---
tag: Y0K0K5NGWW
kind: concept
name: Connected sum
msc: [57N99, 57R65]
sources:
  - url: https://en.wikipedia.org/wiki/Connected_sum
same_as:
  wikidata: Q1890389
  wikipedia: 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.

## Explanations

### L1 human

Take two surfaces, for example two doughnut surfaces.
Punch a small round hole in each.
Join the two holes with a short tube, or simply sew the edges of the holes together.
The result is one surface with two handles, the surface of a pretzel with two holes.
Taking the connected sum with a sphere changes nothing, since the sphere with a hole is just a disk that plugs the hole again.