Before placing my question as repeated, read it completely and see that my approach and my question are different from the others.
Let $M_1$ and $M_2$ two n-manifolds, ($n>2$). I have to proof that $\pi_1(M_1 \# M_2)$ is isomorphic to $\pi_1(M_1)*\pi_1(M_2)$.
My attempt is to use van Kampen. So let $U\cong M_1-{m_1} $ and $V \cong M_2-{m_2}$, where $U$ and $V$ are opens in $M_1\#M_2$, and $m_i \in M_i$, for $i=1,2$. Furthermore, $U \cap V \cong S^{n-1} \times R$. By van Kampen, we have $$ \pi_1(M_1 \# M_2) \cong \pi_1(U) *_{\pi_1(U\cap V)} \pi_1(V) \cong \pi_1(U)* \pi_1(V) $$
My question is: Is true that $\pi_1(U) \cong \pi_1(M_1)$? This is false if $n=2$, a simple example is a torus $T$, where $\pi_1(T) \cong Z\times Z$ but $\pi_1(T-{p_1}) \cong Z*Z$.
EDIT
I think this is true, because we have more dimension to contract a loop into a point.
My attempt is to use the van Kampen again. Let $B$ a coordinate ball contaning $m_1$, take $U_1 = B$ and $V_1 = M_1- m_1$. Both of this sets is open and path connected, furthermore, $U_1\cap V_1$ is homeomorphic to $R^n-\left\{0\right\}$. But how to conclude from this that the groups $\pi_1(M_1)$ and $\pi_1 (U)$ are isomorphic?