Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces.
More generally, one can also join manifolds together along identical submanifolds; this generalization is often called the fiber sum. There is also a closely related notion of a connected sum on knots, called the knot sum or composition of knots.
Connected sum at a point.
A connected sum of two "m"-dimensional manifolds is a manifold formed by deleting a ball inside each manifold and gluing together the resulting boundary spheres.
If both manifolds are oriented, there is a unique connected sum defined by having the gluing map reverse orientation. 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.
However, there is a canonical way to choose the gluing of formula_1 and formula_2 which gives a unique well defined connected sum. Choose embeddings formula_3 and formula_4 so that formula_5 preserves orientation and formula_6 reverses orientation. Now obtain formula_7 from the disjoint sum
by identifying formula_9 with formula_10 for each unit vector formula_11 and each formula_12. Choose the orientation for formula_7 which is compatible with formula_1 and formula_2. The fact that this construction is well-defined depends crucially on the disc theorem, which is not at all obvious. For further details, see 
The operation of connected sum is denoted by formula_16; for example formula_17 denotes the connected sum of formula_18 and formula_19.
The operation of connected sum has the sphere formula_20 as an identity; that is, formula_21 is homeomorphic (or diffeomorphic) to formula_22.
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 formula_23 of tori and some number formula_24 of real projective planes.
Connected sum along a submanifold.
Let formula_1 and formula_2 be two smooth, oriented manifolds of equal dimension and formula_27 a smooth, closed, oriented manifold, embedded as a submanifold into both formula_1 and formula_29 Suppose furthermore that there exists an isomorphism of normal bundles
that reverses the orientation on each fiber. Then formula_31 induces an orientation-preserving diffeomorphism
where each normal bundle formula_33 is diffeomorphically identified with a neighborhood formula_34 of formula_27 in formula_36, and the map
is the orientation-reversing diffeomorphic involution
on normal vectors. The connected sum of formula_1 and formula_2 along formula_27 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 formula_27 and on the choice of formula_31.
Loosely speaking, each normal fiber of the submanifold formula_27 contains a single point of formula_27, and the connected sum along formula_27 is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along formula_27 is often called the fiber sum.
The special case of formula_27 a point recovers the connected sum of the preceding section.
Connected sum along a codimension-two submanifold.
Another important special case occurs when the dimension of formula_27 is two less than that of the formula_36. Then the isomorphism formula_31 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 formula_55; it follows that the choice of embeddings can be canonically identified with the group of homotopy classes of maps from formula_27 to the circle, which in turn equals the first integral cohomology group formula_57. So the diffeomorphism type of the sum depends on the choice of formula_31 and a choice of element from formula_57.
A connected sum along a codimension-two formula_27 can also be carried out in the category of symplectic manifolds; this elaboration is called the symplectic sum.
Local operation.
The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of formula_27. This implies, for example, that the sum can be carried out on a single manifold formula_22 containing two disjoint copies of formula_27, with the effect of gluing formula_22 to itself. For example, the connected sum of a two-sphere at two distinct points of the sphere produces the two-torus.
Connected sum of knots.
There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a one-manifold, then the connected sum of two knots is just their connected sum as a one-dimensional manifold. However, the essential property of a knot is not its manifold structure (under which every knot is equivalent to a circle) but rather its embedding into the ambient space. So the connected sum of knots has a more elaborate definition that produces a well-defined embedding, as follows.
This procedure results in the projection of a new knot, a connected sum (or knot sum, or composition) of the original knots. For the connected sum of knots to be well defined, one has to consider oriented knots in 3-space. To define the connected sum for two oriented knots:
The resulting connected sum knot inherits an orientation consistent with the orientations of the two original knots, and the oriented ambient isotopy class of the result is well-defined, depending only on the oriented ambient isotopy classes of the original two knots.
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. The unknot is the unit. The two trefoil knots are the simplest prime knots. Higher-dimensional knots can be added by splicing the formula_65-spheres.
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.
If one does not take into account the orientations of the knots, the connected sum operation is not well defined on isotopy classes of (nonoriented) knots. To see this, consider two noninvertible knots "K, L" which are not equivalent (as unoriented knots); for example take the two pretzel knots "K" = "P"(3,5,7) and "L" = "P"(3,5,9). Let "K"+ and "K"− be "K" with its two inequivalent orientations, and let "L"+ and "L"− be "L" with its two inequivalent orientations. There are four oriented connected sums we may form:
The oriented ambient isotopy classes of these four oriented knots are all distinct. And, when one considers ambient isotopy of the knots without regard to orientation, there are two distinct equivalence classes: { "A" ~ "B" } and { "C" ~ "D" }. 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.