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 $n\geq 2$.

Question

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 $\pi_1(U_2)$.

Answer 1

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.