# Connected sum

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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
This article was adapted from an original article by V.I. Malykhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article