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INTRODUCTION to DE RHAM COHOMOLOGY Contents 1 INTRODUCTION TO DE RHAM COHOMOLOGY REDMOND MCNAMARA Abstract. We briefly review differential forms on manifolds. We prove ho- motopy invariance of cohomology, the Poincar´elemma and exactness of the Mayer{Vietoris sequence. We then compute the cohomology of some simple examples. Finally, we prove Poincar´eduality for orientable manifolds. Contents 1. Introduction 1 2. Smooth Homotopy Invariance 4 3. Mayer{Vietoris 5 4. Applications 7 5. Poincar´eDuality 7 Acknowledgments 10 References 10 1. Introduction Definition 1.1. A fiber bundle E over a topological space B with fiber a given topological space F is a space along with a continuous surjection π such that for each x 2 E there exists a neighborhood π(x) 2 U ⊂ B and a homeomorphism φ such that the following diagram commutes, φ π−1(U) / U × F π i Ô U where i is the usual map, i(p; v) = p. We call F the fiber and B the base space. Example 1.2. The trivial bundle over a base space B with fiber F is just B × F with the maps π = i and φ = id. Definition 1.3. A section s: B ! E of a fiber bundle is a continuous map with the property that π(s(p)) = p for all p 2 B. If B and E are smooth manifolds, we require that s is smooth. Date: August, 2014. 1 2 REDMOND MCNAMARA Definition 1.4. A vector bundle is a fiber bundle where the fiber is a vector space and with the additional properties that: (1) For all x 2 E and all corresponding U; V ⊂ B and φ, we have ◦ −1 φ (x; v) = (x; gUV (v)) for some invertible linear function gUV depend- ing only on U and V . (2) For all U; V; W , as above, we have gUU = id and gUV ◦ gVW ◦ gWU = id. Intuitively, any three transition maps must agree. Remark 1.5. In this paper, we concern ourselves with vector bundles over manifolds. Definition 1.6. A tangent vector v on a n-dimensional manifold M at a point p is an equivalence class of all smooth curves fγαg such that γα :(−1; 1) ! M, γα(0) = p and given some chart (U; f) such that p 2 f(U) we have −1 −1 D(f ◦ γα) f −1(p) = D(f ◦ γβ) f −1(p); for all γα ∼ γβ where D(·) f −1(p) is just the usual derivative on Euclidean space at f −1(p). The space of all vectors at a point p is called the tangent space of M at p, denote TpM. The space of all vectors at all points on the manifold is the tangent bundle TM and a section of the tangent bundle is a vector field on M. Proposition 1.7. For all M an n-dimensional manifold and p 2 M we have ∼ n TpM = R . Proposition 1.8. The tangent bundle is a vector bundle. Definition 1.9. For a vector bundle E over the base space B with fiber F , we have the dual bundle E∗ over base space B with fiber F ∗, a E∗ ≡ π−1(p)∗ p2B a ≡ fy : y ◦ φ−1 is a linear map on fpg × F g p2B We define π∗(π−1(p)) = fpg and φ∗(y) = y ◦ φ. The new transition maps are ∗ T −1 given by gUV = (gUV ) . The cotangent bundle is the dual of the tangent bundle, (TM)∗ ≡ T ∗M. Definition 1.10. Define the kth exterior power of a vector space F with itself O ΛkF ≡ F= ∼; k where v1 ⊗· · ·⊗w ⊗· · ·⊗w ⊗· · ·⊗vk ∼ 0 and of course two elements of the exterior power are equivalent if their difference is 0. For the element fv1 ⊗· · ·⊗vkg we write k m k+m v1 ^ · · · ^ vk and define ^:Λ F ⊕ Λ F ! Λ F by ^(v1 ^ · · · ^ vk; vk+1 ^ · · · ^ vk+m) ≡ v1 ^ · · · ^ vk+m: INTRODUCTION TO DE RHAM COHOMOLOGY 3 Definition 1.11. For a vector bundle E over the base space B with fiber F we define the kth exterior power a ΛkE ≡ Λkπ−1(p) p2B a ≡ f(x1; : : : ; xk): xi 2 E; π(xi) = π(xj)8i; jg ∼; p2B where (x1; : : : ; xk) ∼ (y1; : : : ; yk) whenever φ(x1) ^ · · · ^ φ(xk) = φ(y1) ^ · · · ^ φ(yk): k with maps Λ φ(x1; : : : ; xk) = (p; φ(x1) ^ · · · ^ φ(xk)) and π = i ◦ φ. The transition k th maps Λ gUV are given by taking the k exterior power of the operator gUV in the traditional, vector-space sense. Remark 1.12. The dual and the exterior product not only define new objects in a category to associated objects. (We would say elements and set except the collection of all vectors spaces or all vector bundles is not properly a set since it's too big). Both the dual and the exterior product are functors because they also send arrows to arrows. (Again, we would say functions except we may want to look at certain kinds of functions like strictly linear functions or may want to use objects which aren't sets). This is true on bundles as well as on vector spaces. Given a linear function f : V ! W , we get a function f ∗ : W ∗ ! V ∗ where f ∗(v)(w) = v∗(f(w)). Similarly, given a linear function f : V ! W , we get another function Λkf :ΛkV ! k Λ W by f(v1 ^ · · · ^ vk) = f(v1) ^ · · · ^ f(vk). Definition 1.13. The space of k-forms on a manifold M, denoted Ωk(M) or Ωk when the manifold is clear, is the space of sections of ΛkT ∗M. Definition 1.14. Pick a basis v1; : : : ; vn for a fiber R at a point p 2 M. Define the exterior derivative d:Ωk ! Ωk+1 at p by X @a d(a(p)w1 ^ · · · ^ wk) ≡ pvm ^ w1 ^ · · · ^ wk: @vm 1≤m≤n This extends linearly to sums. Definition 1.15. A chain complex is a sequence of abelian groups C0;C1;C2;::: and a sequence of homomorphism called boundary maps @1 @2 @3 C0 − C1 − C2 −· · · th with the property that @k ◦ @k+1 = 0 for all k. Then the k homology group is defined by Hk ≡ ker(@k)=im(@k+1). Notice that this is well defined since the groups are abelian and @k ◦ @k+1 = 0 implies ker(@k) ⊃im(@k+1). A cochain complex is a sequence of abelian groups C0;C1;C2;::: and a sequence of homomorphisms called coboundary maps 0 1 2 C0 −!δ C1 −!δ C2 −!·δ · · with the property that δk+1 ◦ δk = 0. Then the kth cohomology group is defined by Hk ≡ ker(δk)=im(δk−1). Lemma 1.16. The exterior derivative is a homomorphism with the property that d2 = 0. 4 REDMOND MCNAMARA Definition 1.17. The De Rham cohomology of a manifold is the cohomology generated by the cochain complex Ω0 −!d Ω1 −!d Ω2 −!·d · · If ! 2 ker(d) we say ! is closed. If ! 2im(d) we say ! is exact. For the rest of the paper, we deal with De Rham cohomology. 2. Smooth Homotopy Invariance Definition 2.1. Given a map f : M ! N, the pullback map f ∗ :Ωk(N) ! Ωk(M) is defined by ∗ −1 −1 f (!) p(v1; : : : ; vk) ≡ ! f(p)(Df v1; : : : ; Df vk): Proposition 2.2. The pullback commutes with the exterior derivative i.e. the following diagram commutes. d Ωk(N) / Ωk+1(N) f ∗ f ∗ d Ωk(M) / Ωk+1(M) Proposition 2.3. The pullback commutes with the wedge product i.e. the following diagram commutes. ^ Ωk(N) ⊕ Ωm(N) / Ωk+m(N) f ∗ f ∗ ^ Ωk(M) ⊕ Ωm(M) / Ωk+m(M) Definition 2.4. We say a map h: M ×I ! N is a smooth homotopy if it is smooth. We say two maps are smoothly homotopic, f : M ! N and g : M ! N, if there exists a smooth homotopy h with h(x; 0) = f(x) and h(x; 1) = g(x). We say two manifolds M and N are smooth homotopy equivalent if there exist smooth maps f : M ! N and g : N ! M such that g ◦ f and f ◦ g are smoothly homotopic to the identity maps on M and N. Theorem 2.5. If two manifolds M and N are smoothly homotopy equivalent then their kth cohomology groups are isomorphic for all k. Proof. Suppose f : M ! N and g : N ! M as above, with h: M × I ! M such that h(x; 0) = x and h(x; 1) = g(f(x)). Let ! 2 Hk((g ◦ f)(M)). Then there exists a pullback h∗(!) ≡ η + dt ^ α; where η 2 Hk(M × I), α 2 Hk−1(M × I), neither η nor α contain a dt term and t is the coordinate in I. Furthermore, for all t0 2 I, h(·; t0): M ! M induces the INTRODUCTION TO DE RHAM COHOMOLOGY 5 pullback ∗ h (·; t0)(!) = η t=t0 Since η varies smoothly in time, we can apply the fundamental theorem of calculus, Z 1 ∗ ∗ @η h (·; 1)(!) − h (·; 0)(!) = η t=1 − η t=0 = dt 0 @t We have two exterior derivative operators which we will temporarily distinguish, dM and dM×I . Because ! is closed and the pullback commutes with the exterior derivative, @η 0 = h∗(d (!)) = d (h∗(!)) = dt ^ ( − d (α)): M M×I @t M @η So @t = dM (α). Combining these last two equations yields, Z 1 Z 1 ∗ ∗ h (·; 1)(!) − h (·; 0)(!) = dM (α)dt = dM αdt 0 0 which is an exact form. Therefore, (g ◦ f)∗ = h∗(·; 1) is the identity map on Hk so f ∗ and g∗ are inverses so they are isomorphisms.
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