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Let $X$ be the topological space given by considering three spheres pairwise tangent. Which is its fundamental group?

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    Don't you have any idea? You might have at least an idea of the geometric meaning. Further more: Do you have a specifc sphere in mind, e.g. 1-sphere, or do you want to consider the general n-sphere? It could also be a 1-sphere, 2-sphere and 4-sphere for instance. You see, you might specify your question. – 2017-01-03
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    I want to know what happens when we have three 2-spheres in $\mathbb{R}^3$. My idea is the following: if we call $X_1,X_2,X_3$ the three spheres, I can take $U=X_1 \setminus \{p\}$, where $p$ is the intersection point between $X_1$ and $X_2$ and $V=X_2 \cup X_3$ and then to use Van Kampen theorem. Then, I repeat the argument for $V$ and so on... – 2017-01-03

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(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 $\pi_1(X)=\mathbb Z$.

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    I would like to see some computation. Can you use Van Kampen theorem to show your claim, please? – 2017-01-03
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    you try to do it, and show me your approach, I'll try to help you...but I'm too lazy to do it explicitly using Van-Kampen. – 2017-01-03
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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 $\mathbb Z$.