Configuration Spaces of Graphs
Configuration spaces
Definition: Let $X$ be a topological space and $n\in\mathbb{N}$, then the configuration space of $n$ ordered particles in $X$ is defined as the set $$\mathrm{Conf}_n(X) = \{ (x_1, \ldots, x_n) \,|\, \text{$x_i \neq x_j$ for $i\neq j$} \}$$ endowed with the subspace topology of $X^n$.
The symmetric group $\Sigma_n$ acts by permuting indices.
Graphs
Definition: A graph $G$ is a 1-dimensional CW-complex.
It consists of sets
- $V(G)$ of 0-dimensional cells called vertices and
- $E(G)$ of 1-dimensional cells called edges.
Main result
Theorem (Chettih-L)
If $T$ is a finite tree, then $H_\bullet (\mathrm{Conf}_n(T); \mathbb{Z})$ is free and generated by products of disjoint basic classes.Definition: A basic class is a class in the image of the map $$ H_1(\mathrm{Conf}_n(\iota))\colon H_1(\mathrm{Conf}_n(G)) \to H_1(\mathrm{Conf}_n(T)) $$ induced by a piecewise linear embedding $\iota\colon G\hookrightarrow T$, where $G$ is a star graph or the H-shaped graph.
Main result
Theorem (Chettih-L)
If $T$ is a finite tree, then $H_\bullet (\mathrm{Conf}_n(T); \mathbb{Z})$ is free and generated by products of disjoint basic classes.A combinatorial model
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.
A combinatorial model
One dimensional cubes move one particle from a vertex to an edge.
Move mouse along the interval
A combinatorial model
A two dimensional cube moves two particles from two distinct vertices onto edges.
Move mouse along the square
A combinatorial model
Theorem (L)
If $G$ is a finite graph, then $\mathrm{Conf}_n(G)$ deformation retracts to a cube complex of dimension $\mathrm{min}\{n, |V|\}$, where $V$ is the set of essential vertices.Therefore, for graphs with exactly one essential vertex the combinatorial model is itself a graph and it is easy to determine its homology.
Example: $\mathrm{Conf}_2(Y)\simeq S^1$
Move mouse along the combinatorial model
A 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.
Configuration spaces with sinks
The map collapsing a subgraph $$G \to G/H$$ does not induce a map on configuration spaces.
Configuration spaces with sinks
To solve this, we introduce sinks to our graphs. Sinks are vertices where particles are allowed to collide.
Definition: For a graph $G$ and a subset $Z$ of the vertices, define $$ \mathrm{Conf}_n(G,Z) := \{ (x_1,\ldots,x_n) | \text{$x_i\neq x_j$ or $x_i\in Z$} \}. $$
Configuration spaces with sinks
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.
Configuration spaces with sinks
Theorem (Chettih-L)
If $G$ is a finite graph and $Z$ a subset of the vertices, then $\mathrm{Conf}_n(G,Z)$ deformation retracts to a cube complex of dimension $\mathrm{min}\{n, |V| + |E_\mathrm{sink}| \}$, where $V$ is the set of essential vertices and $E_\mathrm{sink}$ is the set of edges incident to two sinks.Example: $\mathrm{Conf}_2([0,1], \{0,1\})\simeq S^1$
Move mouse along the combinatorial model
Explicit calculations
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 $r(k,n) = 1 + \frac{(n+k-2)!}{(k-1)!} (n(k-2) - k + 1)$.Explicit calculations
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 $r(n) = 1 + (n-2) 2^{n-1}$.Explicit calculations
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 $H_*(\mathrm{Conf}_n(\mathrm{H}))$.
The Mayer-Vietoris spectral sequence
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 $\mathrm{Conf}_n(G,Z)$.
The Mayer-Vietoris spectral sequence
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.
The H-shaped graph
As open cover for the graph $\mathrm{H}$ we take $V_1$ and $V_2$ defined as follows:
This defines a spectral sequence $E^*_{\bullet,\bullet}$.
The H-shaped graph
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 H-shaped graph
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 $$ \bigoplus_{q_1+q_2=q} H_{q_1}( \mathrm{Conf}_{S_1}(V_1) )\otimes H_{q_2}( \mathrm{Conf}_{S_2}(V_2)). $$
The H-shaped graph
This implies that only the three bottom rows of $E^1_{\bullet,\bullet}$ are non-trivial.
The H-shaped graph
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.
$\cong H_\bullet(\mathrm{Conf}(\mathrm{H},V(\mathrm{H}))$
$\cong H_\bullet(\mathrm{Conf}([0,1],\{0,1\}))$
The H-shaped graph
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.
The H-shaped graph
This describes the infinity page.
The H-shaped graph
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.
Thanks for listening!
Configuration Spaces of GraphsDaniel LütgehetmannBerlin Mathematical SchoolYoung Topologist Meeting 2017