Is it possible to embed de Rham cohomology of a two-dimensional closed surface of genus $g\geq 2$ into the differential graded algebra of differential forms (with de Rham differential and wedge product) on the surface as a differential graded subalgebra (in a way that is compatible with canonical projection from closed forms to cohomology, associating to a closed form its cohomology class)?



Formality of de Rham algebra for two-dimensional closed surfaces
The answer is *no*. Suppose that $\alpha\_1,\ldots,\alpha\_g,\beta\_1,\ldots,\beta\_g$ were closed $1$-forms on $M$ such that their cohomology classes were a basis of $H^1(M)$ and they satisfied $\alpha\_i\wedge\alpha\_j = \beta\_i\wedge\beta\_j = 0$ while $\alpha\_i\wedge\beta\_j = \delta\_{ij}\ \gamma$ where $\gamma$ is a single $2$-form whose cohomology class is nonzero. Then $\gamma$ cannot vanish identically. 


Let $U\subset M$ be the open set on which $\gamma$ is nonzero. Then none of the $\alpha\_i$ or $\beta\_i$ can vanish on $U$. Since $\alpha\_1\wedge\alpha\_2 = \beta\_1\wedge\alpha\_2=0$ while $\alpha\_2\not=0$ on $U$, it follows that $\alpha\_1$ and $\beta\_1$ are multiples of $\alpha\_2$ on $U$. But this implies that $\alpha\_1$ and $\beta\_1$ must be linearly dependent on $U$ as well, which implies that $\alpha\_1\wedge\beta\_1 = 0$. 


