I want to solve the following exercise from the book Basic Topology by M.A. Armstrong.
Prove that the dunce hat is simply connected using Van Kampen's Theorem.
I know that the dunce hat can be obtained from a triangle as shown in wikipedia
This triangle can be decomposed into two spaces K and J where K is a disc inside the triangle and J is the remaining space. The fundamental group of K is trivial. The space J can be deformation retracted to its boundary.
The intersection $K \cap J$ seems to be an annulus.
In order to proceed I need to make statements about the fundamental groups of J and $K \cap J$. Since the latter is an annulus its fundamental group is isomorphic to the group of integers.
I am however having problems to see what the fundamental group of J is. I believe it must be something trivial since the free product of $\pi_1(K)$ and $\pi_1(J)$ must be trivial in order for the dunce hat to be simply connected but I'm not entirely sure what the identification space J looks like after applying the deformation retraction and identifying the sides.
Best Answer
The fundamental group of $J$ is not trivial but it deformation retracts to the boundary of the triangle which is a circle after the identification (you can glue the triangle to see that it will become a circle). Hence $\pi_1(J)$ is also the group of integers. Using Van Kampen, the fundamental group of the dunce hat is the amalgamated product $\mathbb{Z}*_\mathbb{Z} 0$ where the map from $\mathbb{Z}$ to itself is the map $a\mapsto a+a-a = a$ , i.e. the identity map. Hence the product is $0$ so the dunce hat is simply connected.