Topology

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

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

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Uα(j) , . . , Uα(k+2) ) = 0 + j=i+1 Since in this sum, for a given pair a < b each term φ(Uα(0) , . . , Uα(a) , . . , Uα(b) , . . , Uα(k+2) ) appears exactly twice and with cancelling signs. ˇ Thus (Cˇ ∗ (X; {Uα }), δ) forms a cochain complex. We define the Cech cohomology of X with respect to the open cover {Uα } to be the cohomology of this cochain complex, ˇ ∗ (X; {Uα }) = H ∗ (Cˇ ∗ (X; {Uα }), δ). 3. Let X be any topological space and let {X} be the open cover consisting of the single open set X.

Proof. The proof uses the tools developed in the proof of Mayer-Vietoris. Let B = X \ K. Then A ∩ B = A \ K, so we need to show the inclusion induces H∗ (B, A ∩ B) ∼ = H∗ (X, A). Notice that the interiors of A and B form an open cover of X. Thus, using the techniques of the Mayer-Vietoris proof, we can subdivide a chain in S∗ (X) to a chain which is small with respect to the open cover {intA, intB}, and the inclusion S∗small (X) ֒→ S∗ (X) induces an isomporphism in homology. Recall that S∗small (X) is generated by singular simplices whose 37 images lie in either intA or intB.

Usβ(k) ) (Hδ)(φ)(Vβ(0) , . . , Vβ(k) ) = j=0 j k (−1)i φ(Urβ(0) , . . , Urβ(i) , . . , Urβ(j) , Usβ(j) , . . , Usβ(k) ) (−1)j = i=0 j=0 k (−1)i+1 φ(Urβ(0) , . . , Urβ(j) , Usβ(j) , . . , Usβ(i) , . . , Usβ(k) ) . + i=j Thus, k (−1)j (−1)j φ(Urβ(0) , . . , Urβ(j−1) , Usβ(j) , . . , Usβ(k) ) (δH + Hδ)φ(Vβ(0) , . . , Vβ(k) ) = j=0 + (−1)j+1 φ(Urβ(0) , . . , Urβ(j) , Usβ(j+1) , . . , Usβ(k) ) This sum telescopes leaving, (δH + Hδ)φ(Vβ(0) , . . , Vβ(k) ) = φ(Usβ(0) , . . , Usβ(k) ) − φ(Urβ(0) , .

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