I am reading Segal book on Loop groups, and he mentions the following theorem:
$$ \exp: Vect(S ^1) \rightarrow Diff(S ^1) $$
the map taking a vector field to the diffeomorphism obtained by flowing along it's flow for unit time, is neither locally one-one nor locally onto.
I find this highly non-trivial and though the book does give a proof I am having trouble understanding it.
(Am I interpreting the $\exp$ map wrong?)
For example, suppose I look at a diffeomorphism which does not move any point more than a small angle $\epsilon$, then I can give a canonical unit time homotopy from this diffeomorphism to identity which I could differentiate to get a vector field. Which seems to imply that the $\exp$ map is locally surjective. What am I getting wrong here?
Also he says that the map which rotates $S ^1$ by an angle of $2 \pi/n$ is hit more than once by $\exp$. One vector field is easy, the constant vector field $ 2 \pi / n \frac{\partial}{\partial \theta} $ at each point. I am unable to concretely see any other vector field in the pre image.
Thanks,