I'm interested in computing the fundamental group of the twisted loop space $$\Omega_f(M)=\{ \gamma \in C^{\infty}(\Bbb R,M) \mid \gamma(s+1)=f\gamma(s)\}$$ where $f \in \text{Aut}(M,x_0)$, for example a diffeomorphism with a fixed point $x_0$.

The twisted loop space is part of a fibration $$\Omega_{x_0}M \to \Omega_f(M) \xrightarrow{ev_0} M$$ where the map $ev_0$ is the evaluation map at $s=0$ and $\Omega_{x_0}M$ is the loop space (based at $x_0$).

The relevant piece of the l.e.s. homotopy gives $$\pi_2(M,x_0)\xrightarrow{\partial} \pi_1(\Omega_{x_0}M,x_0)\to \pi_1(\Omega_f(M),x_0) \to \pi_1(M,x_0)\to \pi_0(\Omega_{x_0}M)\to \pi_0(\Omega_f(M))$$

The last map should be an injection, hence we have a surjection $\pi_1(\Omega_f(M),x_0) \twoheadrightarrow \pi_1(M,x_0)$. I don't quite understand the effect of the boundary map from the second homotopy group of $M$, so I cannot really go on. As far as I know, if $M$ is simply connected then $\pi_1(\Omega_f(M),x_0) \cong \pi_2(M,x_0)$. If $\delta=0$ that would be the case but I cannot verify it.

Is this computation done somewhere so I can check how's done?

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