# 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 , be oriented (compact) -manifolds and let be the -dimensional unit disc. Let be an orientation-preserving imbedding, . Now paste together (identify) the boundaries of and by means of to obtain the connected sum of and . The orientation of is that of and the differentiable structure of is uniquely determined independent of . Up to a diffeomorphism, the operation of taking connected sums is associative and commutative. The -dimensional sphere serves as a zero element, i.e. is diffeomorphic to the -dimensional manifold .

#### References

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

**How to Cite This Entry:**

Connected sum. V.I. Malykhin (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Connected_sum&oldid=13972