The operation of connected sum has the sphere, S, as identity, so M # S = M for any surface M. Connected sum with a torus is equivalent to adding a handle. Every surface is the connected sum of either the sphere, projective plane, or Klein bottle with zero or more tori.
For example, the connected sum of two tori is the two-handled torus. The connected sum of two projective planes is the Klein bottle.
8/12/94 email@example.com --
The Geometry Center