Fundamental group of connected sum is free product of fundamental groups in dimension at least three

Statement
Suppose $$n$$ is a natural number that is at least equal to 3. Suppose $$M_1$$ and $$M_2$$ are (possibly homeomorphic, possibly not) $$n$$-dimensional connected manifolds and $$M_1 \# M_2$$ is their fact about::connected sum. We then have the following relation between the fact about::fundamental groups of the manifolds:

$$\pi_1(M_1 \# M_2) = \pi_1(M_1) * \pi_1(M_2)$$

In other words, the fundamental group of the connected sum is the fact about::free product of the fundamental groups.

Related facts

 * Homology of connected sum: In most dimensions, it is the connected sum of the homology groups, but in the zeroth, top and second highest dimension, it could behave somewhat differently.
 * Fundamental group of wedge sum relative to basepoints with neighborhoods that deformation retract to them is free product of fundamental groups
 * Connected sum of simply connected manifolds is simply connected

Facts used

 * 1) uses::Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two
 * 2) uses::Seifert-van Kampen theorem
 * 3) uses::n-sphere is simply connected for n greater than 1: We in fact apply this to the sphere of dimension $$n - 1$$, and we need $$n - 1 \ge 2$$, which is why we need $$n \ge 3$$.