Research Catchup

I’m debating about having this web space be a regular log of whatever mathematics I look at on a given day. Part of the goal will be to jot down notes on my research. I don’t know why that’s a goal, what it will accomplish, but nevertheless… Anyway, if that’s going to be the case, I thought I should get some notation going.

So, I have some understanding of $\Sigma^{\infty} C(m,V)$, the (suspension spectrum of the) space of configuration of $m$ points in a (real) vector space $V$ (with inner product). Namely, I can write it as

$\textrm{holim}_{\rho\in P} \Sigma^{\infty}(\hom(m,V)-\hom(m/\rho,V))$

Here $\hom$ just means set functions from the set $m=\{1,\ldots,m\}$ to $V$. The category $P$ that the homotopy limit is taken over is the poset of set partitions of $m$ (besides the “discrete” partition, where nothing is related to anything else, besides itself). With this in mind, the notation $\hom(m/\rho,V)$ stands for maps from $m$ to $V$ that factor through $\rho$. For shorthand, I will let $\textrm{nlc}(\rho,V)$ be the set of maps $m\to V$ that do not factor through $\rho$ (that is, $\{f|\exists x\equiv y\mod \rho\ni f(x)\neq f(y)\}$). Here “nlc” stands for “non-locally constant” (a notational choice that might make more sense later, read on). Thus, I write

$\Sigma^{\infty} C(m,V)\simeq \textrm{holim}_{\rho\in P} \Sigma^{\infty} \textrm{nlc}(\rho,V)$

I also have some understanding of $\Sigma^{\infty} \textrm{Mor}(\mathbb{R}^n,V)$, the (suspension spectrum of the) space of linear inclusions. It can be written tantalizingly similarly, as

$\textrm{holim}_{0\neq E\leq \mathbb{R}^n} \Sigma^{\infty}(\hom(\mathbb{R}^n,V)-\hom(\mathbb{R}^n/E,V)$

Here $\hom$ stands for linear maps, and we are picking out the linear maps that are non-constant on linear subspaces $E$ of $\mathbb{R}^n$. We might as well write this as $\textrm{nlc}(E,V)$, to parallel our earlier notation.

The idea of my research is to blend these two approaches and be able to understand $\textrm{Emb}(m\times \mathbb{R}^n,V)$ (well, its suspension spectrum…). I have one sort of easy way to understand it, as a product, but the understanding is not natural enough (in a strict sense). What I’m currently working on is the more natural understanding.

Given a disjoint union of affine spaces, say $A=\coprod_{i\in s} A_i$, let us say that an equivalence relation $\sim$ on $A$ is locally affine if there are vector spaces $E_i\leq V(A_i)$ ($V(A_i)$ is the vector space underlying the affine space $A_i$) such that $\sim$ factors through $A/E=\coprod_i A_i/E_i$. That is to say, if $x\sim y$ ($x\neq y$) and $x,y\in A_i$, then whenever $a,b\in A_i$ have $a-b\in \langle x-y\rangle$ (that is, the line $ab$ is parallel to the line $xy$), then $a\sim b$. We say that a locally affine partition is complete if each $E_i=V(A_i)$ (that is, the partition factors through the underlying set $s$). Thus, complete partitions are in 1-1 correspondence with set partitions of the underlying set $s$. As another bit of notation, for any equivalence relation $\rho$ on $A$, there is a unique finest coarsening of $\rho$ that is locally affine, and we denote this locally affine coarsening by $\overline{\rho}$.

Now, we make a category, $\mathscr{C}$, of complete locally affine partitions. The objects are tuples $\rho=(\rho,A_{\rho},s,\Lambda)$ where $A_{\rho}=\coprod_{i\in s}A_i$, $\Lambda$ is a set partition of the finite set $s$, and $\rho$ is the complete partition corresponding to the set partition $\Lambda$. This notation is overdetermined, of course. What you need to determine an object is… a finite set $s$ with a partition $\Lambda$ and a dimension function $d:s\rightarrow \mathbb{N}_0$. Intuitively, then, $A_i$ will be $\mathbb{R}^{d(i)}$. Maps $\rho\to \rho'$ in this category will be locally affine maps $f:A_{\rho}\to A_{\rho'}$ such that $\overline{f(\rho)}\leq \rho'$. Perhaps a word on the notation… Given any old set partition, and a function out of that set, you can take the image of the partition. This gives you a relation (not necessarily transitive) on the target set, and you then take it’s transitive closure to get an equivalence relation on the target. For us, we then take the locally affine coarsening of this image relation. The inequality $\leq$ means that $\overline{f(\rho)}$ is a coarser partition than $\rho'$. This is equivalent to saying that the underlying set partition for $\overline{f(\rho)}$ is coarser than the underlying set partition for $\rho'$.

Let $M=M_{m,n}=m\times \mathbb{R}^n=\coprod_m \mathbb{R}^n$. (So, as a reminder, my overall goal is to understand $\textrm{Emb}(M,V)$.)

Next, we get a (contravariant) functor $\textrm{nlc}(-,M)$ from this category $\mathscr{C}$ to spaces. Intuitively, given a $\rho\in \mathscr{C}$, $\textrm{nlc}(\rho,M)$ is the subspace of maps $A_{\rho}\to M$ that are affine on each component, and do not factor through the relation $\rho$ (as in our original two cases). Using this functor, we get the Grothendieck category (I think that’s the right word for it), which I denote by $\mathscr{C}_M=\mathscr{C}\ltimes \textrm{nlc}(-,M)$. Objects are pairs, $(\rho,f)$ where $\rho\in \mathscr{C}$, and $f\in \textrm{nlc}(\rho,M)$. Morphisms… are what you’d guess (being careful about the contravariance of $\textrm{nlc}$).

We also have a functor $\textrm{nlc}(-,V)$ from $\mathscr{C}$ to spaces (it’s the same thing as above, essentially). By factoring first through the projection $\mathscr{C}_M\to \mathscr{C}$ we consider $\textrm{nlc}(-,V)$ as a functor from $\mathscr{C}_M$ to spaces. It is now a standard (or so) result that $\lim_{\mathscr{C}_M}\textrm{nlc}(-,V)$ is equivalent to $\textrm{Nat}_{\mathscr{C}}(\textrm{nlc}(-,M),\textrm{nlc}(-,V))$. Of course, being after (stable) homotopy types, I’m supposed to use $\textrm{holim}$ instead of the normal limit, and end up with what my advisor calls the space of “homotopy natural transformations”. I’m also supposed to stick some $\Sigma^{\infty}$s in there, to get an equivalence with $\Sigma^{\infty} \textrm{Emb}(M,V)$.

Of course, I’m only hopeful that I will get an equivalence. I have not yet shown it’s true. In the above, some details were swept under the rug (surprise, surprise). For example, nearly all of the categories above were “topological”. Meaning, I have a space of objects, and a space of morphisms, and various maps… I’ve got a category object (pair?) in the category of spaces. And so “functor” is something you have to be a little more careful with defining, and then “holim” as well. Nevertheless… consider yourself mostly caught up. If nothing else, on the notation.

More to come? Until then, you can entertain yourself with my advisor’s work, that I am hugely indebted to. It’s on the arxiv.