Do we have an example of a smooth action $S^1 \curvearrowright T^n$ which is faithful, locally free but not free?


I know such an action must induce an injection $\rho:\pi\_1(S^1)\to\pi\_1(T^n)$.
Another related question is: Is the image of $\rho$ saturated?


Thanks in advance!



Faithful locally free circle actions on a torus must be free?
This follows from Theorem 9.3 (page 216) of the book "Compact tranformation groups" by Bredon (note that it is freely available online).


More is true: Any effective group action of a torus on a torus is free. The proof starts along the lines you mentioned via fundamental group considerations.


The result is originally due to Connor and Montgomery:


Conner, P. E., and Montgomery, DC. Transformation groups on a K(n, l), I. Michigan Math. J. 6 (1959), 405-412.


