Happy New Year.
In a similar spirit of question Motion planning algorithm, we consider a path connected topological space $X$, and equip its free loop space $X^{S^1}$ with the open compact topology. We call a loop motion planning algorithm of $X$, any continuous section $s:X\times X \longrightarrow X^{S^1}$ of the evaluation $$ev: X^{S^1} \longrightarrow X\times X, \; \gamma\mapsto (\gamma(0), \gamma(1/2))$$ We have shown that such section exists if and only if $X$ is contractible. Hence, we denote $\mathcal{M}^{LP}(X)$ the non empty set of such sections, topologized with the open compact topology (as subset of ${\rm map}(X\times X,X^{S^1})$.
Question: Is it true that $\mathcal{M}^{LP}(X)$ is contractible when $X$ is ?
General Question: The space of sections for a fibration with section E ---> B where E, and B are both contractible is again contractible?
Any remarks or comments are welcome, thank