The Genus of an Object

Every compact, connected, rim-less, double-sided (orientable) surface

is homeomorphic to either a sphere ==> [Genus=0]
or to a connected sum of g tori (simple g-holed donut) ==> [Genus=g]
-- i.e. a sphere with g handles attached to it,
-- or a (fat) disk with g holes through it.

g is also the maximal number of simultaneous closed cut-lines 
   that can be drawn on the surface without partitioning 
   the surface into different disconnected regions.

Euler Characteristic c (=Surface integral of Gaussian curvature / 2p ): 
c = 2 - 2g = V + F - E (for a polyhedral mesh).