nLab PL de Rham theorem

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).

(Bousfield-Gugenheim 76, Theorem 2.2)

Related statements

• PL de Rham complex of smooth manifold is equivalent to de Rham complex

• equivariant PL de Rham theorem

• de Rham theorem

References

• Aldridge Bousfield, Victor Gugenheim, On PL deRham theory and rational homotopy type, Memoirs of the AMS, vol. 179 (1976) (ams:memo-8-179)