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?
When you attach a handle to a closed orientable surface, you can think of it as adding a "hole" or a "handle" to the surface. To visualize this, imagine taking a sphere and poking a hole through it. This hole can be stretched out into a square shape, which represents the ghost of a panel or surface that was removed.
Now, let's consider the effect of attaching a handle on the Euler characteristic and the number of boundary curves of the surface.
The Euler characteristic (denoted as χ) of a surface is a topological invariant that characterizes its shape. For closed orientable surfaces, the Euler characteristic is given by the formula:
χ = 2 - 2g,
where g represents the genus, which is the number of handles attached to the surface.
When you attach a handle, you are effectively increasing the genus by one. This means that χ becomes χ = 2 - 2(g+1), which simplifies to χ = 2 - 2g - 2 = χ - 2.
In other words, attaching a handle decreases the Euler characteristic of the surface by 2.
As for the number of boundary curves, when a handle is attached, it forms a closed loop on the surface. This loop becomes a boundary curve, as it separates the surface into two regions: the outside of the handle and the inside of the handle.
Therefore, attaching a handle increases the number of boundary curves by 1.
To summarize, when you attach a handle to a closed orientable surface:
- The Euler characteristic decreases by 2.
- The number of boundary curves increases by 1.
This classification process allows us to understand the structure of closed orientable surfaces by adding handles to a base sphere and observing the changes in the Euler characteristic and the number of boundary curves.