Showing that $\pi(M \# N) = \pi(M) \ast \pi(N)$ for $n$-dimensional manifolds $M$,$N$

Question

Problem: Let $M$ and $N$ be $n$-dimensional manifolds, where $n > 2$. Let $M \# N$ be their connected sum. Show that $\pi(M \# N) = \pi(M) \ast \pi(N)$.

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RE-EDITED Attempt:

1. 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 $U_2 = N - \{p_n\}$.

2. 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.

3. 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).

4. Then we have the following open covers of $M$ and $N$ respectively:

   $$ M = U_1 \cup U_2 $$ $$ N = V_1 \cup V_2 $$

5. 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 $M \# N$.

6. 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 $M \# N = W_1 \cup W_2$.

7. 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

   $$ \pi_1(U_1 \cap U_2) \cong \pi(V_1 \cap V_2) \cong \pi(W_1 \cap W_2) \cong \{e\} $$

8. 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

   $$ \pi_1(V_1) = \pi_1(N - \{p_n\})\cong\pi_1(N) \cong \pi_1(W_2) $$

9. Then applying Van Kampen on $M \# N = W_1 \cup W_2$ yields that

   $$ \pi_1(W_1) \ast_{\pi_1(W_1 \cap W_2)} \pi_1(W_2) \cong \pi_1(M\#N) $$

10. 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.

Answer 1

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 $\pi_1$.)