Dunce hat is simply connected I'm trying to prove that the dunce cap is simply connected via Seifert- Van Kampen Theorem. I choose to be my open sets $U$ and $V$ the open disk and the punctured surface below, then $U\cap V$ is the annulus.!enter image description here I'm having problems to find the fundamental group of $V$ I need help. Thanks

!Duncehat

I think you have the figure for the Dunce Hat wrong, see above, where all the arrows have the label $a $, say. So you have one $1$-cell, giving $S^1$, and one $2$-cell attached by a map described by $a+a-a$, which gives a group with one generator $a$ and one relation $a+a-a=a$.

Your figure would give the group with generator $a$ and relation $a^3$, as said by others.

The figure is taken from [Topology and Groupoids. ]