De Rham Cohomology
In Chapter 14 , we defined closed and exact differential forms, and showed that every exact form is closed. In this chapter, we explore the converse question: Is every closed form exact? The answer is locally yes, but globally no. The question of which c
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De Rham Cohomology
In Chapter 14 we defined closed and exact forms: a smooth differential form ! is closed if d! D 0, and exact if it can be written ! D d. Because d ı d D 0, every exact form is closed. In this chapter, we explore the implications of the converse question: Is every closed form exact? The answer, in general, is no: in Example 11.48, for instance, we saw a 1-form on the punctured plane that is closed but not exact; the failure of exactness seemed to be a consequence of the “hole” in the center of the domain. For higher-degree forms, the question of which closed forms are exact depends on subtle topological properties of the manifold, connected with the existence of “holes” of higher dimensions. Making this dependence quantitative leads to a new set of invariants of smooth manifolds, called the de Rham cohomology groups, which are the subject of this chapter. Knowledge of which closed forms are exact has many important consequences. For example, Stokes’s theorem implies that if ! is exact, then the integral of ! over any compact submanifold without boundary is zero. Proposition 11.42 showed that a smooth 1-form is conservative if and only if it is exact. We begin by defining the de Rham cohomology groups and proving some of their basic properties, including diffeomorphism invariance. Then we prove an important generalization of this fact: the de Rham groups are in fact homotopy invariants, which implies in particular that they are topological invariants. Next, after computing the de Rham groups in some simple cases, we state a general theorem, called the Mayer–Vietoris theorem, that expresses the de Rham groups of a manifold in terms of those of its open subsets. Using this, we compute all of the de Rham groups of spheres and the top-degree groups of compact manifolds. Then we give an important application of these ideas to topology: there is a homotopically invariant integer associated with any continuous map between connected, compact, oriented, smooth manifolds of the same dimension, called the degree of the map. At the end of the chapter, we prove the Mayer–Vietoris theorem. J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics 218, 440 DOI 10.1007/978-1-4419-9982-5_17, © Springer Science+Business Media New York 2013
The de Rham Cohomology Groups
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The de Rham Cohomology Groups In Chapter 11, we studied the closed 1-form !D
x dy y dx ; x2 C y2
(17.1)
and showed that it is not exact on R2 X f0g, but it is exact on some smaller domains such as the right half-plane H D f.x; y/ W x > 0g, where it is equal to d (see Example 11.48). As we will see in this chapter, that behavior is typical: closed forms are always locally exact, so whether a given closed form is exact depends on the global shape of the domain. To capture this dependence, we make the following definitions. Let M be a smooth manifold with or without boundary, and let p be a nonnegative integer. Because d W p .M / ! pC1 .M / is linear, its kernel and image are linear subspaces. We define Zp .M / D Ker
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