How can I see that the connected sum $\mathbb{P}^2 \# \mathbb{P}^2$ of the projective plane is homeomorphic to the Klein bottle?
I'm not necessarily looking for an explicit homeomorphism, just an intuitive argument of why this is the case. Can we see it using fundamental polygons?
Best Answer
Here's an answer more in the spirit of the question. All figures should be read from upper left to upper right to lower left to lower right.
Fig 1: A Klein bottle...gets a yellow circle drawn on it; this splits it into two regions, which we reassemble, at which point they're obviously both Mobius bands:
To see that $P^2$ minus a disk is a Mobius band, look at the following. In the upper left is $P^2$, drawn as a fundamental polygon with sides identified. In the upper right, I've removed a disk. The boundary of the now-missing disk is drawn at the lower left as a dashed line, and the two remaining parts of the edge of the fundamental polygon are color-coded to make the matching easier to see. In the lower right, I've morphed things a bit, and if you consider the green-followed-by-red as a single edge, you can see that when you glue the left side to the right, you get a M-band.