# Homeomorphisms of the sphere mapping a geodesic triangulation to another one

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.

• The question in the edited title doesn't seem to be the same as the one in the body (and is much easier). – HJRW Feb 13 '17 at 6:56
• I have improved the title – j.c. Feb 14 '17 at 21:53
• I agree with the change of title. It is better, but I thought of (S,T) as a structure, not an object; it was ambiguous. – Francois Laudenbach Feb 16 '17 at 22:01