should say this is for D, even. And that's a similar answer if D is odd. And this means, yeah, free graded commutative algebra. Okay, so there's no reason to expect this to be an equivalence. This is, let's say, far from an equivalence. Far from weak equivalence, because the keyboardism category is very far from being a groupoid. But you can still say it's still kind of a surprisingly good map. So that's what I want to talk about the rest of this lecture. Not exactly if you take the full keyboardism category, but there are some subcategories with some remarkable properties that in the notes I call CDK. And K here is a number. It's at least negative one. And okay, it can be anything, but it's not interesting if it's bigger than D. It's the subcategory with the same object set, but fewer morphisms. So the morphisms from M0 to M1 are the morphisms in the morphisms here that satisfy that the inclusion of M1 into W is K-connected. Inclusion of outgoing boundary is a K-connected map. Induced map on homotopic groups is search-active on pi K and bi-jective on pi less than K. Okay, then you first check that it is a subcategory, that this is preserved under composition. Put that on the exercise sheets. So for K equals one, for example, K equals zero. Let's take that. And just so I can draw it, let's take D to be two. So the keyboardisms are surfaces. The objects are one manifolds. So some number of circles. K equals zero saying that the inclusion of the outgoing boundary should be zero-connected. Zero-connected means induced projection on pi zero, which, if you say it in words, is just saying that each path component of W should touch the outgoing boundary. So this kind of thing is not allowed because this element represents something in pi zero that does not come from pi zero of M1. So you can have this and you also cannot have this kind of thing. But that's what that condition says. Sometimes called positive boundary subcategory. Yeah, and if K is equal to one, the inclusion should be bijection on pi zero and a surjection on pi one and so on. Somehow, this is, well, maybe, I think, find it a little bit surprising how useful this subcategory is. It's very useful for this question to get information about morphism spaces. So it could be that the particular W you're interested in has this property of being K-connected relative to the outgoing boundary. So then I could use this restrictive bortism category. Let me continue talking a little bit about this case, but I'll say things that kind of apply more generally. That's kind of remarkable property. Or, well, there's something that's possible in this category that's not possible in the full kabordism category. Okay, so let's first say C2 zero of anything, comma. So I write S1 for some particular, there's sort of many things. I pick one of them. So if the outgoing bound is connected, pi zero is a single element, and then this map being subjective is just saying that W should also be connected. Well, we have a good classification theorem for connected surfaces. So say, okay, so M zero is a one manifold, so it's some number of, up to diffeomorphism. It's like some number of circles. So W, if it's connected, so you could take a pair of pants with K-many legs, and then you could connect some of that with G-many carbide of S1 cross S1. That's what classification of orientable surfaces tells you that it has to be diffeomorphic to that if it's orientable, and otherwise it's this connect some G-many carbide of RP2. So that's how many components this space has, and then each component is a classifying space for bundles of this type with fixing the boundary. Right, in the full kabordism category, this would be some, like, classification of surfaces, but this is a really classification of connected surfaces. If you want to write a list of all surfaces, there's some combinatorics of how do you, what do the components look like. So this is one thing. Now, this space is a functor. Two, one, M zero, M one is a functor. It's a space. It's also a functor of M one in C two one, and M zero in C two, oops, it's a contour and functor of the first entry. I want to take a very particular co-limit as in M, in the outgoing boundary, which I take to be a circle. So pick amorphism, whose defiomorphism type is zero one, cross S one, connect some RP two. Think of that as amorphism from this copy of S one to itself. So choose an element in C two zero, S one comma S one, defi-o two, this thing, RP two with two disks removed. So that's an endomorphism here. Composing with that gives a map from this space to itself, just because it's a category, and then do that infinitely many times. I haven't told you why I'm doing all this, but let's just do it and see what happens. Composing with this thing always brings you from this kind of component into that kind of component. Somebody mind closing the door. So gluing non-orientable to a connected orientable makes it non-orientable. So if I do that infinitely many times, I can just ignore these things. Let's take homotopical limit, this map's orientable, path components, C two zero, M zero, circle to a non-orientable, does that immediately, and on these components it increases G. I've mapped the component of the mapping space with G many copies of RP two to the component where we have G plus one many copies of RP two. So that's if I do it one time. If I take the direct limit, first of all these just become irrelevant, and all of these start looking the same. I just get Z many copies of the same space, co-limit over gluing this thing infinitely many times. Either take homotopic co-limit on the space level or take co-limit of homology, C two zero, M zero comma S one. This is canonically the same as... Okay, here I have this G here is in the natural numbers, but if you take co-limit over increasing it, it kind of becomes Z. We get Z many copies, and all these components become the same. They all become some sort of infinite genus surface. Yep, right. Well, you need the boundary in order to have the map. So this complicated space with... Okay, it's not that complicated, but the set of path components are sort of two copies of the natural numbers, and each component looks different. It has to do with either genus G surface or kind of this sort of non-orientable surface. In the limit, instead, I get Z many path components, and all of them look the same. So by this, I just mean... Yeah, okay, co-limit. Yep, so you can rewrite one torus connects some RP2 is diffeomorphic to three copies of RP2. Yeah, that's why this direct system maps, in each step it maps this type of component into that type of component, and if you keep doing that, you can just skip them to begin with by how co-limits work. Well, okay. So... Okay, I lost some information. I might have been interested in bdif of this type of surface, and I have some infinite genus thing instead. But maybe sometimes it's good to lose some information if you can then say something more. So... Okay, that's one thing. It's still a functor of M0. If you have k many things, you can have... You can glue some... ...cobotism-oriented or not. Stick that to this thing. So M0 goes to this. It's a functor that... Well, it does something. It likely kind of changes these z-components around, maybe. If what you're gluing has some genus or some oil characteristic, it might change the number of boundary components. The... So at this point, we want to use a theorem of a consequence of a theorem of Nasser Leval called homological stability, one-orientable surfaces. That implies that this functor sends all morphisms to isomorphisms. This functor, any glue in d20, one isomorphism. To protect co-limit of this homology, I think should be surprising. I mean, by now there are many such theorems, but a priori this has sort of no right to be true. Nevertheless, that's true. So I guess you could phrase this as being an invertible field theory. You call this maybe a representable field theory because it's harm into an object. But I don't know that that point of view is helpful at this point. Anyway, this is true. And the z is what's left over of the Euler characteristic of the morphisms. Yes, so I start out with something that has multiple components, and I connect some with rp2. That maps some components into some other components, but it does not glue all the components together. Not the z, but it's another theorem, which I was secretly involved in saying this, that this is also... But that's kind of a two-dimensional coincidence. This is also B of the mapping class group, that surface. Yeah, so you could say that. Yes, I mean, that's okay. Okay, maybe I'm not... Maybe I should have said several theorems. There's a theorem of Gromain that this is true, and Nassel's theorem is actually about group homology of mapping class groups. Yeah, so I guess that's why I said it's a consequence of what she actually proves. Yeah, what she actually proves is about group homology of mapping class groups, and she also proves something in kind of finite Euler characteristic, that if you just connect some one copy of rp2, that's a homologized morphism in a range. But after gluing these infinitely many copies of rp2, her theorem implies that gluing anything else induces a nice morphism on homology.