and

**rational homotopy theory** (equivariant, stable, parametrized, equivariant & stable, parametrized & stable)

**Examples of Sullivan models** in rational homotopy theory:

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.

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)

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

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