Contents

and

# 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 $k$ 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 $k$ and cochain complex for singular cohomology with coefficients in $k$

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

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

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

(for $X$ any topological space).

## 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.