Fundamental group of three spheres pairwise tangent Let $X$ be the topological space given by considering three spheres pairwise tangent. Which is its fundamental group?
 A: Anubhav's answer is correct, but here's an alternative approach. On each of the spheres draw an arc joining the two points where the sphere is tangent to the two other spheres.  These three arcs form a curvilinear triangle, which is homeomorphic to a circle and therefore has fundamental group $\mathbb Z$. Now build your space $X$ by starting with this triangle and attaching the three spheres one at a time. Each sphere has trivial fundamental group and is being attached along an arc, which also has trivial fundamental group. So van Kampen's theorem tells you that the fundamental group after attaching the three spheres is the same as it was before, namely $\mathbb Z$.
A: (I'm trying to give a hand waving argument because I want to avoid all calculations. I assume here you mean sphere as $S^2$.)
I am trying to give an alternative visualization of your describing space $X$ as follows... Observe a torus, and mark 3 disjoint parallel circle on it (meridian). Now if you squeeze these 3 circles each to a point then the space you'll obtain is essentially $X$ (upto homeomorphis). Now we know the fundamental group of torus. And in our new construction I'm just killing one off the generator corresponding to the meridian. So the new fundamental group of this is generated by only the longitudinal circle. So $\pi_1(X)=\mathbb Z$. 
