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By Morgan J.W., Lamberson P.J.

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Uα(k+1) ) δ(φ)(Uα(0) , . . , Uα(k+1) ) = i=0 This definition is understood to hold only in the case where Uα(0) ∩ . . ∩ Uα(k) = ∅. ,Uα (k+1 ) = 0. Since δ is linear, it is an abelian group homomorphism. Next we check that δ2 = 0, so we have defined a cochain complex. Symbolically, the computation is very similar to the computations made for our previous cochain complex constructions. 2. δ2 = 0 Proof. Assume that Uα(0) ∩ . . ∩ Uα(k+2) = ∅. 57 Then, k+2 (−1)i δφ(Uα(0) , . . , Uα(i) , . . , Uα(k+2) ) δ(δφ)(Uα(0) , .

Here is the theorem that compares ordered and oriented homology. 1. The map C∗ord (K) → C∗or (K) is a chain map. It induces an isomorphism on homology. Proof. We begin by showing that the map is a chain map. If σ : ∆n → K is an ordered n-simplex whose image has dimension less than n − 1, then it is clear that σ and all its 53 faces map to the zero element in C∗or (K). Hence, on these elements the maps commute with the boundary maps. Suppose that the image of σ has dimension n − 1. Then σ maps to zero in C∗or (K).

Uα(1) , . . , Uα(i) ) = sign(π)φ(Uα(π(1)) , . . , Uα(π(i)) ) for π ∈ Σn+1 . 12. The Cech cohomology of an open cover is the singular cohomology of the geometric realization of the nerve of the open cover. 13. Let K be a simplicial complex. The Cech cohomology of K is identified with the singular cohomology of |K| in a manner compatible with simplicial mappings. Proof. Given a simplicial complex K we define an open covering {Uv } of |K| whose open sets are indexed by the vertices of K. For a vertex v of K we consider the open star Uv .

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Algebraic topology by Morgan J.W., Lamberson P.J.

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