Let $c: S^1 \to M$ be a smooth contractible loop (not necesarily an embedding, or even an immersion) on the connected, compact symplectic manifold $(M,\omega)$ (if this helps somehow, $c$ is a $1$-periodic orbit of a $1$-periodic time-dependent Hamiltonian $H_t$). The book by Audin, Damian claims that there exists a neighbourhood $U$ of $c(S^1)$ that retracts (and by this, they mean "deformation retracts") onto $c(S^1)$.

What is needed, however, is the fact that there exists a neighbourhood $U$ of $c(S^1)$ such that $H^2(U;\mathbb{R})=0$. My question is: why is this true?

If $\dim(M)=2$, then this follows by considering $p \notin c(S^1)$ and taking $U:=M\backslash\{p\}$. Since $U$ is a non-compact connected $2$-manifold, it follows that $H^2(U;\mathbb{R})=0$. However, I can't see how to proceed in generality, since tubular neighbourhoods (which I think is what the authors had in mind) seem elusive etc.