Connected sum

Summary

The connected sum of two connected $n$-manifolds is formed by cutting a small open ball out of each. The two resulting boundary spheres are then glued together. The result is again a connected $n$-manifold, written $M \# N$. The operation is the basic way to build new manifolds from old ones. The classification of surfaces and the prime decomposition of $3$-manifolds are stated in terms of it.