Since fedja's excellent comment on Joseph's question on probing a manifold with geodesics remained uncommented (especially by topologists), I'd like to make a question out of it:

Conjecture: Given a 2-dimensional manifold and two closed curves on it which intersect transversally in exactly one point. Then the two curves cannot be homotopic.

(An immediate consequence of this would be that living on a surface with two such curves, one would know, that it is not homeomorphic to the sphere.)

How to prove this conjecture (if it's true)?