A Dichotomy for Finite Abstract Simplicial Complexes
<|reference_start|>A Dichotomy for Finite Abstract Simplicial Complexes: 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]<|reference_end|>
@article{meyer2024a,
    title={A Dichotomy for Finite Abstract Simplicial Complexes},
    author={Sebastian Meyer},
    journal={arXiv preprint arXiv:2408.08199},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.08199},
    primaryClass={math.GN cs.CC}
}