Question

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)?

Answer 1

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 $\alpha_1\wedge\beta_1 = 0$.