Combinatorial Alexander Duality -- a Short and Elementary Proof

Details Combinatorial Alexander Duality -- a Short and Elementary Proof Combinatorial Alexander Duality -- a Short and Elementary Proof

2007-10-05

arxiv.org/abs/0710.1172v3

Discrete Comput. Geom. 42(4) (2009), 586-593

Mathematics

Combinatorics

7 pages, 2 figure; v3: the sign function was simplified

Scientific paper

10.1007/s00454-008-9102-x

Let X be a simplicial complex with the ground set V. Define its Alexander
dual as a simplicial complex X* = {A \subset V: V \setminus A \notin 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 self-contained proof.

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