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 $S^2$.
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.
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!