Introduction to Algebraic Topology and Algebraic Geometry by U. Bruzzo

By U. Bruzzo

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6. Leray’s theorem for Cech cohomology. If an open cover U of a topoloˇ gical space X is suitably chosen, the Cech cohomologies H • (U, F) and H • (X, F) are isomorphic. Leray’s theorem establishes a sufficient condition for such an isomorphism to hold. Since the cohomology H • (U, F) is in generally much easier to compute, this ˇ turns out to be a very useful tool in the computation of Cech cohomology groups. ip = Ui0 ∩· · ·∩Uip , i0 . . ip ∈ I. 24. (Leray’s theorem) Let F be a sheaf on a paracompact space X, and let U be an open cover of X which is acyclic for F and is indexed by an ordered set.

Rn minus a point, 3. the circle S 1 , 4. the torus T 2 , 5. a punctured torus, 6. a Riemann surface of genus g. 2. 1. The relative homology complex. Given a topological space X, let A be any subspace (that we consider with the relative topology). We fix a coefficient ring R which for the sake of conciseness shall be dropped from the notation. For every k ≥ 0 there is a natural inclusion (injective morphism of R-modules) Sk (A) ⊂ Sk (X); the homology operators of the complexes S• (A), S• (X) define a morphism δ : Sk (X)/Sk (A) → Sk−1 (X)/Sk−1 (A) which squares to zero.

On the other hand if c ∈ Bk (X, A) we have c = ∂b + c with b ∈ Sk+1 (X) and c ∈ Sk (A), so that qk (c) = 0 implies 0 = qk ◦ ∂b = ∂ ◦ qk+1 (b), which in turn implies c ∈ Sk (A). To prove the surjectivity of qk , just notice that by definition an element in Bk (X, A) may be represented as ∂b with b ∈ Sk+1 (X). As for the second row, we have Sk (A) ⊂ Zk (X, A) from the definition of Zk (X, A). If c ∈ Sk (A) then qk (c) = 0. If c ∈ Zk (X, A) and qk (c) = 0 then c ∈ Sk (A) by the definition of Zk (X, A).

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