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Introduction to De Rham Cohomology

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Introduction to de Rham cohomology Pekka Pankka December 11, 2013 Preface These are lecture notes for the course \Johdatus de Rham kohomologiaan" lectured fall 2013 at Department of Mathematics and Statistics at the Uni- versity of Jyv¨askyl¨a. The main purpose of these lecture notes is to organize the topics dis- cussed on the lectures. They are not meant as a comprehensive material on the topic! These lectures follow closely the book of Madsen and Tornehave \From Calculus to Cohomology" [7] and the reader is strongly encouraged to consult it for more details. There are also several alternative sources e.g. [1, 9] on differential forms and de Rham theory and [4, 5, 3] on multi- linear algebra. 1 Bibliography [1] H. Cartan. Differential forms. Translated from the French. Houghton Mifflin Co., Boston, Mass, 1970. [2] W. Greub. Linear algebra. Springer-Verlag, New York, fourth edition, 1975. Graduate Texts in Mathematics, No. 23. [3] W. Greub. Multilinear algebra. Springer-Verlag, New York, second edi- tion, 1978. Universitext. [4] S. Helgason. Differential geometry and symmetric spaces. Pure and Applied Mathematics, Vol. XII. Academic Press, New York, 1962. [5] S. Helgason. Differential geometry, Lie groups, and symmetric spaces, volume 80 of Pure and Applied Mathematics. Academic Press Inc. [Har- court Brace Jovanovich Publishers], New York, 1978. [6] M. W. Hirsch. Differential topology, volume 33 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. Corrected reprint of the 1976 original. [7] I. Madsen and J. Tornehave. From calculus to cohomology. Cambridge University Press, Cambridge, 1997. de Rham cohomology and charac- teristic classes. [8] W. Rudin. Real and complex analysis. McGraw-Hill Book Co., New York, third edition, 1987. [9] L. W. Tu. An introduction to manifolds. Universitext. Springer, New York, second edition, 2011. 2 Contents 1 Alternating algebra 5 1.1 Some linear algebra . 5 1.2 Multilinear maps . 9 1.3 Alternating multilinear maps . 12 2 Differential forms 24 2.1 Exterior derivative . 26 2.2 Pull-back of forms . 29 3 De Rham cohomology 31 3.1 Poincar´elemma . 34 3.2 Chain complexes . 36 3.3 Long exact sequence . 41 3.4 Homotopy . 46 4 Applications 52 5 Manifolds and bundles 56 5.1 Topological manifolds and bundles . 56 5.2 Smooth manifolds . 63 5.3 Tangent bundle . 65 5.4 Cotangent bundle . 71 5.5 Exterior bundles . 73 5.6 Exterior derivative . 75 5.7 de Rham cohomology . 76 6 Orientable manifolds 78 7 Integration 82 7.1 Euclidean case . 82 7.2 Manifold case . 86 7.3 Integration on domains with smooth boundary . 90 3 8 Poincar´eduality 95 8.1 Special case . 96 8.2 Exact sequence for compactly supported cohomology . 97 8.3 Exact sequence for dual spaces . 99 8.4 Main steps . 100 4 Chapter 1 Alternating algebra n Let v1; : : : ; vn be vectors in R . The volume of the parallelepiped n P (v1; : : : ; vn) = ft1v1 + ··· + tnvn 2 R : 0 ≤ ti ≤ 1; i = 1; : : : ; ng is given by jdet [v1 ··· vn]j : Whereas the absolute value of the determinant is independent on the order of vectors, the sign of det [v1 ··· vn] depends on the order of (v1; : : : ; vn). The quantity det [v1 ··· vn] is therefore sometimes called \signed volume". The role of the sign is to detect the so-called \orientation" of vectors (v1; : : : ; vn). The notion of volume and \signed volume" based on the determinant allows us to consider n-dimensional objects in n-dimensional space. In this section, we discuss multilinear algebra which allows us to consider volumes (and \signed volumes") of k-dimensional (linear) objects in n-dimensional space when k < n. We discuss these geometric ideas in later sections and develop first the necessary linear theory. 1.1 Some linear algebra We begin by recalling some facts from linear algebra; see e.g. [2, Chapter I & II] for a detail treatment. Let V be a (real) vector space, that is, we have operations +: V × V ! V; (u; v) 7! u + v; ·: R × V ! V; (a; v) 7! av; for which the triple (V; +; ·) satisfies the axioms of a vector space. 5 Linear maps and dual spaces Given vector spaces V and V 0, a function f : V ! V 0 is a called a linear map if f(u + av) = f(u) + af(v) for all u; v 2 V and a 2 R. A bijective linear map is called a (linear) isomorphism. 0 Given linear maps f; g : V ! V and a 2 R, the mapping f + ag : V ! V 0; v 7! f(v) + ag(v); is a linear map V ! V 0. Thus the set Hom(V; V 0) of all linear maps V ! V 0 is also a vector space under operations +: Hom(V; V 0) × Hom(V; V 0) ! Hom(V; V 0); (f; g) 7! f + g; 0 0 ·: R × Hom(V; V ) ! Hom(V; V ); (a; f) 7! af: ∗ An important special case is the dual space V = Hom(V; R) of V . Theorem 1.1.1 (Dual basis). Let V be a finite dimensional vector space. ∗ ∼ n n Then V = V . Furthermore, if (ei)i=1 is a basis of V , then ("i)i=1, where "i : V ! R is the map 1; i = j " (e ) = δ = i j ij 0; i 6= j; is a basis of V ∗. Proof. Exercise. ∗ The basis ("i)i of V in Theorem 1.1.1 is called dual basis (of (ei)i). Note that bases of V and V ∗ do not have the same cardinality (i.e V and V ∗ do not have the same \dimension") if V is infinite dimensional! Induced maps Given f 2 Hom(V; W ) and a linear map ': U ! V , we have a composition f ◦ ' 2 Hom(U; W ): ' U / V f f◦' W The composition with fixed map ' induces a linear map as formalized in the following lemma. 6 Lemma 1.1.2. Let U; V; W be vector spaces. Let ': U ! V be a linear map. Then '∗ : Hom(V; W ) ! Hom(U; W ); f 7! f ◦ ' is a linear map. Moreover, if ' is an isomorphism then '∗ is an isomor- phism. Similarly, we may also consider f 2 Hom(U; V ) and a fixed linear map : V ! W : f U / V f◦ W Lemma 1.1.3. Let U; V; W be vector spaces. Let : V ! W be a linear map. Then '∗ : Hom(U; V ) ! Hom(U; W ); f 7! ◦ f is a linear map. Moreover, if is an isomorphism then ∗ is an isomor- phism. The mapping '∗ in Lemma 1.1.2 is so-called pull-back (under '). The mapping ∗ is called as push-forward. As an immediate corollary of Lemma 1.1.2 we have the following observation. Corollary 1.1.4. Let U and V be vector spaces and ': U ! V a linear map. Then '∗ : V ∗ ! U ∗; f 7! f ◦ ' is a linear map. Moreover, '∗ is an if ' is an isomorphism. Subspaces and quotients A subset W ⊂ V of a vector space V is a subspace of V if u + aw 2 W for all u; v 2 W and a 2 R.A coset of an element v 2 V with respect to W is the subset v + W = fv + w 2 V : w 2 W g: The relation ∼W on V , given by the formula u ∼W v , u − v 2 W , is an equivalence relation with equivalence classes fv + W : v 2 V g. The set V=W of these equivalence classes has a natural structure of a vector space given by operations +: V=W × V=W ! V=W; ((u + W ); (v + W )) 7! (u + v) + W; ·: R × V=W ! V=W; (a; v + V=W ) 7! (av) + W; that is, (u + W ) + a(v + W ) = (u + av) + W 7 for all u; v 2 W and a 2 R. The space V=W is called the quotient space of V (with respect to W ). Note that the mapping π : V ! V=W; v 7! v + W; is linear. The mapping p is called quotient (or canonical) map. A fundamental fact of on linear mappings is that the kernel and the image are vector spaces, that is, let f : V ! V 0 be a linear map between vector spaces, then ker f = f −1(0) = fv 2 V : f(v) = 0g Imf = f[V ] = ff(v) 2 V 0 : v 2 V g are subspaces of V and V 0, respectively. Theorem 1.1.5 (Isomorphism theorem). Let f : V ! V 0 be a linear map between vector spaces V and V 0. Then the mapping ': V= ker f ! Imf, v + W 7! f(v) + W , is a linear isomorphism satisfying f V / Imf : =∼ p ' # V= ker f where p: V ! V= ker f is the canonical map. The proof is left as a voluntary exercise. Products and sums Given a set I (finite or infinite) and a vector space Vi for each i 2 I, the Q product space i2I Vi has a natural linear structure given by 0 0 (vi)i + a(vi)i = (vi + avi)i 0 Q where vi; vi 2 Vi and a 2 R. For I = ;, we declare i2I Vi = f0g. I Q I If Vi = Vj =: V for all i; j 2 I, denote V = i2I V . Note that, V is, in fact, the space of all functions I ! V . Note also that, for n 2 N, we have V n = V × · · · × V =∼ V f1;:::;ng = fall functions f1; : : : ; ng ! V g: The sum of two vector spaces V ⊕ W has (at least) three different mean- ings in the literature.
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