Graphical configuration spaces, Contractads and Formality

TL;DR

This work develops a comprehensive contractadic framework for graphical configuration spaces, extending the little disks operad to the little disks contractad 𝒟_d and its framed variant. It constructs Fulton–MacPherson-type compactifications in the graphical setting, connects cohomology to twisted Lie contractads, and proves formality results in dimensions 1 and 2 and for chordal graphs, while identifying obstructions in cycles. A rich interplay is established among twisted algebras, logarithmic geometry, and combinatorial models (posets and acyclic directions), culminating in a combinatorial model AD_d that recovers 𝒟_d in favorable cases. The work also develops a robust apparatus for modular and graphical compactifications, with applications to cohomology, Euler characteristics, and stratifications, and lays groundwork for higher-dimensional and non-chordal analyses.

Abstract

Given a finite simple connected graph $Γ$, the graphical configuration space $\mathrm{Conf}_Γ(X)$ is the space of collections of points in $X$ indexed by the vertices of $Γ$, where points corresponding to adjacent vertices must be distinct. When $X=\mathbb{R}^d$ and the points are replaced by small disks, the resulting spaces for all possible graphs fit together into an algebraic structure that extends the little disks operad, called the little disks contractad $\mathcal{D}_d$. In this paper, we investigate the homotopical and algebraic properties of the little disks contractad $\mathcal{D}_d$. We construct and study Fulton-MacPherson compactifications of graphical configuration spaces, which provide a convenient model for $\mathcal{D}_d$ within the class of compact manifolds with boundary. Using these and wonderful compactifications, we prove that $\mathcal{D}_d$ is formal in the category of (Hopf) contractads for $d=1$, $d=2$, and for chordal graphs for any $d$. We also identify the first obstructions to coformality in the case of cyclic graphs. In addition, we give a combinatorial description of the cell structure of $\mathcal{D}_2$ and present applications to the study of graphical configuration spaces $\mathrm{Conf}_Γ(X)$ using the language of twisted algebras.

Figures (5)

• Figure 1: Examples of partitions of graphs and associated contractions.
• Figure 2: Example of configurations in $\operatorname{\mathcal{D}}_2(\operatorname{\mathsf{C}}_4)$.
• Figure 3: Disks composition in $\operatorname{\mathcal{D}}_2$
• Figure 4: List of complete multipartite graphs on 4 vertices.
• Figure 6: Stable curves in $\overline{\EuScript{M}}_{0,\operatorname{\mathsf{K}}_3}, \widetilde{\EuScript{M}}_{0,\operatorname{\mathsf{K}}_{3}}$ and $\widetilde{\EuScript{M}}^{\operatorname{\mathrm{fr}}}_{0,\operatorname{\mathsf{K}}_{3}}$

Theorems & Definitions (123)

• Theorem A
• Theorem B
• Theorem C
• Theorem D
• Theorem E
• Theorem F
• Definition 1.8
• Definition 1.9
• Proposition 1.10
• proof