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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!

Is it true that every faithful and locally smooth action $S^1 \curvearrowright T^n$ is 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!

Do we have an example of a smooth action $S^1 \circlearrowright T^n$ 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!

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!

Do we have an example of an action $S^1 \circlearrowright T^n$ 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!

Do we have an example of a smooth action $S^1 \circlearrowright T^n$ 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!