Fundamental group of two linked spheres

Question

I want to calculate the fundamental group of this space: $$X=\{(x,y,z)\in\mathbb{R}^3:(x^2+y^2+z^2-100)[(x-10)^2+y^2+z^2-1]=0\}.$$ I’m not good with this kind of exercises: I know the Seifert-van Kampen theorem and the basic results about covering spaces.

Answer 1

$X$ is equal to the union of two spheres which intersect in a circle. You can apply Van Kampen's Theorem and compute $\pi_1(X)$ using the amalgamated product. An easier way to get the result is observing that $X \simeq S^2 \vee S^2 \vee S^2$. This is because the two halves of the smaller sphere can be homotoped into two spheres by contracting the circle of intersection with the larger sphere to a point.

Answer 2

We have $$X=\{(x,y,z):x^2+y^2+z^2=100\}\cup\{(x,y,z):(x-10)^2+y^2+z^2=1\}$$ These two spheres intersect in the circle, so after a little visualization we can see that $X\cong S^2\vee S^2\vee S^2$. Applying Seifert-van Kampen twice to obtain the result.