Let ${k}$ be a field. The commutative cochain problem over ${k}$ is to assign (contravariantly) functorially, to every simplicial set ${K_\bullet}$, a commutative (in the graded sense) ${k}$-algebra ${A(K_\bullet)}$, which is naturally weakly equivalent to the algebra ${C^*(K_\bullet, k)}$ of singular cochains (with ${k}$-coefficients). We also require that ${A(K_\bullet) \rightarrow A(L_\bullet)}$ is a surjection whenever ${L_\bullet \subset K_\bullet}$. Recall that ${C^*(K_\bullet, k)}$ is an associative algebra, but it is not commutative; the commutativity only appears after one takes cohomology. The commutative cochain problem attempts to find an improvement to ${C^*(K_\bullet, k)}$.

If ${k}$ has finite characteristic, this problem cannot be solved, owing to the existence of nontrivial cohomology operations. (The answers at this MathOverflow question are relevant here.) However, there is a solution for ${k = \mathbb{Q}}$, given by the polynomial de Rham theory. In this post, I will explain this. (more…)