1. Problems
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Relative Homology
Problem 1.02.
[John Wiltshire-Gordon] Given graphs $\Gamma$ and $\Gamma'$, where $\Gamma$ is obtained from $\Gamma'$ via an edge deletion, can we compute $H_i(\overline{\operatorname{Conf}}_n(\Gamma'),\overline{\operatorname{Conf}}_n(\Gamma))$.
If $\Gamma$ is obtained from $\Gamma'$ via an edge contraction, we have an induced map on the level of chain complexes. Can we compute the homology of the mapping cone of this map on chain complexes? -
Poincaré Polynomial
Problem 1.04.
[Nicholas Proudfoot] What do Poincaré polynomials of ordered or unordered configuration spaces look like? Do they have internal zeroes? Are the coefficients unimodal/log concave? Are they real rooted? In the case of the ordered configuration space, can we understand the coefficients as $S_n$-representations? -
Are braid groups CAT$(0)$?
Graph braid groups are CAT$(0)$ which can be seen by looking at their action on the universal cover of their respective configuration space.Problem 1.06.
[Aaron Abrams] Can we use this fact to determine if braid groups are CAT$(0)$ by looking at some sort of limits of configuration spaces on graphs. -
Generation of $\pi_1(\operatorname{Conf}_n(\Gamma))$.
Problem 1.08.
[Tomasz Maciazek] Can we generate graph braid group using $Y$-exchanges and 1-particle cycles? Specifically, for $\pi_1(\operatorname{Conf}_n(\Theta_3))$, if we quotient out the 1-particle cycles, are the relations the classical braid relations? The latter question has been confirmed for three, four, and 5 particles. -
Verdier Duality for $\operatorname{Conf}_n(\Gamma)$
Given a graph configuration space, we can consider its one point compactification by adding a single point where two particles are allowed to collide.Problem 1.1.
[Nir Gadish] How does this compactified version relate to the original configuration space? Can we understand Verdier duality for $\operatorname{Conf}_n(\Gamma)$? -
Genus 1 groups.
A genus 1 group is one where every relator is a commutator. They sit somewhere between right-angled artin groups and commutator-relator groups.Problem 1.12.
[Aaron Abrams] Is the 2-string braid group of a non-planar graph genus 1 and how can we tell if a group is genus 1 or not?
Ko and Park showed that for if the 3 string braid group of a graph is genus 1, then the graph doesn’t contain $\Theta_4$ as a topological minor. They conjecture that if $\Gamma$ doesn’t contain $\Theta_4$ as topological minor, then the $n$-string braid group of $\Gamma$ is genus 1. -
Embedding genus of a graph.
Problem 1.14.
[John Wiltshire-Gordon] Can we use the homology of configuration spaces of graphs to see if there can be an $S_2$-equivariant map $\operatorname{Conf}_n(K_7) \to \operatorname{Conf}_2(S^1 \times S^1)$? Can we generalize in order to detect min or max embedding genus of a graph. -
Using the cohomology ring structure to detect embedding genus.
Problem 1.16.
Can we compute the cup product structure or Steenrod operations on $H^*(\operatorname{Conf}_n(\Gamma))$? Can we use this structure to say anything about the min or max embedding genus of $\Gamma$? -
Torsion in homology of configuration space of complete bipartite graph.
We see that there is 2-torsion in $H_1(\overline{\operatorname{Conf}}_{2}(K_{3, 3}))$.Problem 1.18.
[Ben Knudsen] In general, is there $r+1$-torsion in $H_1(\overline{\operatorname{Conf}}_{r+1}(K_{r+2, r+2}))$? -
Quasi-isometry type.
Problem 1.2.
[Aaron Abrams] What can we say about the quasi-isometry type of configuration spaces of graphs? For example, all of the 2 string braid groups of trees are quasi-isometric. Can we classify graphs up to quasi-isometry of their 2 string braid groups? -
Underlying matroid of the graph.
Problem 1.22.
[Alexandru Suciu] What properties of the configuration spaces on a graph depend only on the underlying matroid of the graph? -
Non-k equals.
Problem 1.24.
[Safia Chettih] Can we compute the homology and homotopy groups for configuration spaces where collisions of $k$ or fewer points are allowed? -
Leray Spectral Sequence.
Problem 1.26.
[John Wiltshire-Gordon] What does the Leray spectral sequence look like for the inclusion $\operatorname{Conf}_n(\Gamma) \hookrightarrow \Gamma^n$? -
Leray spectral sequence of embedding into a surface.
Problem 1.28.
[Eric Ramos] What does the Leray spectral sequence look like for a map $\operatorname{Conf}_n(\Gamma) \to \operatorname{Conf}_n(\Sigma_g)$ induced by an embedding $\Gamma \to \Sigma_g$. -
Pseudo-manifolds.
Problem 1.3.
[Aaron Abrams] The $n$-string ordered (or unordered) configuration space on $K_{2n+1}$, $K_{n+1,n+1}$, or $K_{n-1,n+1}$ is a manifold away from the codimension 3 skeleton. Can we investigate this phenomenon further? -
Topological Complexity
Problem 1.32.
[Alexandru Suciu/Michael Farber] What is LS category and topological complexity of $\operatorname{Conf}_n(\Gamma)$ or $\overline{\operatorname{Conf}}_n(\Gamma)$ (in particular when $n$ is at least twice the number of essential vertices)? The answer is already know for trees and other special graphs. -
At what point do things stabilize?
Problem 1.34.
[Gabriel Drummond-Cole] Fix $i$ and $\Gamma$. We know that for $n \gg 0$, $H_i(\overline{\operatorname{Conf}}_n(\Gamma))$ stabilizes. When does it stabilize?
Alternatively, can we answer similar questions about stability phenomena when we instead fix $n$ and $\Gamma$? What if we fix $i$ and $n$? -
Module structure.
Problem 1.36.
[Ben Knudsen] Can we explicitly compute $\bigoplus_n H_i(\overline{\operatorname{Conf}}_n(\Gamma))$ as a $Z[E]$-module? -
Random graphs.
Problem 1.38.
[Maya Banks] Given a random graph $\Gamma$ (from your favorite model), what is the expected value of $\operatorname{rk}H_i(\operatorname{Conf}_n(\Gamma))$?
When does a certain property occur with probability 1? -
Leading coefficient of the growth polynomial of the first betti number.
Problem 1.4.
[Byung Hee An] Given a graph $\Gamma$, is there any geometric meaning behind the growth polynomial for the first betti number of $\overline{\operatorname{Conf}}_n(\Gamma)$?
Byung Hee An has a conjecture for the value of leading coefficient of this polynomial. -
Graph invariants that are functorial with respect to deletions and contractions.
Problem 1.42.
[Nicholas Proudfoot] What graph invariants are functorial with respect to deletions and contractions? In particular, can we find invariants that play nicely with the graph minor category?
Cite this as: AimPL: Configuration spaces of graphs, available at http://aimpl.org/configgraph.