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 an
fold torus. The genus
of an
fold torus is
and the Euler characteristic is
Hence the Euler characteristic of a sphere with n handles is