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PL de Rham theorem

Contents

Contents

Idea

The PL de Rham theorem is the variant of the de Rham theorem with the smooth de Rham complex replaced by the PL de Rham complex.

Statement

Proposition

(PL de Rham theorem)

Let kk be a field of characteristic zero (such as the rational numbers, real numbers or complex numbers).

Then the evident operation of integration of differential forms over simplices induces a quasi-isomorphism between the PL de Rham complex with coefficients in kk and cochain complex for singular cohomology with coefficients in kk

Ω PLdR (X)C (X;k) \Omega^\bullet_{PLdR}(X) \underoverset{}{\simeq}{\longrightarrow} C^\bullet(X; k)

and hence an isomorphism from PL de Rham cohomology to ordinary cohomology with coefficients in kk (such as rational cohomology, real cohomology, complex cohomology):

H PLdR (X)H (X;k) H^\bullet_{PLdR}(X) \underoverset{}{\simeq}{\longrightarrow} H^\bullet(X; k)

(for XX any topological space).

(Bousfield-Gugenheim 76, Theorem 2.2)

References

Last revised on September 25, 2020 at 15:34:18. See the history of this page for a list of all contributions to it.