Abstract
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.
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Björner, A., Tancer, M. Note: Combinatorial Alexander Duality—A Short and Elementary Proof. Discrete Comput Geom 42, 586–593 (2009). https://doi.org/10.1007/s00454-008-9102-x
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DOI: https://doi.org/10.1007/s00454-008-9102-x