The n-Category Café

Configuration spaces and polynomials

The topological spaces we’ll consider are configuration spaces. Let $M$ be any topological space (typically a manifold) and write $\text{Conf}_n(M)$ for the space of ordered tuples of $n$ distinct points in $M$. The space $\text{Conf}_n(M)$ has a natural free action by $S_n$, the symmetric group on $n$ elements, that simply permutes the labels. The space of orbits, (the colimit of this action) is denoted by $\text{Conf}_n(M)/S_n$ and is the space of unordered tuples of $n$ distinct points in $M$.

Configuration spaces are basic mathematical objects. For example

$$\text{Conf}_n(\mathbb{C})/S_n = \text{Poly}_n(\mathbb{C})$$

is the space of monic, degree $n$, square-free polynomials. Here, the tuple of unordered points corresponds to the roots of the polynomial. The square-free condition guarantees that the roots are distinct. The spaces $\text{Poly}_n(\mathbb{C})$ are algebraic varieties. For example:

$$\text{Poly}_2(\mathbb{C}) = \{ z^2 + b z + c \mid b^2 - 4 c \neq 0\}$$ $$\text{Poly}_3(\mathbb{C}) = \{ z^3 + b z^2 + c z + d \mid b^2 c^2 - 4 c^3 - 4 b^3 d - 27 d^2 + 18 b c d \neq 0\}.$$

There are many reasons to care about the topology of these spaces. Even basic questions, such as whether $\text{Conf}_n(M)$ is connected, can have non-trivial applications to robotics. (Aside: I hereby propose “configuration” as the collective noun for a group of robots.) Another point of interest is that a loop in the space $\text{Conf}_n(\mathbb{C})/S_n$ defines a braid with $n$-strands. Indeed, the fundamental group $\pi_1(\text{Conf}_n(\mathbb{C})/S_n)$ of endpoint-preserving homotopy classes of loops is the braid group on $n$-strands.

It will be important in what follows for us to learn how to visualize this — the visualization is easiest if you replace $\mathbb{C}$ with a small disk. Picture $n$ distinct points in the plane, perhaps each in a different color, but as we’re visualizing the space of unordered configurations, you don’t have to remember which point has which color. We’ll visualize a path in this space as a “movie” in which we can watch the points move around. Each “frame” is a point in the configuration space. If we stack all the frames on top of each other in a direction orthogonal to the disk, the ambient space is a cylinder. As the colored points move around, they trace strings from the top of the cylinder to the bottom. Because the points never collide, the strings never intersect. Hence, a loop defines a braid. A loop in the space of ordered configurations is called a pure braid, meaning that each strand starts and ends in the same relative position.

The fundamental problem is to understand the topology of the spaces of configurations of ordered and unordered points. A standard way to try to do this is to compute the cohomology of the space. It’s okay if you don’t know exactly what this means. The point is that to any topological space $X$ one can define a sequence of rational vector spaces $H^i(X; \mathbb{Q})$ for each $i \geq 0$. These algebraic invariants capture a significant portion of the topological information about the space. For fixed $i$, our aim is to reduce infinitely many computations to finitely many by investigating how these vector spaces depend on $n$.

Case study: first cohomology

Let us consider this problem for $i=1$. Elements of $H^1(\text{Conf}_n(\mathbb{C});\mathbb{Q})$ are represented by homomorphisms $\phi\colon\pi_1(\text{Conf}_n(\mathbb{C})) \to \mathbb{Q}$. To define such a $\phi$, we must assign an algebraic invariant (a rational number) to each loop in $\text{Conf}_n(\mathbb{C})$ in a natural way; here this means that the invariant must be additive with respect to concatenation of loops, so that the map $\phi$ is a homomorphism.

One idea is to use the winding number. For each pair $1 \leq i,j \leq n$ with $i \neq j$, define $\omega_{ij} \colon \pi_1(\text{Conf}_n(\mathbb{C})) \to \mathbb{Q}$ by

$\omega_{ij}(\alpha) =$ winding number of $j$ around $i$ while performing $\alpha$.

Note that $\omega_{ij}=\omega_{ji}$. Benson described a beautiful intuition for this observation that he learned from his advisor, William Thurston, who always encouraged him to project himself into the space. Imagine $i$ and $j$ are dance partners holding hands. As they turn around the dance floor, $i$ performs one clockwise rotation each time $j$ does so and conversely.

Proposition. $\{\omega_{ij} \mid 1 \leq i \lt j \leq n \}$ is a linearly independent in $H^1(\text{Conf}_n(\mathbb{C}))$.

Linear independence is implied by the existence of loops that produce a non-trivial winding number for some pair $(i,j)$ while fixing the remaining points. Moreover:

Theorem (Arnol’d 1968). $H^1(\text{Conf}_n(\mathbb{C});\mathbb{Q})$ is spanned by $\{\omega_{ij} \mid 1 \leq i \lt j \leq n \}$. Hence $H^1(\text{Conf}_n(\mathbb{C});\mathbb{Q}) \cong \mathbb{Q}^{\left(\begin{array}{c} n \\ 2 \end{array}\right)}$.

This is exactly the sort of closed form we might hope for. This leads to the next question:

Question: What is $H^1(\text{Conf}_n(\mathbb{C})/S_n; \mathbb{Q}) = H^1(\text{Poly}_n(\mathbb{C});\mathbb{Q})$?

As before, we seek a numerical invariant for loops of unordered configurations. One such invariant is the total winding number

$\alpha \mapsto$ total winding number of $\alpha = \sum_{i \neq j} \omega_{ij}(\alpha).$

It turns out this is the only invariant.

Corollary. $H^1(\text{Poly}_n) \cong \mathbb{Q} = \mathbb{Q}\{$total winding number$\}$.

Proof: $H^1(\text{Poly}_n) = H^1(\text{Conf}(\mathbb{C})/S_n)$ = $S_n$-fixed vectors in $H^1(\text{Conf}_n(\mathbb{C}))$ = $\mathbb{Q}$-span of the total winding number. $\square$

Homological stability

More generally:

Theorem (Arnol’d 1968, F Cohen 1972). For any $i \geq 0$, $H^i(\text{Poly}_n)$ does not depend on $n$ for $n \geq i$.

This theorem implies that the spaces $\text{Poly}_n$ satisfy homological stability. While configurations of unordered points satisfy homological stability, configurations of ordered points do not. In fact, for each $i$

$$\dim H^i(\text{Conf}_n(\mathbb{C})) \to \infty, n \to \infty.$$

The reason that homological stability always fails for the spaces of ordered configurations has to do with representation theory. Because the symmetric group $S_n$ acts on $\text{Conf}_n(\mathbb{C})$, $S_n$ also acts on the vector space $H^i(\text{Conf}_n(\mathbb{C}))$.

A fact from representation theory is that (except for $n=4$) any vector space that admits a non-trivial $S_n$ action must have dimension at least $n-1$. (Technically this is only true if I can also exclude alternating representations. Exercise: prove that $H^1(\text{Conf}_n(\mathbb{C}))$ does not contain any alternating representations.) In particular

$$\dim H^i(\text{Conf}_n(\mathbb{C})) \to \infty, n \to \infty$$

and homological stability always fails. But somehow homological stability should hold morally. For example, up to permuting indices, there’s only one $\omega_{ij} \in H^1(\text{Conf}_n(\mathbb{C}))$. A more precise statement is that each $H^1(\text{Conf}_n(\mathbb{C}))$ is the same representation of $n$. From the basis described in Arnol’d’s theorem above, it is just the representation where $S_n$ acts in the expected way on unordered pairs of distinct elements of $n$.

Representation theory of the symmetric groups

Here is a quick primer on representation theory. A representation is a $\mathbb{Q}$-vector space with a linear group action, or, equivalently, a functor from the one-object groupoid whose morphisms correspond to group elements to the category $\mathbf{Vec}_{\mathbb{Q}}$. A representation is irreducible if it has no invariant subspaces. By Maschke’s theorem, every representation decomposes into a direct sum of irreducibles. Thus, the goals of representation theory are to:

classify all the irreducible representations of a given group

understand how to decompose a given representation into irreducibles

Here, we’re interested in representations of symmetric groups, for which we have the following beautiful theorem.

Theorem (Young 1900). Irreducible representations of $S_n$ correspond to partitions of $n$.

For example:

The trivial representation of $S_n$ corresponds to the partition $n$.

The standard representation of $S_n$ (the orthogonal complement of the fixed vector in the permutation representation of $S_n$ on $\mathbb{Q}^n$) corresponds to the partition $(n-1)+1$.

We want notation for irreducible representations that is independent of $n$. For this, we denote the irreducible representation of $S_n$ corresponding to the partition

$$(n-5) + 3 + 1 + 1$$

of $n$ by $V(3,1,1)$. So $V(0)$ is the trivial representation, and $V(1)$ is the standard representation.

Representation stability?

Maschke’s theorem says that

$$H^i(\text{Conf}_n(\mathbb{C})) = \oplus_{\text{partitions} \lambda} V(\lambda)^{c_{n,\lambda}}$$ for some indices $c_{n,\lambda}$. The main problem is to find the $c_{n,\lambda}$.

Example. $H^1(\text{Conf}_n(\mathbb{C})) = V(0) \oplus V(1) \oplus V(2)$ for all $n \geq 3$. Here the trivial representation $V(0)$ is the collection of fixed vectors, the span of the total winding number. The subspace $V(0) \oplus V(1)$ is the permutation representation of some collection of basis vectors $\{u_1,\ldots, u_n\}$ of $H^1(\text{Conf}_n(\mathbb{C}))$. These basis vectors have geometric meaning: $u_i(\alpha)$ is the total winding of the loop $\alpha$ around the point $i$.

This example implies that first cohomology of ordered configuration spaces is representation stable: the index $c_{n,\lambda}$ of the irreducible representation $\lambda$ is independent of $n$ for $n$ sufficiently large. Tom and Benson wondered whether this phenomenon would also occur for the higher cohomology groups. They recruited David Hemmer to compute $H^2(\text{Conf}_n(\mathbb{C}))$ for low $n$. Unfortunately, his calculations, aided by some computer algebra package whose name I didn’t catch, did not stabilize. But thinking this theory was too beautiful not to be true, Benson pressed him to try again, and it turned out that there was a bug in the software, relating to a failure to account for the outer automorphism of $S_6$. Once this was corrected, Hemmer calculated that the second cohomology groups appear to stabilize for $n \geq 8$.

This lead to the following result:

Theorem (Church-Farb 2010). Fix $i \geq 0$. Then $$H^i(\text{Conf}_n(\mathbb{C})) = \oplus_{\text{partitions} \lambda} V(\lambda)^{c_{n,\lambda}}$$ and $c_{n,\lambda}$ does not depend on $n$ for $n \geq 4i$.

Applications

Now it turns out that the stable representations are quite complicated. See their paper. But nevertheless, this stability phenomenon has led to useful computations with numerous practical applications. I’ll mention just one: to combinatorial statistics for polynomials over a finite field $\mathbb{F}_q$. More details can be found in this paper, joint with Jordan Ellenberg.

By the Grothendieck-Lefschetz theorem, the multiplicity of a given irreducible representation in $H^i(\text{Conf}_n(\mathbb{C}))$ corresponds to a point count in the variety $\text{Poly}_n(\mathbb{F}_q)$ weighted by the character of the representation, and this correspondence is linear. Thus each index encodes some “statistic” for monic, square-free polynomials over $\mathbb{F}_q$. Representation stability implies that these statistics are asymptotically stable.

For example:

The multiplicity of the trivial representation $V(0)$ corresponds to the number of degree $n$ square-free polynomials, which is $q^n - q^{n-1}$.

The multiplicity of the standard representation $V(1)$ corresponds to the expected number of linear factors, which is $1 - \frac{1}{q} + \frac{1}{q^2} - \cdots \pm \frac{1}{q^{n-2}}$.