Let $c: S^1 \to M$ be a contractible smooth loop. Is it true that $c(S^1)$ has a neighbourhood $U$ such that $H^2(U;\mathbb{R})=0$?
If $c$ is an embedding this is clear by tubular neighbourhood, but I don't know how to tackle the general case. I suspect the result to be true without the contractibility assumption, but I only need it in this case for my purposes.