Archive for February, 2010

A homotopy limit description of the embedding functor

February 25, 2010

(Approximate notes for a talk I gave this afternoon.)

Setup

So according to the title, I should be telling you about {\text {Emb}(M,N)}, as a functor of manifolds {M} and {N}. That’s perhaps a bit ambitious. I’ll only be thinking about {M=\coprod _{m} D^{n}}, a disjoint union of closed disks, and I’ll actually fix {M}. And instead of letting {N} range over any manifolds, it’ll just be {V}, ranging over real vector spaces.

By taking derivatives at centers, we obtain a homotopy equivalence from {\text {Emb}(M,V)} to something I’ll maybe denote {\text {AffEmb}_{0}(\coprod _{m} \mathbb {R}^{n},V)}. This is componentwise affine (linear followed by translation) maps {\coprod _{m} \mathbb {R}^{n}\to V} whose restriction to {\epsilon }-balls around the 0s is an embedding. I may use {\underline{m}=\{1,\ldots ,m\}}, and write {\underline{m}\times \mathbb {R}^{n}}. And I’ll actually send everything to spectra, instead of topological spaces, via {\Sigma ^{\infty }}.

So really I’ll be talking about

\displaystyle  \Sigma ^{\infty }\text {AffEmb}_{0}(\underline{m}\times \mathbb {R}^{n},V)

as a functor of {V}. I’ll be lazy and write it {\Sigma ^{\infty }\text {Emb}(M,V)}, having fixed an {m} and {n} to give {M=\underline{m}\times D^{n}}.

Useful Cases

The first useful case is when {M=\underline{m}} (i.e., {n=0}). Then embeddings are just configuration spaces, {F(m,V)}. I’ve talked before about a homotopy limit model in this case, but let me remind you about it.

The category I have in mind is something I’ll denote {\mathcal{P} (m)}. Objects will be non-trivial partitions of {\underline{m}}, and I’ll probably denote them {\Lambda }, perhaps writing {\Lambda \vdash \underline{m}}. Non-trivial means that some equivalence class is larger than a singleton. I’ll write \Lambda\leq \Lambda' if {\Lambda } is finer than {\Lambda'}, meaning that whenever {x\equiv y\pmod{\Lambda }}, then {x\equiv y\pmod{\Lambda '}}.

The functor I want is something I’ll denote {\text {nlc}(-,V)} and call “non-locally constant” maps. So {\text {nlc}(\Lambda ,V)} is the set (space) of maps {f:\underline{m}\to V} such that there is an {x\equiv y\pmod{\Lambda }} where {f(x)\neq f(y)}. Equivalently, maps which don’t factor through {\underline{m}/\Lambda }.

Depending on which order you make your poset {\mathcal{P} (m)} go, {\text {nlc}(-,V)} is contravariant, and you can show

\displaystyle  \Sigma ^{\infty }F(m,V)\simeq \text {holim}_{\mathcal{P} (m)}\Sigma ^{\infty }\text {nlc}(-,V).

The second useful case is when {M=D^{n}} (i.e., {m=1}). Then the space of embeddings is homotopy equivalent to the space of injective linear maps. You can obtain a homotopy limit model in this case that looks strikingly similar to the previous case. Namely, you set up a category of “linear” partitions (equivalent to modding out by a linear subspace), and take the holim of the non-locally constant maps functor, as before.

I like to think of both cases as being holims over categories of kernels, and the non-locally constant maps of some kernel are maps that are known to fail to be not-injective for some particular reason. Embeddings fail to be not-injective for every reason.

But there’s another model I want to use in what follows. My category will be denoted {\mathcal{L} (n)}, and objects in the category will be vector spaces {E} with non-zero linear maps {f:E\to \mathbb {R}^{n}}. Morphisms from {f:E\to \mathcal{R}^{n}} to f':E'\to \mathbb{R}^n will be surjective linear maps \alpha:E\to E' with f=f'\circ \alpha. You might think of the objects as an abstract partition ({E}) with a map to {\mathbb {R}^{n}}, which then determines a partition of {\mathbb {R}^{n}}, by taking the image.

The functor out of this category is something I’ll still denote {\text {nlc}(-,V)}. On an object {f:E\to \mathbb {R}^{n}} it gives all non-constant affine maps {E\to V}. Arone has shown

\displaystyle  \Sigma ^{\infty }\text {Inj}(\mathbb {R}^{n},V)\simeq \text {holim}_{\mathcal{L} (n)}\Sigma ^{\infty }\text {nlc}(-,V).

Product Structure

The space of embeddings we are considering splits, as

\displaystyle  \text {Emb}(M,V)\simeq F(m,V)\times \text {Inj}(\mathbb {R}^{n},V)^{m}.

We know a homotopy limit model of each piece of the splitting, and might hope to combine them into a homotopy limit model for the product. This can, in fact, be done, using the following:

Lemma: If {\Sigma ^{\infty }X_{i}\simeq \text {holim}_{\mathcal{C}_{i}}\Sigma ^{\infty }F_{i}}, {i=1,\ldots ,k}, then

\displaystyle  \Sigma ^{\infty }\prod _{i} X_{i} \simeq \text {holim}_{*_{i}\mathcal{C}_{i}}\Sigma ^{\infty }*_{i} F_{i}.

Here {*} denotes the join. For categories, {C*D} is {(C_{+}\times D_{+})_{-}}, obtained by adding a new initial object to each of {C} and {D}, taking the product, and removing the initial object of the result.

Proof (Sketch): Consider the case {k=2}. The idea is to line up the squares:

and

Both of which are homotopy pullbacks. The equivalence of the lower-right corners follows because join is similar enough to smash, which plays nicely with {\text {holim}}.

So, anyway, applying this lemma and perhaps cleaning things up with some homotopy equivalences, we obtain an equivalence

\displaystyle  \Sigma ^{\infty }\text {Emb}(M,V)\simeq \text {holim}_{\mathcal{P} (m)*\mathcal{L} (n)^{*m}}\Sigma ^{\infty }\text {nlc}(-,V).

Objects in the category consist of a partition {\Lambda \vdash \underline{m}} along with, for {i\in \underline{m}}, linear {f_{i}:E_{i}\to \mathbb {R}^{n}}. To tidy up a little bit, I’ll denote this category {\mathcal{J}=\mathcal{J}(M)}, for join. The functor takes an object as above and returns the set of componentwise affine maps {\coprod _{i} E_{i}\to V} such that either (a) the map is non-constant on some component, (b) when restricted to the image of {0:\underline{m}\hookrightarrow \coprod E_{i}}, the map is non-locally constant with respect to {\Lambda }.

There you have it, a homotopy limit description for the embedding functor.

But not a particularly nice one. If we had an embedding {M\hookrightarrow M'}, then we’d have map {\text {Emb}(M,V)\leftarrow \text {Emb}(M',V)}. It’d be really swell if this map was modelled by a map {\mathcal{J}(M)\to \mathcal{J}(M')} of the categories we are taking homotopy limits over. But that’s not going to happen. What can go wrong? Non-trivial partitions of {M}, when sensibly composed with the map to {M'}, may become trivial, and thus not part of the category. This is, essentially, because several components of {M} might map to a single component of {M'}. If {M} has two components, and {M'} one, say, where do you send the object consisting of the non-trivial {\Lambda \vdash \underline{2}} paired with some 0 vector spaces?

A More Natural Model

We sort of need to expand the category we take the homotopy limit over, and make it a more natural construction. We actually have an indication on how to do this from the discussion, above, in the case of linear injective maps from a single component. Perhaps we can find a proper notion of “abstract partition”, pair such a beast with a map to {M}, sensibly define non-locally constant, and get what we want. Let’s see how it goes…

An affine space is, loosely, a vector space that forgot where its 0 was. There is, up to isomorphism, one of any given dimension, just like for vector spaces; I’ll denote the one of dimension {d} by {A^{d}}, say. That should be enough of a description for now.

Let me define a Complete Affine Partition (CAP), {\rho }, to be a partition of a disjoint union of affine spaces, such that equivalence classes contain components. That is, everybody that’s in the same component is in the same equivalence class. Given a {\rho }, I’ll denote by {A(\rho )} the underlying component-wise affine space. The data that determines a {\rho } is: a finite set {s} (the set of components), a partition of {s}, and a dimension function, {d:s\to \mathbb {N}_{0}} (non-negative integers). With this information, {A(\rho )} is {\coprod _{i\in s}A^{d(i)}}.

By a refinement {\alpha } from {\rho } to {\rho'}, denoted {\alpha :\rho \to \rho'}, I will mean an affine map {\alpha :A(\rho )\to A(\rho')} so that the “affine closure” of the partition {\alpha (\rho )} is coarser than {\rho'}. I don’t want to spend too much time talking about the affine closure operation, on partitions of a component-wise affine space. If {\rho } and {\rho'} have a single component, a refinement is just a surjective affine map (recall before we had surjective linear maps in {\mathcal{L} (n)}). If {\rho } and {\rho'} have dimension function 0, so basically {\rho =\Lambda } and {\rho'=\Lambda'} (partition of possibly distinct finite sets), a refinement just means {\alpha (\Lambda ) \geq \Lambda'}.

We’re now ready to define a category, which I’ll denote {\mathcal{C}=\mathcal{C}(M)}. The objects will be pairs of: a CAP, {\rho }, along with a non-locally constant affine {f:A(\rho )\to M} (subsequently denoted {f:\rho \to M}). A morphism from {f:\rho \to M} to {f':\rho'\to M} will be a refinement {\alpha :\rho \to \rho'} such that {f=f'\circ \alpha }. This should look familiar to the {\mathcal{L} (n)} construction.

The functor I’ll consider still deserves to be called {\text {nlc}(-,V)}, and it takes {f:\rho \to M} to the set of non-locally constant affine maps {A(\rho )\to V}. We’d really like to be able to say

\displaystyle  \Sigma ^{\infty }\text {Emb}(M,V)\simeq \text {holim}_{\mathcal{C}(M)}\Sigma ^{\infty }\text {nlc}(-,V).

It seems sensible to try to do so by showing that

\text {holim}_{\mathcal{C}(M)}\Sigma ^{\infty }\text {nlc}(-,V)\simeq \text {holim}_{\mathcal{J}(M)}\Sigma ^{\infty }\text {nlc}(-,V),

since {\mathcal{J}(M)\subseteq \mathcal{C}(M)}, and we know the homotopy limit over {\mathcal{J}(M)} has the right homotopy type. This is our goal.

Semi-direct Product Structure

I’ll use the semi-direct product notation for the Grothendieck construction, as follows. Recall that for a category {D}, and a functor {F:D\to E}, the Grothendieck construction is a category, which I’ll denote {D\ltimes F}, whose objects are pairs {(d,x)} where {x\in F(d)}. Morphisms {(d,x)} to {(d',x')} are morphisms {h:d\to d'} such that {F(h)(x)=x'}. Of course, my functors are all contravariant as defined, so you have to mess about getting your arrows right. Best done in private.

I claim that {\mathcal{C}} can be written as a Grothendieck construction. Actually, it can in a few ways. The obvious way is to set {\mathcal{U}} to be the category of CAPs {\rho }, paired with refinements. The functor you need is then {\text {nlc}(-,M)}. You find that {\mathcal{C}=\mathcal{U}\ltimes \text {nlc}(-,M)}.

But there’s another way to slice it. Let {\mathcal{U}_{0}} be the category of CAPs {\rho }, along with functions {f_{0}:\rho \to \underline{m}}. Now the functor you need is not all non-locally constant maps to {M}, but only those that are lifts of {f_{0}}. You might denote this set {\text {nlc}_{f_{0}}(\rho ,M)}. I’m tired of all the notation, so let me let {\mu } denote this non-locally constant lifts functor. We have, then {\mathcal{C}=\mathcal{U}_{0}\ltimes \mu }.

While I’m simplifying notation, let me also write {\nu } for {\Sigma ^{\infty }\text {nlc}(-,V)}. Notice that it is actually a functor from {\mathcal{U}}, and thus from {\mathcal{U}_{0}}.

Let’s return to the category {\mathcal{J}(M)} again. It has the same structure. In fact, we just need to pick out of {\mathcal{U}_{0}} the subcategory of CAPs whose set of components is {\underline{m}}, and where {f_{0}} is the identity on {\underline{m}}. Calling this subcategory {\mathcal{R}}, we have {\mathcal{J}=\mathcal{R}\ltimes \mu }.

Summarizing all the notation, our goal is to show that

\displaystyle  \text {holim}_{\mathcal{U}_{0}\ltimes \mu }\nu \simeq \text {holim}_{\mathcal{R}\ltimes \mu }\nu .

The first thing I’d like to do is use twisted arrow categories to re-write things, so perhaps I should tell you about these categories first. If {D} is a category, the twisted arrow category, {{}^{a}D} has objects the morphisms of {D}. Morphisms from {d_{1}\to d_{2}} to {d_{1}'\to d_{2}'} are commuting squares

If {F} and {G} are contravariant functors from {D}, one can check that {(d_{1}\to d_{2})\mapsto \text {mor}(F(d_{2}),G(d_{1}))} is a covariant functor from {{}^{a}D}. I’ll denote it {G^{F}}. One can show that

\displaystyle  \text {holim}_{D\ltimes F}G\simeq \text {holim}_{{}^{a}D}G^{F}.

Using this, we’re hoping to show

\displaystyle  \text {holim}_{{}^{a}\mathcal{U}_{0}}\nu ^{\mu }\simeq \text {holim}_{{}^{a}\mathcal{R}}\nu ^{\mu }.

Proof Outline

We’ve got {{}^{a}\mathcal{U}_{0}\supseteq {}^{a}\mathcal{R}}. Between them lies a category I’ll denote {\mathcal{U}_{0}\to \mathcal{R}}, consisting of arrows {u\to r} with {u\in \mathcal{U}_{0}}, {r\in \mathcal{R}}. Morphisms are “twisted” commuting squares, as they should be, as part of the twisted arrow category. One can reduce the holim over {{}^{a}\mathcal{U}_{0}} to one over {\mathcal{U}_{0}\to \mathcal{R}}, and from there to one over {{}^{a}\mathcal{R}}.

To reduce from {\mathcal{U}_{0}\to \mathcal{R}} to {{}^{a}\mathcal{R}}, one can show that for all {u\to r\in \mathcal{U}_{0}\to \mathcal{R}}, the over-category {{}^{a}\mathcal{R}\downarrow (u\to r)} is contractible. In fact, this result seems to rely very little on our particular {\mathcal{R}} and {\mathcal{U}_{0}}, and doesn’t depend on the functors, {\mu }, {\nu }, or {\nu ^{\mu }}.

For the reduction from {{}^{a}\mathcal{U}_{0}} to {\mathcal{U}_{0}\to \mathcal{R}}, one shows that for all {u_{1}\to u_{2}\in {}^{a}\mathcal{U}_{0}}, we have

\displaystyle  \text {mor}(\mu (u_{2}),\nu (u_{1}))\stackrel{\sim }{\rightarrow } \text {holim}_{\substack {(u_1\to u_2)\to (u\to r)\\ \in (u_1\to u_2)\downarrow (\mathcal{U}_0\to \mathcal{R})}}\text {mor}(\mu (r),\nu (u)).

Essentially this shows that {\nu ^{\mu }}, as a functor from {{}^{a}U_{0}}, is equivalent to the right Kan extension of it’s restriction to a functor from {\mathcal{U}_{0}\to \mathcal{R}}. And the homotopy limit of a right Kan extension is equivalent to the homotopy limit of the restricted functor.

It is in this second reduction, {{}^{a}\mathcal{U}_{0}} to {\mathcal{U}_{0}\to \mathcal{R}}, that we rely on information about our categories and functors ({\mu }, in particular). Pick your object {u_{1}\to u_{2}\in {}^{a}\mathcal{U}_{0}}. You can quickly reduce the crazy over-category above to just {u_{2}\downarrow \mathcal{R}}. Now remember {u_{2}} is a CAP with a function to {\underline{m}}. I’ll denote it {f_{0}:u_{2}\to \underline{m}}. If this function is locally constant (all objects within an equivalence class get sent to the same point), then you sort of replace {u_{2}} with an object obtained by taking affine and direct sums of it’s components. The result is an object of {\mathcal{R}}, but from the perspective of {\mu }, the two objects give equivalent spaces of lifts. Alternatively, if {f_{0}:u_{2}\to \underline{m}} is non-locally constant, then every lift {u_{2}\to M} is non-locally constant, and so {\mu (u_{2})\simeq *}.

This all works out to be useful in the whole proof. But I’ll maybe save all that for another day.

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Approximating Functions of Spaces

February 25, 2010

The branch of mathematics known as topology is concerned with the study of shapes. Whereas shapes in geometry are fairly rigid objects, shapes in topology are much more flexible; topologists refer to them as “spaces.” If one space can be flexed and twisted and not-too-drastically mangled into another space, topology deems them to be the same. It becomes much more difficult, then, to tell if two spaces are different. A primary goal in topology is to find ways to distinguish spaces.

Another fundamental question in topology is concerned with the ways to put one space into another space – to understand the functions between spaces. Each space is a collection of points. A function from space X to space Y is a way to assign points in X to points in Y. If X is a collection of students, and Y a collection of tables, then a function from X to Y is a way to assign each student to a table. In topology, we don’t allow just any function from X to Y. While the spaces are flexible, we have to be careful not to separate points from X that are close to each other. Using the students and tables example, we might think about two students holding hands as being close. These students could be placed at the same table, or perhaps neighboring tables, but cannot be separated across the room. A function that doesn’t separate points too much is called “continuous,” and these are the types of functions topologists consider; topologists tend to call them “maps.”

It turns out that these two primary questions of topology are actually related. If one wants to determine how similar shapes X and Y are, one might begin by introducing a third space, Z, and asking about the maps from Z to X and from Z to Y. If the collection of maps are the same in both cases, one expects that X and Y are similar, at least somewhat. More information can be obtained by replacing Z by another space W, and repeating the process. Typically the spaces Z and W are fairly well-understood spaces, like circles and spheres.

Spaces, and the maps between them, can be quite complicated in general. By restricting to various types of spaces, or types of maps, one is able to make significant progress. One important class of spaces consists of what are called “manifolds.” Intuitively, a manifold is a space which, when viewed from quite close, looks flat (like a line, or a plane), and has no corners. If you were a tiny ant, walking along on a mathematician’s idealized sphere, for example, you might get the impression that you were walking on a giant sheet of paper. Indeed, a similar viewpoint of our own world was common in the not too distant past.

Circles and spheres, and lines and planes themselves, make good examples of manifolds to keep in mind. In fact, lines and planes, and the higher dimensional “Euclidean” spaces, are the fundamental building blocks for manifolds. The defining property of a manifold is that when you get close enough, you are looking at a Euclidean space. Manifolds are essentially spaces obtained by gluing together Euclidean spaces. An interesting example, known as the Möbius strip, can be modeled by taking a strip of paper, introducing a half-twist, and taping the ends together. A tiny ant crawling along on the resulting object would have a hard time noticing that it isn’t just crawling along a strip of paper.

If one’s attention is restricted to studying manifolds, instead of more general spaces, it makes sense to also restrict the types of maps under consideration. General continuous maps need not respect the information about manifolds that makes manifolds a nice class of spaces (they are reasonably “smooth”). We replace, then, all continuous maps with a more restricted class of maps which preserve the structure of manifolds. A particularly nice such class consists of those maps known as “embeddings.” An embedding will not introduce corners in manifolds, and also will not send two points to the same point (embeddings would place only one student at each table, in the earlier example).

When studying manifolds, then, a topologist may be concerned with the collection of embeddings between two manifolds. If the manifolds are called M and N, then we might denote the embeddings of M into N by E(M,N). This is then a function itself – a function of two variables, M and N. If we fix one of the variables, say we only think about M being a circle, we still have a function of one variable, and have made our study somewhat easier.

Leaving M fixed, how do the values E(M,N) change as N changes? Said another way, if we modify N slightly, what is the effect on E(M,N)? If it is difficult to find E(M,N), how can it be approximated? How can the function itself be approximated?

These questions are strikingly similar to questions asked in calculus. Given a function that takes numbers in and spits numbers out (y=e^x, for example) what happens to the output values (y) if the input value (x) is changed slightly? If we know about the value at a particular point (e^0=1), what can be said about values nearby (e^{1/2}, say)? The answers to these questions lie with the derivative, and its “higher” analogues (the derivative of the derivative, and so on). If one knows about the derivatives of a function at a point, one can create “polynomial” approximations to the function, near that point.

It turns out that something quite similar happens when studying the embedding function (and other functions like it). Some sense can be made of derivatives, polynomials, and best approximations, all in the context of functions of spaces (instead of functions of numbers).

I have been studying the embedding function, and its polynomial approximations, when M is fixed. I let M be a collection of disjoint Euclidean spaces of any dimension; so I might take M to be 3 lines and 2 planes, all separate from each other. I also restrict my attention to E(M,N) only when N itself is a Euclidean space. Since any manifold is built out of Euclidean spaces, the cases I consider are important building blocks to understanding more general embedding functions.

Previous work has already covered some of the cases I consider. If M is a finite collection of points, the collection of embeddings is called a “configuration space.” Loosely, this case covers the idea that embedding may not bring two points together, and is somewhat of a “global” situation. Another case is when M only has one piece, say a single line. Here, one is exploring more the notion that embeddings may not introduce corners, a “local” situation. In both of these cases, the best polynomial approximations for the embedding functions have been identified. Moreover, useful descriptions of the approximations have been obtained.

In the more general situation I consider, I have been interacting with both aspects of embeddings. Since my spaces, M, may have many pieces, I am involved in global aspects of embeddings. Since my M may have pieces of any dimension, I am involved in local aspects of embeddings. Unifying the description of the approximations in these two cases has been my task.

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A somewhat different, perhaps more elementary version of this is also available.