The symmetric group $\Sigma_n$ acts by permuting indices.

It consists of sets

- $V(G)$ of 0-dimensional cells called
*vertices*and - $E(G)$ of 1-dimensional cells called
*edges*.

We will work with a combinatorial cube complex model of $\mathrm{Conf}_n(G)$, analogous to Świątkowski's model in the unordered case.

Its zero dimensional cubes are given by configurations where the particles on each edge are equidistant.

One dimensional cubes move one particle from a vertex to an edge.

Move mouse along the interval

A two dimensional cube moves two particles from two distinct vertices onto edges.

Move mouse along the square

Therefore, for graphs with exactly one essential vertex the combinatorial model is itself a graph and it is easy to determine its homology.

Move mouse along the combinatorial model

If the number of particles is at least $2|V|$, then the dimension of this combinatorial model is equal to the homological dimension of the space.

The map collapsing a subgraph $$G \to G/H$$ does __not__ induce a map on configuration spaces.

To solve this, we introduce *sinks* to our graphs. Sinks are vertices where particles are allowed to collide.

Collapsing a subgraph $H$ of $G$ now induces a map $$ \mathrm{Conf}_n(G) \to \mathrm{Conf}_n(G/H, H/H). $$

Collapsing and computing the homology of small graphs with sinks allows us to compute the homology of configurations in trees.

Move mouse along the combinatorial model

For star graphs, the combinatorial model is good enough to compute the homology:__Theorem__ (Ghrist)For $n\ge 1$ and $k\ge 3$ we have $$\tilde{H}_q(\mathrm{Conf}_n(\mathrm{Star}_k)) \cong \begin{cases} \mathbb{Z}^{r(k,n)} & \text{$q$ = 1,}\\ 0 & \text{else,} \end{cases}$$ where $r(k,n) = 1 + \frac{(n+k-2)!}{(k-1)!} (n(k-2) - k + 1)$.

For the interval with sinks, we get:__Proposition__ (Chettih-L)For $n\ge 1$ we have $$\tilde{H}_q\left(\mathrm{Conf}_n([0,1], \{0\})\right) =0,$$ and $$\tilde{H}_q\left(\mathrm{Conf}_n([0,1], \{0,1\})\right) \cong \begin{cases} \mathbb{Z}^{r(n)} & \text{$q$ = 1,}\\ 0 & \text{else,} \end{cases}$$ where $r(n) = 1 + (n-2) 2^{n-1}$.

It turns out that these are the only building blocks we need to describe the homology $H_*(\mathrm{Conf}_n(T))$ for any finite tree $T$.

To illustrate the approach of our proof, we will now compute $H_*(\mathrm{Conf}_n(\mathrm{H}))$.

Let $n\ge 1$, $G$ a finite graph, $Z$ a subset of the vertices and $\{V_i\}_{i\in I}$ an open cover of $G$.

For each map $$\phi\colon \{1,\ldots,n\}\to I$$ define the open subset $U_\phi$ of $\mathrm{Conf}_n(G,Z)$ as $$ U_\phi = \bigcap_{1\le j\le n} \pi_j^{-1}( V_{\phi(j)}), $$ where $\pi_j\colon\mathrm{Conf}_n(G,Z)\to G$ projects to the $j$-th particle.

This defines an open cover $\{ U_\phi \}$ of $\mathrm{Conf}_n(G,Z)$.

Associated to this open cover $\{ U_\phi \}$ of $\mathrm{Conf}_n(G,Z)$ there is the Mayer-Vietoris spectral sequence given by $$ E^1_{p,q} = \bigoplus_{\phi_0,\ldots,\phi_p} H_q( U_{\phi_0} \cap\cdots\cap U_{\phi_p}) $$ converging to the homology $H_*(\mathrm{Conf}_n(G,Z))$.

Instead of computing the pages directly, we compare the spectral sequences for different graphs and different sets of sinks.

As open cover for the graph $\mathrm{H}$ we take $V_1$ and $V_2$ defined as follows:

This defines a spectral sequence $E^*_{\bullet,\bullet}$.

Each intersection of sets $U_\phi$ is a disjoint union of spaces of the form $$ \mathrm{Conf}_{S_1}(V_1) \times \mathrm{Conf}_{S_I}(V_1\cap V_2) \times \mathrm{Conf}_{S_2}(V_2) $$ for some $S_1\sqcup S_I \sqcup S_2 = \{1,\ldots,n\}$.

The connected components of $\mathrm{Conf}_{S_I}(V_1\cap V_2)$ are contractible and determined by the order of the particles $S_I$. By the Künneth theorem, the $q$-th homology of those products is given by $$ \bigoplus_{q_1+q_2=q} H_{q_1}( \mathrm{Conf}_{S_1}(V_1) )\otimes H_{q_2}( \mathrm{Conf}_{S_2}(V_2)). $$

This implies that only the three bottom rows of $E^1_{\bullet,\bullet}$ are non-trivial.

We now compute the bottom row $E^2_{\bullet,0}$.

Entries are sums of modules $$ H_0(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_0(\mathrm{Conf}_{S^j_2}(V_2)) $$ These groups don't change if we turn the vertices into sinks.

$\cong H_\bullet(\mathrm{Conf}(\mathrm{H},V(\mathrm{H}))$

$\cong H_\bullet(\mathrm{Conf}([0,1],\{0,1\}))$

We now compute the first row $E^2_{\bullet,1}$.

Entries are sums of modules $$ H_1(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_0(\mathrm{Conf}_{S^j_2}(V_2)) $$ and $$ H_0(\mathrm{Conf}_{S^j_1}(V_1))\otimes H_1(\mathrm{Conf}_{S^j_2}(V_2)) $$ These groups don't change if we turn one of the vertices into a sink.

This describes the infinity page.

This gives a generating set of $H_*(\mathrm{Conf}_n(\mathrm{H}))$ in terms of classes in the configuration spaces of the $\mathrm{Y}$-graph and the interval with two sinks.

The general proof proceeds in the same way by glueing star graphs to trees.