The following theorem is well-known:



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> **Theorem.** Let $M$ be a smooth manifold. A smooth covector field $\omega \in \Omega^1(M)$ is conservative, that is, $$\int\_{\mathbb{S}^1}f^\*\omega = 0 \qquad \forall f \in C^\infty(\mathbb{S}^1,M),$$ if and only if $\omega$ is exact.
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For a proof, see for example Theorem 11.42 in the brilliant book on smooth manifolds by @JackLee. I wonder if one can generalise this result as follows:


Let $\omega \in \Omega^k(M)$ such that $$\int\_{\mathbb{S}^k}f^\*\omega = 0 \qquad \forall f \in C^\infty(\mathbb{S}^k,M).$$ Does then also follow that $\omega$ is exact?



Generalisation of conservative covector fields
Consider a 2-torus. The second homotopy is trivial, so the spheres can't feel the homology, or cohomology. The same for any manifold obtained by quotienting a simply connected manifold by a cocompact discrete group action, so all surfaces of genus 2 or more.


