I am trying to compute the fundamental group of the sphere with n points identified by using Seifert Van Kampen. I split up my surface into $U =$ the top half of the sphere including the n points and $V =$ the lower half of the sphere (not including the points), where $U \cap V$ is a cylinder.
$U$ turns out to be homotopy equivalent to the the wedge of n petals, so
$\pi_1(U)$= free group on $n-1$ generators, $\pi_1(V) = \{1\}$ ( as $V$ is a disk) and $\pi_1(U \cap V) = \pi_1(S^1) = \mathbb{Z}$.
I am having trouble applying van Kampen to find the relators of $U \cup V$.
Best Answer
Hint: The inclusion maps $$i : U \cap V \to U \quad\text{and}\quad j : U \cap V \to V $$ induce group homomorphisms $$i_* : \mathbb{Z} = \pi_1(U \cap V) \to \pi_1(U) = F_{n-1} $$ and $$j_* : \mathbb{Z} = \pi_1(U \cap V) \to \pi_1(V) = \{1\} $$ You must compute these homomorphisms, as Van Kampen's Theorem suggests.
Added: Here's more information regarding your comment where you ask how to visualize the first homomorphism.
The second homomorphism $j_*$ is obviously the trivial homomorphism.
The first homomorphism $i_*$ is also trivial. To see why, notice that the map $i$ factors as $$U \cap V \mapsto \text{the boundary circle of the top hemisphere} \hookrightarrow \text{the top hemisphere} \mapsto U $$ where the first map of this sequence is a deformation retraction of the cylinder $U \cap V$ onto its central circle (which is the boundary circle of the top hemisphere), and last map is the quotient map under which the $n$ points are identified to a single point. Since the top hemisphere is simply connected, it follows that $i_*$ is the trivial homomorphism. (Thanks to @BennyZack for this observation in the comments.)