Consider the standard Riemannian 2-sphere $S$, equipped with a geodesic triangulation $T$. Let $L(S,T)$ be the space of homeomorphisms of $S$ which map $T$ to a geodesic triangulation. What is the homotopy type of $L(S,T)$? A similar question has been solved by E. Bloch, R. Connelly, D. Henderson, Topology 23 (1984), 161-175, when $S$ is a simplicial disc in the plane, and it is a "tour de force"!
This question is on behalf of Jean Cerf, my former advisor, who is interested in S. Cairns' work.