I have a stupid and probably naive question about one line in the book of Milnor about Morse theory. What does exactly means if $v \in T_pM$ then there is an associated vector field $\tilde v $ ?
I have a kind of vague idea of what could be this vector fields (identify $T_pM$ and $T_qM$ for $p,q$ close and show that this does not depends of this identification, and that we can extend this to all $M$).
Milnor just means that $\tilde{v}$ is any vector field such that $\tilde{v}_p=v$. Any $v\in T_p M$ can be extended to a vector field. See Lee, Smooth Manifolds, page 84, Lemma 4.5 for a proof.
Also, where he says "Poisson bracket" he meant to say "Lie bracket".