de Rham theorem

In mathematics, more specifically in differential geometry, the de Rham theorem says that the ring homomorphism from the de Rham cohomology to the singular cohomology given by integration is an isomorphism.

The Poincaré lemma implies that the de Rham cohomology is the sheaf cohomology with the constant sheaf $\mathbb {R}$. Thus, for abstract reason, the de Rham cohomology is isomorphic as a group to the singular cohomology. But the de Rham theorem gives a more explicit isomorphism between the two cohomologies; thus, connecting analysis and topology more directly.

Statement

The deRham homomorphism is well defined because for any ${\widetilde {c}},{\widetilde {c}}'$ representing the same homology class ${\widetilde {c}}-{\widetilde {c}}'=\partial {\widetilde {b}}$, so $\int _{\widetilde {c}}\omega -\int _{{\widetilde {c}}'}\omega =0'$ and if $\omega$ is an exact form then $\int _{\widetilde {c}}\omega =0$

The key part of the theorem is a construction of the de Rham homomorphism. Let M be a manifold. Then there is a map

$k:\Omega ^{p}(M)\to S_{{\mathcal {C}}^{\infty }}^{p}(M)$

from the space of differential p-forms to the space of smooth singular p-cochains given by

$\omega \mapsto \left(\sigma \mapsto \int _{\sigma }\omega \right).$

Stokes' formula implies: $k\circ d=\partial \circ k$; i.e., $k$ is a chain map and so it induces:

$[k]:\operatorname {H} _{\textrm {deRham}}^{*}(M)\to \operatorname {H} _{\mathrm {sing} }^{*}(M)$

where these cohomologies are the cohomologies of $\Omega ^{*}(M)$ and $S_{{\mathcal {C}}^{\infty }}^{*}(M)$, respectively. As it turns out, $[k]$ is a ring homomorphism and is called the de Rham homomorphism.

Finally, the theorem says that the induced homomorphism $[k]$ is an isomorphism (i.e., bijective).

There is also a variant of the theorem that says the de Rham cohomology of M is isomorphic as a ring with the Čech cohomology of it. This Čech version is essentially due to André Weil.

Singular-homology version

There is also a version of the theorem involving singular homology instead of cohomology. It says the pairing

$(\omega ,\sigma )\mapsto \int _{\sigma }\omega$

induces a perfect pairing between the de Rham cohomology and the (smooth) singular homology; namely,

$\operatorname {H} _{\mathrm {deRham} }^{*}(M)\to \operatorname {H} _{*}^{\mathrm {sing} }(M)^{*},\,[\omega ]\mapsto \left([\sigma ]\mapsto \int _{\sigma }\omega \right)$

is an isomorphism of vector spaces.

This theorem has the following consequence (familiar from calculus); namely, a closed differential form is exact if and only if the integrations of it over arbitrary cycles are all zero. For a one-form, it means that a closed one-form $\omega$ is exact (i.e., admits a potential function) if and only if $\int _{\gamma }\omega$ is independent of a path $\gamma$. This is exactly a statement in calculus.

Current version

There is also a current (a differential form with distributional coefficients) version of the de Rham theorem, which says the singular cohomology can be computed as the cohomology of the complex of currents. This version is weaker in the sense that the isomorphism is not a ring homomorphism (since currents cannot be multiplied and so the space of currents is not a ring).

References

• Griffiths, Phillip; Harris, Joseph (1994). Principles of Algebraic Geometry. doi:10.1002/9781118032527. ISBN 978-0-471-05059-9.
• Warner, Frank W. (1983). Foundations of Differentiable Manifolds and Lie Groups. New York: Springer. ISBN 0-387-90894-3.
• Weil, André (1952). "Sur les théorèmes de de Rham". Commentarii Mathematici Helvetici. 26: 119–145. doi:10.1007/BF02564296. S2CID 124799328.