Dunce hat is simply connected

Question

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.

I'm having problems to find the fundamental group of $V$

I need help.

Thanks

Answer 1

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. ]

Answer 2

I am not sure about what is the U∪V, and I guess it's the triangle with three edges identified as in your picture. Then you've give the U and V. The fundamental group of U is trivial. And V can be deformation retract to the edges identified,i.e, a circle, thus the fundamental group of V is Z. And U∩V is annulus as you pointed out. So the fundamental group of it is also Z. Now you may use S.V.K theorem to conclude that the fundamental group of U∪V is Z/3Z. Because the generator of π（U∪V） is three times of the generator of π（V）.