| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

1. Problems

    1. 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.