Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds.

This construction plays a key role in the classification of closed surfaces.

Although the construction uses the choice of the balls, the result is unique up to homeomorphism.

One can also make this operation work in the smooth category, and then the result is unique up to diffeomorphism.

There are subtle problems in the smooth case: not every diffeomorphism between the boundaries of the spheres gives the same composite manifold, even if the orientations are chosen correctly.

For example, Milnor showed that two 7-cells can be glued along their boundary so that the result is an exotic sphere homeomorphic but not diffeomorphic to a 7-sphere.

The fact that this construction is well-defined depends crucially on the disc theorem, which is not at all obvious.

The classification of closed surfaces, a foundational and historically significant result in topology, states that any closed surface can be expressed as the connected sum of a sphere with some number

a smooth, closed, oriented manifold, embedded as a submanifold into both

Suppose furthermore that there exists an isomorphism of normal bundles that reverses the orientation on each fiber.

, and the map is the orientation-reversing diffeomorphic involution on normal vectors.

is then the space obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism.

The sum is often denoted Its diffeomorphism type depends on the choice of the two embeddings of

is simply the connected sum as described in the preceding section, performed along each fiber.

a point recovers the connected sum of the preceding section.

of normal bundles exists whenever their Euler classes are opposite: Furthermore, in this case the structure group of the normal bundles is the circle group

; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from

to the circle, which in turn equals the first integral cohomology group

The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of

This implies, for example, that the sum can be carried out on a single manifold

For example, the connected sum of a 2-sphere at two distinct points of the sphere produces the 2-torus.

There is a closely related notion of the connected sum of two knots.

So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.

Under this operation, oriented knots in 3-space form a commutative monoid with unique prime factorization, which allows us to define what is meant by a prime knot.

Proof of commutativity can be seen by letting one summand shrink until it is very small and then pulling it along the other knot.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots.

This fact follows from additivity of knot genus; another proof relies on an infinite construction sometimes called the Mazur swindle.

In higher dimensions (with codimension at least three), it is possible to get an unknot by adding two nontrivial knots.

To see that A and B are unoriented equivalent, simply note that they both may be constructed from the same pair of disjoint knot projections as above, the only difference being the orientations of the knots.

Similarly, one sees that C and D may be constructed from the same pair of disjoint knot projections.