# Examples of fundamental groups which we can use the Van-Kampen Theorem

I'm starting to study the Van-Kampen Theorem and I realized there are some questions we can use this theorem directly. I find this theorem difficult for a beginner. Do some of you know some questions which can be solved directly using The Van-Kampen theorem? I think it's important to have examples for beginners.

There is a similar question here: Good exercises to do/examples to illustrate Seifert - Van Kampen Theorem. In spite of the title, the question was more about the understanding of the author of the concept of this theorem than the examples themselves, and almost every answer was focused in this part.

Thanks

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 I don't understand why the answers to the other question are not what you are looking for. I see several examples of calculations of fundamental groups using the theorem. – Grumpy Parsnip Nov 16 '12 at 3:05

One nice application of Seifert- van Kampen is that it offers and easy proof that $S^n$ is simply connected for $n \ge 2$. We can cover $S^n$ by two open discs $D_1$, $D_2$, whose intersection is an open band about the equator of $S^n$. Because discs are simply connected $S^n$ is simply connected, since $\pi_1(S^n)$ must be generated by the generators for $\pi_1(D_1)$ and $\pi_1(D_2)$.

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A very useful application of Van Kampen theorem is to graphs. It is shown by Allen Hatcher, Algebraic Topology, in the example 1.22 at page 43.

If you have a finite graph, you can always extract a maximal tree, so the fundamental group of the graph is the free groups with many generators as many edges that not lies in the maximal tree.

In the Hatcher's example you have to choose $A_\alpha$ adding to $T \cup e_\alpha$ a small one-side-open "piece" of each other edges, so what you obtain is an open subset of the graph that deformation retracts to $T \cup e_\alpha$. So the intersection of two or more $A_\alpha$'s does not contain loops and so is contractible.