de Rham cohomology

In the case that there are indeed “holes” present, we can still relate them to closed and exact forms. The de Rham cohomology groups are simple to construct; similar to the singular homology groups \({H_{n}(X)\equiv\textrm{Ker}\partial_{n}/\textrm{Im}\partial_{n+1}}\), they are the quotient groups \({\textrm{Ker d}_{n}/\textrm{Im d}_{n-1}}\), or the closed \({n}\)-forms modulo the exact \({n}\)-forms. Thus a de Rham cohomology class is a coset of closed forms that differ by an exact form.

Consider the integral of a closed \({k}\)-form \({\varphi}\) over a \({k}\)-cycle \({c}\) in \({M}\), i.e. \({\int_{c}\varphi}\) where \({\mathrm{d}\varphi=0}\) and \({\partial c=0}\). Stokes’ theorem says that this number is invariant if either \({c}\) changes by a boundary or \({\varphi}\) changes by an exact form:

\(\displaystyle \int_{c+\partial V}\varphi=\int_{c}\varphi+\int_{V}\mathrm{d}\varphi=\int_{c}\varphi \)

\(\displaystyle \int_{c}\varphi+\mathrm{d}\psi=\int_{c}\varphi+\int_{\partial c}\psi=\int_{c}\varphi \)

This integral can thus be viewed as a mapping from a de Rham cohomology coset represented by the closed \({k}\)-form \({\varphi}\) to the real functions on a homology coset represented by the \({k}\)-cycle \({c}\). But this last is just the singular cohomology groups \({H^{k}(M;\mathbb{R})}\), so that we have a mapping \({\int_{c}\varphi:H_{\textrm{de Rham}}^{k}\rightarrow H^{k}\left(M;\mathbb{R}\right)}\). The de Rham theorem states that this mapping is an isomorphism, so that the de Rham and singular cohomology groups with real coefficients are identical for manifolds.

This allows us to deduce information about forms from topological properties. For example, if a manifold \({M}\) has Betti number \({b_{k}=0}\), then \({H^{k}=0}\) and so every closed \({k}\)-form on \({M}\) is exact.

For manifolds, our intuitive picture of \({n}\)-cycles as “closed surfaces within a space” is quite literal. Every closed oriented submanifold \({C^{k}}\) of \({M^{n}}\) defines a \({k}\)-cycle, and a converse is provided by Thom’s theorem: every \({k}\)-cycle with real coefficients in \({M^{n}}\) is homologous to a real \({k}\)-chain \({\Sigma r_{i}V_{i}^{k}}\) of closed oriented submanifolds \({V_{i}^{k}\subset M^{n}}\).

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