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Two Proofs of the De Rham Theorem

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Two Proofs of the De Rham Theorem TWO PROOFS OF THE DE RHAM THEOREM ANDREW FANOE Contents Introduction 1 1. The Basics 3 1.1. The calculus of di®erential forms 3 1.2. Elementary homological algebra 8 1.3. The DeRham complexes 10 1.4. The Poincar¶elemma 12 2. Simplicial Complexes 15 2.1. Basic Concepts 15 2.2. Simplicial Homology 17 3. A Simplicial Approach to the De Rham Theorem 19 3.1. A Simplicial Approach to the De Rham Theorem 19 4. Singular Cohomology Theory 29 4.1. Singular Homology and Cohomology 29 4.2. The General Mayer-Vietoris Principle 31 5. Andr¶eWeil's Approach to the De Rham Theorem 32 5.1. Double Complexes 32 5.2. De Rham Theorem 34 References 38 Introduction The main goal of this paper is to state and prove the De Rham Theorem in two di®erent ways. We will work exclusively in the realm of smooth manifolds, and we will discuss various di®erent ways of associating cohomology groups to a smooth manifold. Of primary concern for us will be the language of di®erential forms. At this point, we wish to give the reader some geometric intuition for the De Rham Theorem, since the treatment in the paper will be largely technical. In the discussion that follows, we will consider the question of when a planar vector ¯eld on R2 is a gradient vector ¯eld. That is, we wish to decide when, for a vector ¯eld F~ (x; y) = P (x; y)i + Q(x; y)j de¯ned on a region D ½ R2, there exists a function f(x; y) on D such that @f @f rf := i + j = P i + Qj @x @y 1 2 ANDREW FANOE If this happens, we will say that F~ is exact. We see that by the equality of mixed partial derivatives that if F~ = P i + Qj is exact, then @P @Q = @y @x In general, we say that if this condition is satis¯ed, then the vector ¯eld is closed. We wish to know if a closed vector ¯eld is exact. It turns out that if we have a closed vector ¯eld F~ on a convex subset D ½ R2, then the vector ¯eld is exact on D. Also, in this case, we have that Z Z F~ ¢ ddr = P dx + Qdy = 0; ~r = xi + yj; C C for any closed path C in D. Now, consider the vector ¯eld F~ (x; y) given by ¡y x F~ (x; y) = i + j x2 + y2 x2 + y2 With this de¯nition, it is easily checked that, on R2 ¡ (0; 0) @P @Q y2 ¡ x2 = = ; @y @x (x2 + y2)2 so that F~ is closed. However, it can be shown that F~ is not exact. Furthermore, if we let S1 denote the unit circle, then we have Z F~ d~r = 2¼ S1 The di®erence between these two cases is that in the ¯rst case we had a convex subset, and in the other case we have the set R2 ¡ (0; 0). We see that this particular vector ¯eld is not a gradient because it has strange behavior near zero. We'd like to say the problem is that (0; 0) is missing from the space, or that it is in some sense a \hole" in our space. Of course, there are a priori many di®erent ways to de¯ne holes as subsets of R2. One way is to look at integrals over closed paths of closed vector ¯elds. If we happen to ¯nd a closed vector ¯eld with integral not equal to 0 over a closed path, then we can say that this path surrounds a hole in our space. Similarly, we could say that if we have a closed path which we cannot homotopically shrink to a point, then we say this path surrounds a hole. There are of course other possible de¯nitions. Intuitively, in more abstract settings, the various di®erent ways of de¯ning holes in a space are described by the various de¯nitions of the cohomology of a manifold, and the De Rham theorem says that all of these methods of measuring holes are the same. In e®ect, the De Rham theorem tells us that it does not matter how we compute how many holes there are in our manifold, we will always get the same answer. To prove this striking theorem, we will ¯rst discuss in great detail the language of dif- ferential forms, stating many important theorems, de¯ning the De Rham cohomology, and proving the Poincar¶eLemma. In section 2 we will discuss the notion of simplicial complexes, and will develop the theory of simplicial homology and cohomology, de¯ning many key concepts that will be used in the proof of the De Rham Theorem. In section 3, we will de¯ne the notion of a smoothly triangulated manifold, and we will also de¯ne a period map which will go from di®erential forms on M to simplicial cochains on the triangulation of M. Finally, we will prove that this map is an isomorphism in cohomology, which is the ¯rst proof of the De Rham Theorem. TWO PROOFS OF THE DE RHAM THEOREM 3 In section 4, we will develop the notion of singular homology and cohomology, and we will discuss the generalized Mayer-Vietoris exact sequence, proving its exactness. Finally, in section 5, we will build up the notion of double complexes, which we will then use to de¯ne another period map which goes from di®erential forms on M to singular cochains on M. We will then prove that this map induces isomorphisms in cohomology, which will be the second proof of the De Rham theorem. We o®er two proofs of the De Rham theorem in this paper because the two proofs represent two widely di®erent views of the subject. The ¯rst proof is given in a very classical setting, and represents the classical point of view, whereas the second proof uses very modern machinery and represents a more modern point of view. Both of these points of view have merit, and so we demonstrate them both. At this point, I would like to thank all the people who made this thesis possible. I would like to thank all the faculty of the University for their help in building up my mathematical career and teaching me all that I know about math. Most of all, I would like to thank my advisor, Professor Liviu Nicolaescu, without whom none of this would have been possible. He has been there since the beginning of this project, helping me put together the paper you see here. 1. The Basics 1.1. The calculus of di®erential forms. We want survey without proofs the basic facts concerning the calculus of di®erential forms on a smooth manifold. For details we refer to [4]. For any smooth manifold M we denote by Vect(M) the vector space of smooth vector ¯elds on M, and by ­k(M) the vector space of di®erential forms of degree k, i.e., maps ! : Vect(M) £ ¢ ¢ ¢ £ Vect(M) ! C1(M); | {z } k 1 such that, for every X1;:::;Xk 2 Vect(M), f1; : : : ; fk 2 C (M) and any permutation σ of f1; : : : ; kg we have ¡ ¢ ! Xσ(1);:::;Xσ(k) = ²(σ)!(X1;:::;Xk); !(f1X1; : : : ; fkXk) = (f1 ¢ ¢ ¢ fk)!(X1;:::;Xk); where ²(σ) 2 f§1g denotes the signature of the permutation σ. By de¯nition, ­0(M) is the space C1(M) of smooth real valued functions on M. We form the graded vector space dimMM ­²(M) := ­k(M): k=0 We say that ! 2 ­²(M) is homogeneous if it belongs to one of the summands ­k(M). For a homogeneous element ! we denote by j!j its degree. A linear map L : ­²(M) ! ­²(M) is called homogeneous of degree q if it maps any homogeneous form ! to a homogeneous form L! and jL!j = j!j + q: The space ­²(M) is an associative R-algebra with respect to the wedge or exterior product ^ : ­²(M) £ ­²(M) ! ­²(M): 4 ANDREW FANOE The product of any two homogeneous forms !, ´ is a homogeneous form ! ^ ´ and j! ^ ´j = j!j + j´j;! ^ ´ = (¡1)j!j¢j´j´ ^ !: A di®erential form of degree k can also be interpreted as a smooth section of the vector bundle ¤kT ¤M, the k-th exterior power of the cotangent bundle T ¤M, k ¤ k ¤ ! : M ! ¤ T M; M 3 x 7! !x 2 ¤ Tx M: The support of ! is de¯ned as the closed set n o k ¤ supp ! := closure x 2 M; !x 6= 0 2 ¤ Tx M : k We denote by ­c (M) the space of smooth di®erential forms of degree k with compact support. The exterior derivative on M is the homogeneous linear operator d : ­²(M) ! ­²(M) of degree 1 uniquely determined by the following conditions. For every pair of homogeneous forms !; ´ we have j!j d(! ^ ´) = (d!) ^ ´ + (¡1) ! ^ (d´): (P ¡) d2 = 0: (1.1) 8f 2 C1(M) = ­0(M); df is the di®erential of f: Any vector ¯eld X 2 Vect(M) determines a homogeneous linear operator of degree ¡1 ² ² iX : ­ (M) ! ­ (M) called the contraction with X and de¯ned by k iX !(X2;:::;Xk) = !(X; X2;:::;Xk); 8! 2 ­ (M);X2;:::;Xk 2 Vect(M): The operator iX satis¯es the odd product rule (P ¡ ). The Lie derivative along a vector ¯eld X 2 Vect(M) is the homogeneous linear operator of degree 0 ² ² LX : ­ (M) ! ­ (M) uniquely determined by the following conditions. ² 8!; ´ 2 ­ (M);LX (! ^ ´) = (LX !) ^ ´ + ! ^ (LX !); (P +) 1 8f 2 C (M);LX f = df(X); (1.2) 1 ¡ ¢ 8® 2 ­ (M);Y 2 Vect(M); (LX ®)(Y ) = LX ®(Y ) ¡ ®([X; Y ]); (1.3) where [X; Y ] denotes the Lie bracket of the vector ¯elds X; Y .
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