A Dichotomy for Finite Abstract Simplicial Complexes

TL;DR

This work establishes a topological dichotomy for the Hom-Complex $Hom_{sc}(A,B)$ of finite abstract simplicial complexes: for fixed $B$, either every $Hom_{sc}(A,B)$ has contractible components (Hom-contractible) or $B$ is Hom-universal, capable of realizing every finite CW-complex up to homotopy via choice of $A$. The dichotomy is tightly linked to algebraic properties of $B$ through Taylor and Siggers polymorphisms, with Hom-universality corresponding to the absence of Taylor polymorphisms and implying a CSPs are NP-complete, while Hom-contractibility corresponds to the presence of such polymorphisms and a CSP in P; moreover, these ideas extend to finite relational structures via pp-interpretations and provide alternate routes to known results like the Hell–Nesetril theorem and connections to social choice theory. The paper develops a robust clone-theoretic framework that translates algebraic symmetries into topological and computational consequences, highlighting deep interactions between topology, universal algebra, and computational complexity. It also sketches broad generalizations, including infinite and pairwise structures, and emphasizes the unifying role of polymorphisms in understanding the geometry of solution spaces and their algorithmic implications.

Abstract

Given two finite abstract simplicial complexes A and B, one can define a new simplicial complex on the set of simplicial maps from A to B. After adding two technicalities, we call this complex Homsc(A, B). We prove the following dichotomy: For a fixed finite abstract simplicial complex B, either Homsc(A, B) is always a disjoint union of contractible spaces or every finite CW-complex can be obtained up to a homotopy equivalence as Homsc(A, B) by choosing A in a right way. We furthermore show that the first case is equivalent to the existence of a nontrivial social choice function and that in this case, the space itself is homotopy equivalent to a discrete set. Secondly, we give a generalization to finite relational structures and show that this dichotomy coincides with a complexity theoretic dichotomy for constraint satisfaction problems, namely in the first case, the problem is in P and in the second case NP-complete. This generalizes a result from [SW24] respectively arXiv:2307.03446 [cs.CC]

Figures (6)

• Figure 1: A graphical visualization of the complexes from Example \ref{['Example1']}. Non-degenerate faces with exactly two elements are drawn as lines. Higher dimensional faces are highlighted in gray. Note that some of them overlap which is not visible in the picture.
• Figure 2: The simplicial complexes from Example \ref{['Example2']}.
• Figure 3: The structure of the Proof. Dashed arrows denote opposite statements. Normal arrows denote implications that will be proved or are already proved. The arrows become a big cycle through all non-dashed boxes when replacing the right side (or left side) by its negations.
• Figure 4: The path $\mathcal{P}_2$ is isomorphic to $\operatorname{GS}(\mathcal{P}_1)$ and $\mathcal{P}_4 \times \mathcal{P}_4$ maps into $\operatorname{GS}(\mathcal{P}_1 \times \mathcal{P}_1)$. To denote this map, the points of $\mathcal{P}_4 \times \mathcal{P}_4$ are labeled with their respective images in $\operatorname{GS}(\mathcal{P}_1 \times \mathcal{P}_1)$.
• Figure 5: The map for $d=3$ is a map from $\mathcal{P}_{6}^3$ to $\operatorname{GS}(\mathcal{P}_1^3)$. To safe some paper and clarity, we only display the number of elements, the image has. So the number 8 which is in brackets represents the fact that the vertex $(3,3,3)$, which is the center of the (hyper-)cube, is mapped to the only subset of $\{0,1\}^3$ with 8 elements. The number at a position $(x_1,x_2,x_3)$ is displayed in brackets, if all coordinates $x_1$ to $x_3$ are $0$, $3$ or $6$.

Theorems & Definitions (133)

• Example 1.1
• Definition 2.1
• Example 2.2
• Definition 2.3: Products, Homomorphisms
• Definition 2.4
• Example 2.5
• Definition 2.6
• Example 2.7
• Definition 2.8
• Definition 2.9