... A closed surface (i.e. a compact surface without boundary) is homeomorphic to either a sphere with g handles (a connected sum of a sphere with g tori) or to a sphere with g M obius bands attached (a connected sum of a sphere with g copies of RP 2 ). A compact surface with boundary is homeomorphic to a closed surface with a nite number of disks deleted. ... Prove that on the sphere with g handles the maximal number of nonintersecting closed curves not dividing this surface is equal to g. IV.11. ...