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36 II Homology and cohomology of manifolds where Z∗ (X, A) = Ker ∂, B∗ (X, A) = Im ∂. Observe that a “relative cycle” [an element of Z∗ (X, A)] is represented by a chain of X whose boundary is a chain in A. This boundary is a cycle in A, and one can easily check that its homology class in A only depends on the relative homology class [in (X, A)] of the original relative cycle. We have thus defined a homomorphism ∂∗ : Hp (X, A) −→ Hp−1 (A). 8 [a “map of pairs” f : (X, A) → (Y, B) consists of two pairs (X, A), (Y, B) and a map f : X → Y such that f (A) ⊂ B].

The boundary V˙ of V is therefore fibred by (r − 1)-spheres. The Leray coboundary map “inflates” (q − r)-cycles of S inside X − S by fibring them by these (r − 1)-spheres. Special case. Φ = c (the family of compact sets of X). Φ|S and Φ|X − S are therefore simply the families of compact sets of S and X − S, and the Leray coboundary can be written δ ∗ : Hq−r (S) −→ Hq−1 (X − S). 6 Currents Throughout this section, X is an oriented differentiable manifold of dimension n. 1 A current j on X is a continuous linear form13 on Ω(c X).

But Φ0 (X) is the vector space of differential functions on X with zero differential. Any such function is constant on every connected component of the manifold X. In this way, a basis for the vector space H 0 (X) is given by the set of connected components of X. 3 Every continuous map f : X → Y induces a homomorphism f ∗ : H ∗ (Y ) → H ∗ (X) between cohomology groups which preserves the degree. The correspondence f f ∗ is a contravariant functor . Corollaries. • If f : X ≈ Y is a homeomorphism, f ∗ : H ∗ (X) ≈ H ∗ (Y ) is an isomorphism; • if X retracts onto A, H ∗ (A) is a direct summand of H ∗ (X).

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