# The fundamental group of the circle with some points identified

I'm thinking about the fundamental group of a circle with some points identified. I mean let $r:\mathbb S^1\to \mathbb S^1$ be a quotient map mapping the point of the circle $(cos \theta, sin \theta )$ to $(cos(\theta+2\pi /n),sin (\theta+2\pi /n))$. Form a quotient space identifying $x$ to $r(x), r^2(x), \ldots,r^{n-1}(x)$.

I need help to find this fundamental group.

Thanks

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You probably mean $(\cos(\theta), \sin(\theta)) \mapsto (\cos(\theta + \frac{2\pi}{n}), \sin(\theta + \frac{2\pi}{n}))$. Anyway, can you identify the quotient space itself? Say, when $n = 2$? –  levap Nov 25 '12 at 3:35
@levap yes of course thank you, I'm editing right now –  user42912 Nov 25 '12 at 3:39
@levap Maybe we have a homeomorphism between a bouquet of n circles and this space, am I correct? –  user42912 Nov 25 '12 at 3:45
Not really. In a bouquet of $n$ circles there is a point at which you "wedge" the circles. Which point would it be in the quotient? When $n = 2$, a fundamental domain for the action is the half circle $(\cos(\theta), \sin(\theta))$ where $0 \leq \theta < \pi$. That is, each equivalence class in the quotient has two members and you can always choose a member in the equivalence class to lie in the half circle. What is the relation between $(\cos(0),\sin(0))$ and $(\cos(\pi), \sin(\pi))$ in the quotient? –  levap Nov 25 '12 at 4:02

This and several related questions illustrate the advantages of having the notion of the fundamental groupoid $\pi_1(X,A)\;\;$ on a set $A$ of base points, since one may want to identify some of the points of $A$. There is an appropriate groupoid construction, due to Philip Higgins, see Categories and Groupoids. Given a groupoid $G\;$ with object set $X$ and a function $f: X \to Y\;\;$ there is a groupoid $f_*(G)\;$, or $U_f(G)\;$ in C&G, Chapter 8, which is universal for morphisms from $G$ whose object map factors through $f\;$. The construction of this generalises that of free groups and of free products of groups.

The application of this to fundamental groupoids is given in Topology and Groupoids, p. 343.

This may seem difficult for beginners; but it follows the idea of choosing algebraic structures which well model the geometry, and has the advantage of dealing with two group theory constructions often given separate expositions.

The philosophical point here is that to model the gluing of spaces, one needs algebraic objects with structure in a range of dimensions starting with $0$, since in homotopy theory identifications in low dimensions have homotopical effects in high dimensions. Groupoids may be seen as having structure in dimensions $0,1\;$. This philosophy does carry over to higher dimensions, but one needs appropriate models.

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There is an easy approach. I hope you have read something about orbit spaces. If it not the case, let's say that they are the result of a group acting on a space, for example the real projective plane is the orbit space resulting from $Z_2$ acting on $S^2$. Your space is the result of $Z_n$ acting on $\mathbb{R}$. So now we need a result to compute the fundamental group. And here it comes, it is Theorem 5.13 in Armstrong, Basic Topology:

As $\mathbb{R}$ is simply connected (path connected with trivial fundamental group) and the second condition in the theorem holds because n is fixed once you decide which space you want to deal with. Then by the quoted result follows that the fundamental group is $Z_n$ .

If something is not clear enough in the approach I propose please let me know.

EDIT: Maybe you wanted to refer to the space who is described below (I quote from this question):

Reading in Sieradski (Introduction to Topology and Homotopy), where this example was coming from. He is talking about the disc $\mathbb{D}^2/eq$, where $eq$ is the equivalence relation coming from $\phi: \mathbb{S}^1 \rightarrow \mathbb{S}^1:z \mapsto z e^{2\pi/m}$ (a rotation of $2\pi/m$ radians. Thus the identification map $q: \mathbb{D}^2 \rightarrow P_m$ wraps the boundary 1-sphere m-times around its image $q(\mathbb{S}^1)$. This $P_m$ is called a pseudo-projective plane of order m.

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