I am watching a differential geometry lecture on youtube. The professor introduced the notion of compact oriented 2-manifolds after explaining the definition of manifold and topology.
I am very new to this field. Can anyone help me understand the following screenshot?
Specifically, I have thee questions.
Why can all vertex be labeled A? How can they all be the same vertex?
Why are we cutting up manifolds this way?
What does the cutting have to do with manifold at all? A manifold is a pair of a topology and a set of charts. When we are cutting it this way, how is the notion of topolgy and charts used?
Imagine a torus (an inner tube). It's an example of a compact oriented $2$-manifold. It's a $2$-manifold because if you lived on a big torus, your neighborhood would look like a flat Euclidean plane, just the way the earth looks flat until you imagine large areas.
Your torus is compact essentially because you can never travel infinitely far away - like the Earth, unlike the Euclidean plane.
It's oriented because moving around never swaps left and right. A Moebius strip is not oriented.
Now how might you build a torus from a flat piece of fabric? Make a rectangle. Sew the left edge to the right edge to make a cylinder. Sew one circular end of the cylinder to the other to make the torus.
If you labelled the fabric as in the first image in your picture the four points labelled $A$ end up in exactly the same place after you sew up the seams. That's why they have the same name before you sewed.
The manifold you are building from these pieces is more complicated than a torus, since it sews all three pieces together, matching edges with the same label. But if you look carefully, all the seams meet at the same place, $A$.
The theorem you are aiming for is the opposite of sewing manifolds from patterns. It's proving that when you define a manifold using abstractions like "topology" and "charts" you end up with an object you could have built by sewing a pattern like the one in your picture.