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The fundamental group of $S^1$ is $\mathbb Z$. Let's also call that space $P_1$. Then we'll build $P_n$ for $n > 1$ by taking $P_{n-1}$ and adjoining a circle to it with the condition that it must intersect exactly one of the already present circles in exactly one point, with the additional condition that no point can be the intersection of more than two circles.

That is to say, a row of n circles, OOOOOO, such that no point is shared by more than two circles, and every circle intersects at least one other circle and at most two other circles.

The fundamental group of $P_2$, which is just the figure-eight, has $\mathbb Z \times \mathbb Z$ as its fundamental group.

What's the fundamental group of $P_n$ for $n > 2$? That is, what's the fundamental group of rows of circles with more than two circles?

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The space is homotopy equivalent to the wedge of $n$ circles, hence the group is free on $n$ generators. Accidentally, for $n=2$ it's $\mathbb{Z}*\mathbb{Z}$, not $\mathbb{Z}\times\mathbb{Z}$!

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  • $\begingroup$ Please, let's not post answers to questions which are clearly off-topic. $\endgroup$ – HJRW Mar 6 '14 at 5:02
  • $\begingroup$ wrt $Z*Z$: yeah, sorry, that's what I meant x) But I don't quite see how it's equivalent to the wedge of n circles. I mean, let's take the space $P_3$ which is just ooo. Suppose I take as base point the intersection between the left and middle circles. There is certainly one loop to the left that's a generator, that's okay; then one loop around the center; then one loop around the right circle. That's fine. But then what about the loop around the right and center circles? And the loop that forms a figure-eight on them? That makes it look like a free group on 5 generators, not 3. $\endgroup$ – Pedro Carvalho Mar 6 '14 at 7:46
  • $\begingroup$ ... but are your 5 generators independent of each other? Try to find a homotopy equivalence from your space to the bouquet of n-circles. Look up the van Kampen theorem as well. Finally, for fun what about the infinite (or the doubly infinite rows of circles???? $\endgroup$ – Tim Porter Mar 6 '14 at 11:53
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Hint: do the obvious thing, use the van Kampen theorem with respect to the open covering of your space by the two open sets $U$ and $V$ such that $U$ is the leftmost circle and a little bit more, and $V$ is all the rest of the circles and a bit more.

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