# Connected sum of simply connected manifolds is simply connected

From Topospaces

## Statement

Suppose is a natural number. Suppose and 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 . Let denote the Connected sum (?) of these manifolds. Then, 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

## Facts used

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

## Proof

*Fill this in later*