# Van Kampen Theorem shows $\pi_1(S^1 \vee S^1)\cong \mathbb{Z} * \mathbb{Z}$ and $\pi_1(S^n) \cong \left\{1\right\}$ for $n\geq 2$.

I do not see how using Van Kampen theorem we obtain:

(a) $$\pi_1(S^1 \vee S^1)\cong \mathbb{Z} * \mathbb{Z}$$ and

(b) $$\pi_1(S^n) \cong \left\{1\right\}$$ for $$n\geq 2$$.

In my lecture notes I have that Van Kampen is the main theorem that helps us compute the fundamental group of a space $$X$$ provided that $$X$$ is expressed $$X=U_1 \cup U_2$$ where $$U_i \subset X$$ for $$i=1,2$$ and $$U_1 \cap U_2$$ are open and path connected and that the fundamental groups $$\pi_1(U_i)$$ and $$\pi_1(U_1 \cap U_2)$$ are known. Moreover, the theorem as stated is:

Van Kampen Theorem: If $$X = U_1 \cup U_2$$ with $$U_i$$ open and path connected, and $$U_1 \cap U_2$$ path connected and simply connected, then the induced homomorphism $$\Phi : \pi_1(U_1)*\pi_1(U_2)\to\pi_1(X)$$ is an isomorphism; where $$*$$ is the free product of the groups $$\pi_1(U_1)$$ and $$\pi_1(U_2)$$.

• So choose appropriate $U_1$ and $U_2$ that cover the space and have easy to compute $\pi_1$. – anomaly Dec 11 '17 at 21:11
• Do you not have any examples in your lecture notes? Nor any book with examples? – Lee Mosher Dec 11 '17 at 21:13
• You will (probably) need a more general version of the theorem for question $2$. – Andres Mejia Dec 12 '17 at 1:46

For the first, choose open sets $U$ and $V$ such that $U$ contains one copy of $S^1$ and $V$ the other, such that $U\cap V$ is a cross at the basepoint, and such that $U$ and $V$ both deformation retract onto such copies of $S^1$. Then Van Kampen says $\pi_1(S^1\vee S^1)$ is the coproduct of $\pi_1(U)$ and $\pi_1(V)$ over $\pi_1(U\cap V)$. This last group is trivial, so you get the free product of $\pi_1(S^1)=\mathbb Z$ with itself.
For the second, take open sets $U$ and $V$ such that $U$ and $V$ are contractible and intersect at an open set that deformation retracts onto $S^{n-1}$. Since $n>1$, this set is connected, and Van Kampen says that $\pi_1(S^n)$ is the coproduct of two trivial groups over trivial maps from $\pi_1(S^{n-1})$. Obviously this is the trivial group.
• Note that the version of Van Kampen's theorem used in the OP cannot be used to argue for question $2$, since it assumes that $U \cap V$ is simply connected. – Andres Mejia Dec 12 '17 at 1:44