Genus in topology

Question

Can somebody provide a formal definition of genus in topology? I find it difficult to imagine what genus is. For example, objects of genus zero are the ones that homeomorphic to a sphere? What about higher genus?

Answer 1

The classification theorem of closed surfaces tells us that any connected closed surface is homeomorphic to one of:

$1)$ The unit sphere

$2)$ The connected sum of $g$ tori (surface of genus $g$)

$3)$ The connected sum of $k$ real projective planes.

(Where $k,g$ are positive integers.)

So if a particular surface falls into the second category (that is to say, is homeomorphic to a connected sum of $g$ tori) we say that it has genus $g$. It can be pictured as the number of "holes" in the surface.