I want to compute the presentation of the fundamental group of the non orientable surfaces $N_h$, thus $\pi_1(N_h)$.
I notated with $N_h$ the sphere with $h$ crosscaps. Herefore I first have to compute the fundamental group of the dunce hat in topological sense.
But how to do this on a good mathematical way ? I know that we can use van Kampen Theorem. Then I want to compute the fundamental group of the projective plane and then try to generalize this to $N_h$ for arbitrary $h\in\Bbb{N}$.
Can someone explain how to use van Kampen Theorem (I know what the meaning is but I thought of applying) and how to solve this exercise ?!
Thank you.