A duality on simplicial complexes
نویسنده
چکیده
The usual definition of finite simplicial complex is a set of non-empty subsets of a finite set, closed under non-empty subset formation. For our purposes here, we will omit the non-emptiness and define a finite simplicial complex to be a down-closed subset of the set of subsets of a finite set. We can, and will suppose that the finite set is the integers 0,. . . ,N . We will denote by K the set 2 of all subsets of N + 1. If S ⊆ K is a finite simplicial complex, then a subset of n + 1 elements in S is called an n simplex. We will write an element of S as [a0, . . . , an] with a0 < · · · < an. We also write [ ] for the unique (−1)-simplex. If σ = [a0, . . . , an] is an n-simplex, we say that a0, . . . , an are the vertices of σ. We will be dealing with the free abelian group generated by the n-simplexes. We will continue to write [a0, . . . , an], for a0 < · · · < an, but we will also denote by [a0, . . . , an] the element sgn p[ap0, . . . , apn] where p is the unique permutation such that ap0 < · · · < apn and also let [a0, . . . , an] = 0 if the vertices are not distinct.
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