Let $(X,x)$ be a pointed connected CW complex, and let $\mathcal L X = Map(S^1, X)$ be its free loop space. We have a fibration $\mathcal L X \to X$ given by evaluating at the basepoint $0 \in S^1$, whose fiber is the pointed loop space $\Omega_{x} X$ based at $x$. Let $\lambda \in \mathcal L X$ be a basepoint; we may assume without loss of generality that $\lambda(0)= x$. Then properly speaking, we have a fibration sequence of pointed spaces:
$$(\Omega_{x} X, \lambda) \to (\mathcal L X, \lambda) \to (X,x)$$
We get a long exact sequence in homotopy. The end of it looks like this:
$$\dots \to \pi_1(X,x) \to \pi_0(\Omega_{x} X, \lambda) \to \pi_0(\mathcal L X, \lambda) \to 0$$
Now, I'm told that $\pi_0(\mathcal L X)$ is in bijection with the conjugacy classes of elements of $\pi_1(X)$. Because the middle term is in canonical bijection with $\pi_1(X, x)$,
- This leads me to guess that the map $\pi_1(X,x) \to \pi_0(\Omega_{x} X, \lambda)$ is $\gamma \mapsto \gamma \lambda \gamma^{-1}$.
If that's the case, then the kernel of this map is the centralizer $Z_{\pi_1(X,x)}(\lambda)$. So the next part of the long exact sequence is:
$$ \dots \to \pi_2(X,x) \to \pi_1(\Omega_x X, \lambda) \to \pi_1(\mathcal L X, \lambda) \to Z_{\pi_1(X)}(\lambda) \to 0$$
I don't know what the map $\pi_2(X,x) \to \pi_1(\Omega_x X, \lambda)$ is:
- The natural guess is that this map is $\beta \mapsto \beta^\lambda$ where this denotes the natural action of $\pi_1(X,x)$ on $\pi_2(X,x)$.
But this map is an isomorphism; if this pattern were to persist to higher homotopy groups, it would imply that $\mathcal L X$ is aspherical, and that can't be true -- for instance, the path components of constant loops contain $X$ as a retract, and $X$ need not be aspherical. So I don't think this guess can be correct.
Question: Are either of the bulleted statements above correct? If not, is there a good description of the map $\pi_{n+1}(X,x) \to \pi_n(\Omega_x X, \lambda)$ in the long exact sequence of the free loop space fibration?