Foundamental group of $n+1$ spheres in $\mathbb{R}^{n+1}$ that touch two by two

Question

How can i calculate the foundamental group of three $S^2$ in $\mathbb{R}^3$ that touches two by two in one point (if you take any two spheres, they touch only in one point) ?

I know that is $\mathbb{Z}$ but i'm not able to formalize it using van Kampen.

How changes the foundamental group, and the application of van Kampen theorem, if you have $n+1$ $S^{n}$ in $\mathbb{R}^{n+1}$ that touches two by two in one point?

Thanks!

Answer 1

Your space is homotopically equivalent to a circle with three spheres attached to it at different points. Like this picture (the filled in circles are spheres, the last one is just a circle):

Now apply Van Kampen's theorem three times and the fact that $\pi_1(S^2) = 0$ to show that the fundamental group of the space is the fundamental group of the circle drawn in black, that is, $\mathbb{Z}$.

Answer 2

If you know that fundamental group depends only on the 2-skeleton, you can glue a 3-cell in the interior of each sphere. From a family $X$ of spheres, you get a family $\tilde{X}$ of closed balls. Now, $\tilde{X}$ retracts by deformation on a circle, so $$\pi_1(X) \simeq \pi_1(\tilde{X}) \simeq \pi_1(\mathbb{S}^1) \simeq \mathbb{Z}.$$