# Connected sum

*of a family of sets*

The union of these sets as a single connected set. The notion of a connected sum arose from the need to distinguish this sort of union from the notion of an unconnected or open-closed sum, that is, a union of disjoint sets such that the only connected subsets are those that are connected subsets of the summands in this union.

#### Comments

There are several obvious ways to implement the vague idea of a connected sum or union of spaces and sets: none particularly canonical. Definitions vary with the kind of objects under consideration.

The connected sum of two differentiable manifolds in differential topology is defined as follows. Let $ M _ {1} $, $ M _ {2} $ be oriented (compact) $ C ^ \infty $- manifolds and let $ D ^ {n} $ be the $ n $- dimensional unit disc. Let $ f _ {i} : D ^ {n} \rightarrow M _ {i} $ be an orientation-preserving imbedding, $ i = 1, 2 $. Now paste together (identify) the boundaries of $ M _ {1} \setminus f _ {1} ( D ^ {n} ) $ and $ M _ {2} \setminus f _ {2} ( D ^ {n} ) $ by means of $ f _ {2} \circ f _ {1} ^ { - 1 } $ to obtain the connected sum $ M _ {1} \# M _ {2} $ of $ M _ {1} $ and $ M _ {2} $. The orientation of $ M _ {1} \# M _ {2} $ is that of $ M _ {i} $ and the differentiable structure of $ M _ {1} \# M _ {2} $ is uniquely determined independent of $ f _ {i} $. Up to a diffeomorphism, the operation of taking connected sums is associative and commutative. The $ n $- dimensional sphere serves as a zero element, i.e. $ M \# S ^ {n} $ is diffeomorphic to the $ n $- dimensional manifold $ M $.

#### References

[a1] | M.W. Hirsch, "Differential topology" , Springer (1976) |

**How to Cite This Entry:**

Connected sum.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Connected_sum&oldid=46474