Knowlpedia

Theorem

de Rham theorem

Integration identifies de Rham cohomology naturally with singular cohomology with real coefficients.

Statement

For every MM, over smooth singular simplices defines a cochain map

I:Ωk(M)⟶Csmk(M;R),I(ω)(σ)=∫Δkσ∗ω.\mathcal I:\Omega^k(M)\longrightarrow C_{\mathrm{sm}}^k(M;\mathbb R), \qquad \mathcal I(\omega)(\sigma)=\int_{\Delta^k}\sigma^*\omega.

The de Rham theorem states that the induced map

HdRk(M)→  ≅  Hk(M;R)H_{\mathrm{dR}}^k(M)\xrightarrow{\;\cong\;}H^k(M;\mathbb R)

is an isomorphism for every kk. Thus the cohomology of the of smooth forms agrees naturally with with real coefficients. No orientation or compactness assumption on MM is required.

Naturality and products

For a f:M→Nf:M\to N, and pullback of singular cochains commute with the de Rham isomorphism. On cohomology, the isomorphism also identifies with cup products, so it is an isomorphism of graded real algebras, not merely of graded Bott and Tu, Chapter I.

Proof architecture

One proof uses the to establish local exactness and to pass from local to global information. Another proceeds by showing that both theories satisfy compatible , checking the result on contractible coordinate neighborhoods, and then gluing. Smooth singular cochains compute the same cohomology as ordinary singular cochains.

Coefficients and compact supports

The target is real singular cohomology. The theorem does not identify de Rham cohomology with integral cohomology: torsion information disappears over R\mathbb R. There is a separate compact-support version relating to singular cohomology with compact supports; its support conditions should not be silently inserted into the ordinary theorem.

References
  1. Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter I, the de Rham theorem, products, and Mayer–Vietoris method.
  2. Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: the chapter on de Rham theory.