Let $M$ be a differential manifold and $\mathcal H^k$ the presheaf of real vector spaces associating to the open subset $U\subset M$ the $k$-th de Rham cohomology vector space: $\mathcal H^k(U)=H^k_{DR}(U)$. Is this presheaf a sheaf?

Of course not! Indeed, given *any* non-zero cohomology class $0\neq[\omega]\in \mathcal H^k(U)$ represented by the closed $k$-form $\omega\in \Omega^k_M(U)$ there exists (by Poincaré's Lemma) a covering $(U_i)_{i\in I}$ of $U$ by open subsets $U_i\subset U$ such that $[\omega]\vert U_i=[\omega\vert U_i]=0\in \mathcal H^k(U_i)$, and thus the first axiom for a presheaf to be a sheaf is violated.

But what about the second axiom?

**My question:**

Suppose we are given a differential manifold M, a covering $(U_\lambda)_{\lambda \in \Lambda}$of $M$ by open subsets $U_\lambda \subset M$, closed differential $k-$forms $\omega_\lambda \in \Omega^k_M(U_\lambda)$ satisfying $[\omega_\lambda]\vert U_\lambda \cap U_\mu=[\omega_\mu]\vert U_\lambda \cap U_\mu\in \mathcal H^k(U_\lambda\cap U\mu)$ for all $\lambda,\mu \in \Lambda$.

Does there then exist a closed differential form $\omega\in \Omega^k(M)$ such that we have for the restrictions in cohomology: $[\omega]\vert U_\lambda=[\omega _\lambda]\in \mathcal H^k(U_\lambda)$ for all $\lambda\in \Lambda$ ?

**Remarks**

- This is an extremely naïve question which, to my embarrassment, I cannot solve.

I have extensively browsed the literature and consulted some of my friends, all brilliant geometers (albeit not differential topologists), but they didn't know the answer offhand. For what it's worth, I would guess (but not conjecture!) that such glueing is impossible. - If the covering of $X$ has only two opens then we can glue.

This follows immediately from Mayer-Vietoris's long exact sequence $$\cdots \to \mathcal H^k(M) \to \mathcal H^k(U_1) \oplus \mathcal H^k(U_2) \to \mathcal H^k(U_1\cap U_2)\to \cdots$$

**Update**

My brilliant friends didn't answer offhand but a few hours later, unsurprisingly, they came back to me with splendid counterexamples! See below.

cohomologysheaves of the De Rham complex in the smooth category (i.e. the quotient sheaves of the respective kernel and image sheaves under $d$). I've actually never thought about it! $\endgroup$