Problem: Let $M$ and $N$ be $n$-dimensional manifolds, where $n > 2$. Let $M \# N$ be their connected sum. Show that $\pi(M \# N) = \pi(M) \ast \pi(N)$.
RE-EDITED Attempt:
Let $U_2$ and $V_2$ be two small open balls from $M$ and $N$ to be removed by the connected sum operation. Let $p_m \in U_2$ and $p_n \in V_2$, and then let $U_1 = M – \{p_m\}$ and $U_2 = N – \{p_n\}$.
We can now view the connected sum $M\#N$ as the quotient of the disjoint union of $M$ and $N$ by an equivalence relation identifying $U_2 -\{p_m\}$ with $V_2-\{p_n\}$ defined by some homeomorphism between the two. Denote $W_1$ and $W_2$ as the respective images of $U_1$ and $V_1$ under the quotient map.
We have that $U_2$ and $V_2$ are open by construction. Furthermore, since $M$ and $N$ are open, we have that $U_1 = M – \{p_m\}$ and $V_1 = N – \{p_n\}$ are also open (since open sets with a point removed are still open).
Then we have the following open covers of $M$ and $N$ respectively:
$$
M = U_1 \cup U_2
$$
$$
N = V_1 \cup V_2
$$
We can then express
$$
M \# N = \underbrace{W_1}_{\text{open}} \cup \underbrace{W_2}_{\text{open}}
$$
as well since $M \cup N = V_1 \cup U_1$ and $W_1$ and $W_2$ are just the images of $U_1$ and $V_1$ under the natural quotient map used to define $M \# N$.
Consider that $W_1 \cap W_2$ is path connected since $W_1 \cap W_2$ is homeomorphic to $U_2 – \{p_m\} \cong V_2 – \{p_n\}$, both of which are punctured disks which deformation retract onto $S^{n-1}$. Since spheres of dimension greater than $2$ are simply connected (hence path connected), we have that $W_1 \cap W_2$ is path connected as well. This will allow us to later apply Van Kampen to $M \# N = W_1 \cup W_2$.
Now we have that
$$
U_1 \cap U_2 = U_2 – \{p_n\} \cong W_1 \cap W_2 \cong V_2 – \{p_m\} = V_1 \cap V_2
$$
so that from above we can say
$$
\pi_1(U_1 \cap U_2) \cong \pi(V_1 \cap V_2) \cong \pi(W_1 \cap W_2) \cong \{e\}
$$
Now since $n > 2$, we have that
$$
\underbrace{\pi_1(U_1) = \pi_1(M – \{p_m\}) \cong \pi_1(M) \cong \pi_1(W_1)}_{\text{removing a point doesn’t change fundamental group for $n > 2$}}
$$
and similarly
$$
\pi_1(V_1) = \pi_1(N – \{p_n\})\cong\pi_1(N) \cong \pi_1(W_2)
$$
Then applying Van Kampen on $M \# N = W_1 \cup W_2$ yields that
$$
\pi_1(W_1) \ast_{\pi_1(W_1 \cap W_2)} \pi_1(W_2) \cong \pi_1(M\#N)
$$
But since (8) yields that
$$
\pi_1(W_1) \ast_{\pi_1(W_1 \cap W_2)} \pi_1(W_2) \cong \pi_1(W_1) \ast_{\{e\}} \pi_1(W_2) \cong \pi_1(W_1) \ast \pi_1(W_2) \cong \pi_1(M) \ast \pi_1(N)
$$
we then have from (9) that
$$
\pi_1(M) \ast \pi_1(N) \cong \pi_1(M\#N)
$$
as desired.
Let’s be a little more precise. The direct sum $M\#N$ is defined as quotient of the disjoint union of $M$ and $N$ by an equivalence relation identifying $U_2 -\{p_m\}$ with $V_2-\{p_n\}$ defined by some homeomorphism between the two. Then denote by $W_1$ the image of $U_1$ under the quotient map and by $W_2$ the image of $V_1$ under the quotient map.
5) $W_1\cap W_2$ is connected because it is homeomorphic to $U_2-\{p_m\}\cong V_2-\{p_n\}$. These are punctured disks, which deformation retract onto spheres. Spheres of dimension greater than $0$ are connected.
(Note that this still works for $n=2$.)
6) See above: everything here intersects in the punctured disk where we glued to create the connect sum. Punctured disks deformation retract to spheres.
7) The space $S^{n-1}$ is definitely not contractible for $n\geq 2$. It is simply connected.
8) Here is where you need the hypothesis $n>2$. For dimension greater than $2$, removing a point does not change the fundamental group. To see why, show that the homomorphism $\pi_1(U_1)\to \pi_1(M)$ induced by the inclusion is actually an isomorphism. The intuition is that if any of the homotopies between generators $\pi_1(M)$ hits the ball you’ve removed, it can be “moved” so it “goes around” the ball.
(Removing a point does change higher homotopy groups — $M$ is not homotopy equivalent to $U_1$ — but does not change $\pi_1$.)