I am trying to identify and find the fundamental group of the surface $M,$ which is constructed by removing the shaded middle section from the surface below and replacing it with a four-holed Klein bottle and matching the holes.
For the fundamental group, I tried using van Kampen, where $U$ is the leftmost part of the surface, the Klein bottle in the middle, and a little bit of the rightmost part, and vice-versa for $V,$ but became stuck.
For simply identifying the surface, should I look at orientability and try to compute Euler characteristic?
Any guidance is appreciated.
Thank you.