Combinatorial Alexander Duality — a Short and Elementary Proof
نویسنده
چکیده
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 i-th 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 selfcontained proof from first principles, accessible to the non-expert.
منابع مشابه
2 1 O ct 2 00 7 COMBINATORIAL ALEXANDER DUALITY — A SHORT AND ELEMENTARY PROOF
Let X be a simplicial complex with the ground set V . Define its Alexander dual as a simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th 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 selfcontained proof.
متن کامل5 O ct 2 00 7 COMBINATORIAL ALEXANDER DUALITY — A SHORT AND ELEMENTARY PROOF
Let X be a simplicial complex with the ground set V . Define its Alexander dual as a simplicial complex X∗ = {σ ⊆ V | V \ σ / ∈ X}. The combinatorial Alexander duality states that the i-th 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 selfcontained proof.
متن کاملNote: Combinatorial Alexander Duality - A Short and Elementary Proof
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 i-th 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 selfcontained proof from first principles, accessible to the non-ex...
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