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I have a sphere with two points identified. I want to compute its fundamental group. I see that this question has already been made here in the site but no answer satisfies me. I want to use Van Kampen's theorem finding the open sets $U$ and $V$ and so on.

Can you give me a detailed answer please?

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Denote the sphere as $S^2$, and use spherical coordinates $\theta,\phi$.

Let the two points $p_1,p_2 \in S^2$ be the north and south poles, so $p_1$ is the point where $\phi=0$ and $p_2$ the point where $\phi=\pi$.

Let $X$ be the quotient of $S^2$ where $p_1,p_2$ have been identified to a single point $P \in X$.

Let $U=X-P$.

Let $\alpha \subset S^2$ be the longitude line $\theta=0$ with endpoints $p_1,p_2$.

Let $\widehat V$ be the neighborhood of $\alpha$ consisting of the union of the north polar cap $\phi < \pi/4$, the south polar cap $\phi > 3 \pi/4$, and the longitudinal strip $-\pi/4 < \theta < \pi/4$. Let $V$ be the image of $\widehat V$ under the quotient map $S^2 \to X$, and so $V$ is obtained from $\widehat V$ by identifying $p_1$ to $p_2$.

Choose the base point $q \in U \cap V$ to be $\theta=0$, $\phi=\pi/2$.

Define two loops $\gamma_1,\gamma_2$ based at $q$ as follows: $\gamma_1$ goes from $q$ up $\alpha$ to the latitude circle $\phi = \pi/8$, then around that circle, then back down $\alpha$ to $q$; and $\gamma_2$ goes from $q$ down $\alpha$ to the latitude circle $\phi = 7\pi/8$, then around that circle, than back up $\alpha$ to $q$. We may choose the directions around those two latitude circles so that $\gamma_1,\gamma_2$ are path homotopic in $U$, in fact both will be path homotopic in $U$ to the equator.

It follows that $\pi_1(U)$ is the infinite cyclic group whose generator is $[\gamma_1]_U=[\gamma_2]_U$ (where $[\cdots]_Z$ denotes path homotopy class in $Z$).

It also follows that $\pi_1(U \cap V)$ is the rank 2 free group with basis $g_1 = [\gamma_1]_{U \cap V}$, $g_2=[\gamma_2]_{U \cap V}$.

In $V$ we may regard $\alpha$ as a closed curve, starting from $q$ and going up to $p_1$, jumping to $p_2$, and then going up to $q$. It follows that $\pi_1(V)$ is the infinite cyclic group with basis $[\alpha]_V$.

The inclusion induced homomorphism $\pi_1(U \cap V) \to \pi_1(V)$ is trivial, because each of $\gamma_1,\gamma_2$ is null homotopic in $V$.

Also, the inclusion induced homomorphism $\pi_1(U \cap V) \to \pi_1(U)$ is clearly surjective.

Applying Van Kampen's Theorem, it follows that $\pi_1(X)$ is the infinite cyclic group generated by $[\alpha]_X$.

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The background to this question is discussed in this mathoverflow discussion. Since two points, say $a,b$, are to be identified, it is as well to start with a model of the whole situation, namely the fundamental groupoid $G=\pi_1(X,C)$ on the set $C$ consisting of the two points $a,b$. Since $X$, presumably the $2$-sphere, is simply connected, $G$ is isomorphic to the groupoid $\mathcal I$ which has two objects $0,1$ and exactly one arrow $\iota:0 \to 1$, as well as $\iota^{-1}: 1 \to 0$, and identities at $0,1$. When you identify $0,1$ of $\mathcal I$ to a single point in the category of groupoids you get the group $\mathbb Z$ of integers.

All this is explained in the book Topology and Groupoids, the third edition of a book published in 1968, i.e. nearly 50 years ago. A translation of some 1984 comments of Grothendieck on the neglect of this use of many base points is given here. See also the account by Philip Higgins in this now downloadable Categories and Groupoids (originallly 1971).

My understanding is that algebraic topology deals with algebraic modelling of geometric situations, and groupoids, which may have many objects, allow the modelling of the identification of several base points. If you are tethered to only one base point, then things are not so clear, and students may get confused.

May 27: Just to add some information. The key point is that is that the category $Gpd$ has a forgetful functor $Ob: Gpd \to Sets$ which is a "cofibration" of categories, meaning that if $G$ is a groupoid and $f: Ob(G) \to Y$ is a function then there is a groupoid $f_*(G)$ and morphism $U_f: G \to f_*(G)$ with a universal property that can be expressed in a pushout diagram of groupoids $$\begin{matrix} Ob(G) & \to & Y\\ \downarrow & & \downarrow\\ G & \to & f_*(G) \end{matrix} $$ The construction of $f_*(G)$ was introduced by Philip Higgins and includes that of free groups and free products of groups.

The relevance of this to the generalised van Kampen Theorem is given in Section 9.1 of Topology and Groupoids.

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