Connected sum of simply connected manifolds is simply connected

Statement

Suppose $n$ is a natural number. Suppose $M_1$ and $M_2$ are Simply connected manifold (?)s (in other words, they are both manifolds that are Simply connected space (?)s, i.e., they are both path-connected spaces with trivial Fundamental group (?)) of dimension $n$. Let $M_1\#M_2$ denote the Connected sum (?) of these manifolds. Then, $M_1\#M_2$ is also a simply connected manifold.

Related facts

Similar facts

• Fundamental group of connected sum is free product of fundamental groups in dimension at least three
• Connected sum of compact manifolds is compact
• Connected sum of orientable manifolds is orientable

Facts used

1. Point-deletion inclusion induces isomorphism on fundamental groups for manifold of dimension at least two
2. Seifert-van Kampen theorem

Proof

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