Euler Characteristic of a Sphere With n Handles and no Holes

A sphere with two handles and no holes is homeomorphic to a two fold torus.

The Euler characteristic of a two fold torus is -2 , so the euler characteristic of a sphere with two handles is also -2. In general a sphere withhandles and no holes is homeomorphic to anfold torus. The genusof anfold torus isand the Euler characteristic isHence the Euler characteristic of a sphere with n handles is