Natural conditions for existence of sections of $g_n:\pi_n(\Omega_0(X,A)) \to \pi_n(A\times A)$ Let $\Omega_0(X,A)$ denote the (component of the constant paths on $A$ of the) space of paths with endpoints on $A$, where $X$ and $A$ are path-connected. There is a map $\Omega_0(X,A) \to A \times A$ which is just the evaluation at the endpoints, and thus we get maps
$$g_n: \pi_n(\Omega_0(X,A)) \to \pi_{n}(A \times A).$$
Are there natural conditions that can be imposed upon $X,A$ and perhaps $n$ such that the above map admits a section, i.e. a map $s_n: \pi_n(A\times A) \to \pi_n(\Omega_0(X,A))$ such that $g_n \circ s_n =1$? I'm particularly interested in the cases $n=1,2$.
An idea would be to try to "force through" a section via a map $s:A \times A \to \Omega_0(X,A)$ that takes $(p,q)$ into a path that connects $p$ to $q$, but this is obviously not even well-defined to begin with. However, this idea seems to suggest that maybe some conditions can be related to imposing that, given a certain $n$, some of the maps $\pi_i(A) \to \pi_i(X)$ induced by the inclusion vanish. In any case, if "natural conditions" is somewhat unclear, then consider it "conditions on the maps induced by the inclusion $i:A \to X$", although I appreciate other situations which may come up.
 A: Ignoring path components, the homotopy fibre of $\Omega(X,A)\rightarrow A\times A$ is equivalent to $\Omega X$. Thus there is a fibration sequence
$$\dots\rightarrow\Omega(\Omega(X,A))\xrightarrow{\Omega e}\Omega A\times \Omega A\xrightarrow{\delta} \Omega X\rightarrow \Omega(X,A)\xrightarrow{e} A\times A$$
where we have identified $\Omega (A\times A)\cong \Omega A\times\Omega A$ in the natural way. It can be shown that the connecting map $\delta$ is homotopic to the sum $\Omega(j\circ pr_1)-\Omega(j\circ pr_2)$, where $j:A\hookrightarrow X$ is the inclusion and the sum is formed using the loop multiplication on $\Omega X$.
Thus, in general, the endpoint evaluation $e$ has no canonical section, even after looping. For instance, if $in_1:A\hookrightarrow A\times A$ is the inclusion into the first factor, then by the above $\delta\circ\Omega(in_1)\simeq\Omega j$, so $\delta$ cannot be null-homotopic unless $\Omega j$ is. Of course a sufficient, but restrictive, condition is that $j$ (or $\Omega j$) be null-homotopic. In this case $\delta\simeq\ast$, so $\Omega(\Omega(X,A))\simeq \Omega A\times\Omega A\times\Omega^2 X$. (Examples: $(X,A)=(D^{n+1},S^n)$, $(X,A)=(S^{n+1},S^n)$)
However, since you are mainly interested in homotopy groups, we do have the following. Under the isomorphisms $\pi_n(\Omega A\times \Omega A)\cong\pi_{n+1}(A)\oplus\pi_{n+1}(A)$ and $\pi_n(\Omega X)\cong\pi_{n+1}(X)$, the homomorphism induced by $\delta$ becomes
$$\delta_*=j_*pr_{1}-j_*pr_{2}:\pi_{n+1}(A)\oplus\pi_{n+1}(A)\rightarrow \pi_{n+1}(X).$$
i.e. $\delta_*(x\oplus y)=j_*x-j_*y.$
In particular, if $j_*:\pi_{n+1}A\rightarrow\pi_{n+1}X$ is the trivial homomorphism, then the map
$$\pi_{n+1}(X)\rightarrow \pi_n(\Omega(X,A))$$
is injective, and
$$\pi_{n+1}(\Omega(X,A))\rightarrow \pi_{n+1}(A)\oplus\pi_{n+1}(A)$$
is surjective. I'm not guaranteeing that this map have a natural splitting, or even a splitting at all. I think it will not in general, even under the assumption on $j_*$ in this degree. Certainly surjectivity is a necessary condition, however. (Example: $(\mathbb{C}P^2,S^2)$. The map $\pi_2S^2\rightarrow\pi_2\mathbb{C}P^2$ is an isomorphism but $\pi_nS^2\rightarrow\pi_n\mathbb{C}P^2$ is trivial when $n\neq2$. Thus $\pi_n\Omega(\mathbb{C}P^2,S^2)\rightarrow\pi_nS^2\oplus\pi_nS^2$ is onto if $n\neq 2$. When $n=2$ it is injective and $\pi_2\Omega(\mathbb{C}P^2,S^2)\cong\mathbb{Z}$.)
Finally I'll comment on the path components. We haven't done anything restrictive in the above, since for instance $\pi_n(\Omega(X,A))=\pi_n(\Omega_0(X,A))$ when $n\geq1$, assuming the basepoint in $\Omega(X,A)$ is the constant loop at the basepoint of $A$.
