posted by ml .
The classification of closed orientable surfaces says that they are all spheres with a certain number of handles attached. For example, a sphere is a sphere with 0 handles. A torus is a sphere with one handle. A two holed torus is a sphere with two handles etc. How does one "attach a handle"? Think of puncturing a surface as "removing a lid". After all if you make a puncture, you can stretch out the hole into a square shape which you can think of as being the ghost of a panel in the construction of the original surface. So the puncture is removing a lid. What does "removing a lid" do to the Euler characteristic and number of boundary curves?