Use Van Kampen Theorem to find the fundamental group of a circle $S^1$ joining a sphere $S^2$

Question

this is a topology question:

Compute the fundamental group of a Christmas ball, obtained by joining a copy of the circle $S^1$ to a copy of the sphere $S^2$.

My thoughts:

Intuitively, the fundamental group should be Z, and a path may jump through the loop several times or not. One open set is the interior of a filled torus with the circle lying on the surface. Another set could be the whole $R^3$ with the closed disk removed. Then the first set contracts to a circle, and the second set contracts to a sphere.

And I'm struggling to write out the exact fundamental group from here. Am I on the right track? Please point out the right direction if not, and help me with computing the fundamental group.

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Updated: using the van kampen theorem

First to clarify, the "join" here means it is the union of the two copies, having a single point in common.

We just learned van kampen theorem, and I'm thinking let X be the Christmas Ball, and let U = $S^1$ and V = $S^2$ which X = U $\cup$ V. However, this won't satisfy that both U and V are open sets, will it?

Could someone please give a formal proof on how to use van kampen theorem to solve this problem?

Any help is greatly appreciated!

Answer 1

Using the van Kampen theorem:

First, note that this is called the wedge sum.

The trick is to "go over a little bit." I don't want to write this down with equations, but here is one formalism:

center $S^1$ about the origin on the $xy$ axis, and $S^2$ at $(2,0,0)$ in $\mathbb R^3$ and the point in which they intersect will be $(1,0,0)$. Equip them with the subspace topology. Then take open balls $A,B$ of radius $3/2$ around $(0,0)$ and $(0,2)$ respectively. In the subspace topology, $U$ will be $A \cap (S^2 \vee S^1)$ and $V$ will be $B \cap(S^2 \vee S^2)$. Note that $U \cap V$ is contractible.

Now apply the Van Kampen theorem.

Answer 2

The trick to applying Van Kampen's Theorem is to take the two sets that you dearly wish to apply it to, namely $S^1$ and $S^2$; next to realize, sadly, that $S^1$ and $S^2$ are not open in $X$ (which you did); and then to realize, very happily that there exists an open neighborhood $U \subset X$ of $S^1$ which deformation retracts to $S^1$, and there exists an open neighborhood $V \subset X$ of $S^2$ which deformation retracts to $S^2$.

To put the icing on the cake, you then realize that $U \cap V$ is path connected, has an easily computed fundamental group, and that the inclusion induced homomorphisms $\pi_1(U \cap V) \mapsto \pi_1(U)$ and $\pi_1(U \cap V) \mapsto \pi_1(V)$ are easily computed. Now you have all the data you need to apply Van Kampen's Theorem.