On the Topology of Discrete Strategies
International Journal of Robotics Research
Algorithmic complexity of finding cross-cycles in flag complexes
Proceedings of the twenty-eighth annual symposium on Computational geometry
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Let X be a simplicial complex with ground set V. Define its Alexander dual as the simplicial complex X *={σ⊆V∣V∖σ ∉ X}. The combinatorial Alexander duality states that the ith reduced homology group of X is isomorphic to the (|V|−i−3)th reduced cohomology group of X * (over a given commutative ring R). We give a self-contained proof from first principles accessible to a nonexpert.