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

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5 DeRham Cohomology In this section we will define a second cohomology theory, the DeRham cohomology of a smooth manifold. Eventually we will prove what is known as DeRham’s theorem, which 54 its says that this cohomology agrees with the singular cohomology defined above for smooth manifolds. Differential forms give a contravariant functor from the category of smooth manifolds and smooth maps to the category of real differential graded algebras. dimM M → Ω∗ (M ) = { ⊕ Ωk (M ), d} k=0 and, (f : N → M ) → (f ∗ : Ω∗ (M ) → Ω∗ (N )) In particular, these differential graded algebras are cochain complexes, and so we can apply the cohomology functor.

If U and V are homeomorphic, then n = m. Proof. 2. Let U be a non-empty open subset of Rn and let x ∈ U . Then Hk (U, U \{x}) is zero except when k = n in which case the relative homology group is Z. 39 Proof. Let U ⊂ Rn be a non-empty open set. Let x ∈ U . If we let K = Rn \ U , then K is closed and K ⊂ Rn \ {x}. So applying excision with X = Rn , A = Rn \ {x} and K = Rn \ U , we have, H∗ (U, U \ {x}) ∼ = H∗ (Rn , Rn \ {x}) ˜ ∗ (Rn ) = 0, and by the homotopy axiom H ˜ ∗ (Rn \ {x}) = Since Rn is contractible, we have H ˜ ∗ (Rn \ {0}) = H ˜ ∗ (S n−1 ) = Z if ∗ = n − 1 and 0 otherwise.

Urβ(k+1) ) = i=0 = δφ(Urβ(0) , . . , Urβ(k) ) = r ∗ δφ((Vβ(0) , . . , Vβ(k) ) ˇ ∗ (X; {Uα }) → H ˇ ∗ (X; {Vβ }). Now, Thus, we have an induced map on cohomology, r ∗ H suppose that s : B → A is another refinement map. We use r and s to define a cochain homotopy H(r,s) : Cˇ k (X, {Uα }) → Cˇ k−1 (X, {Vβ }) such that δH(r,s) + H(r,s)δ = s∗ − r ∗ , and thus r ∗ = s∗ on cohomology. Given φ ∈ Cˇ k (X, {Uα }) define k−1 (−1)i φ(Urβ(0) , . . , Urβ(i) , Usβ(i) , . . , Usβ(k−1) ) H(r,s)φ(Vβ(0) , .

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

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