A discrete model for surface configuration spaces

TL;DR

This work constructs a geometric, cube‑complex model for ordered surface configuration spaces by introducing square‑tiled surfaces $K_{g,b}(n)$ and their duals, and by defining discrete configuration spaces $DF_m(X)$. The central results show that for $m\le n$, $F_m(\Sigma_{g,b})$ is homotopy equivalent to $DF_m(K_{g,b}(n))$ (or to $DF_m(K^{*}_{g,1}(n))$ in the boundary case), via intermediate square configuration spaces $SF_m(K(n),d)$ and a tautological affine Morse–Bott framework on $(K(n))^m$. The approach bridges point configurations, square (disk) configurations, and discrete combinatorial models, enabling explicit, $S_m$‑equivariant CW structures and potential discrete Morse theory analyses of surface braid groups and their (co)homology. The paper also outlines extensions to graphs, higher dimensions, and asymptotic homological computations, highlighting practical pathways for computing and understanding surface configuration spaces. Overall, it provides a constructive, geometrically grounded conduit from continuous surface configuration spaces to tractable discrete models with potential computational and topological payoffs.

Abstract

One of the primary methods of studying the topology of configurations of points in a graph and configurations of disks in a planar region has been to examine discrete combinatorial models arising from the underlying spaces. Despite the success of these models in the graph and disk settings, they have not been constructed for the vast majority of surface configuration spaces. In this paper, we construct such a model for the ordered configuration space of $m$ points in an oriented surface $Σ$. More specifically, we prove that if we give $Σ$ a certain cube complex structure $K$, then the ordered configuration space of $m$ points in $Σ$ is homotopy equivalent to a subcomplex of $K^{m}$

Figures (12)

• Figure 1: The cube complex structure $K_{2,0}(3)$ on $\Sigma_{2,0}$. We have highlighted the sole singular point in red. Note that we have a different indexing convention for surfaces with non-zero genus than we do for the plane. We do this to ensure that if $m\le n$, the $m^{\text{th}}$-discrete ordered configuration space of $K_{g,0}(n)$ is homotopy equivalent to the $m^{\text{th}}$-ordered configuration space of points in $K_{g,0}(n)$.
• Figure 2: The cube complex structure $K_{2,1}(3)$ on $\Sigma_{2,1}$. The boundary of $K_{2,1}(3)$ is colored red. Note the thick black lines that are identified are not part of the cube complex structure, though we draw them to emphasize the relation between $K_{g,1}(n)$ and $K_{g,0}(n)$; see Figure \ref{['fig:K20(3)']}.
• Figure 3: Two $l^{\infty}$-balls of diameter $2$ in $K_{2,0}(3)$. The blue ball looks like a square, whereas the green ball looks like a $12$-pointed star as it contains $p$, the point of non-zero curvature highlighted in red.
• Figure 4: The projection map $\pi$ from $\tilde{K}_{2,0}(3)$ to $K_{2,0}(3)$. Note that we have colored the various cells of $\tilde{K}_{2,0}(3)$ and $K_{2,0}(3)$ so that $\pi$ preserves the coloring. Moreover, the $1$-cells in $K_{2,0}(3)$ that are colored (a color other than black) the same are identified.
• Figure 5: A non-contractible loop in $K_{2,0}(3)$ arising from the concatenation $\gamma^{-1}\gamma'$ of two paths $\gamma$ and $\gamma'$ starting at $p$. If $\gamma'$ is the green path and $\gamma$ is the blue path, we have that $\theta(\gamma'(1))=\frac{\pi}{4}$ and $\theta(\gamma(1))=\frac{13\pi}{4}$ despite $\gamma(1)=\gamma'(1)$. Our restrictions on $U$ allow us to avoid such a situation and the resulting concerns about the well-definedness of $s$.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Proposition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5