I'm watching this video on basic topology and around 9 minutes in they introduce the sphere and how it can be cut along a line from the north to the south pole then distorted into a square, using arrows along the cuts to show how the new boundaries were previously connected. There are two arrows on the sphere, the arrow on top pointing towards the north pole and the one on the bottom towards the south (I suppose both arrows point to both poles on the sphere but my question is about after cutting, at which point I think both poles are truly on opposite ends of the 1-sided surface they live on).
After cutting along this line, these two arrows become four half-arrows which live on the boundary of the square. This makes sense. However, now the all arrows point towards the north pole, as shown below.
I imagine the half-arrows should still point towards their respective poles, which I believe are on opposite sides (top and bottom, as he's showing it) of the square, and I don't understand why this isn't the case.
Just logically, it seems that each point on a cut has to be mapped to at least two points on the new surface, so I think it's possible that my confusion stems from that. Based on the next example (cutting a tetrahedron along 3 edges), it seems that if an arrow $A$ pointed to a point $p$, then after a cut $\varphi$, $\varphi[A]$ consists of disjoint sets $X$ that each point towards elements of $\varphi[p]$, but that's a very specific guess based on just one example. Also I'm not sure it applies to this problem since the cuts on the sphere stop at the poles, whereas the point being tripled here is on several cuts.
Anyway, it seems like the analogous thing for the sphere would be:
What's happening here? Did the arrows always point in the same direction? Or did something weird happen during the cut or distortion? I just can't imagine how to deform the square in the video back into the sphere we started with.
I think it might just be a small mistake in the video: pi.math.cornell.edu/~mec/Winter2009/Victor/part2.htm