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.
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
M1
M2
i1:Dn → M1
i2:Dn → M2
i1
i2
M1n{\#}M2
(M1-i1(0))\sqcup(M2-i2(0))
i1(tu)
i2((1-t)u)
u\inSn-1
0<t<1
M1n{\#}M2
M1
M2
The operation of connected sum is denoted by
\#
The operation of connected sum has the sphere
Sm
Mn{\#}Sm
M
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
g
k
Let
M1
M2
V
M1
M2.
\psi:
| N | |
| M1 |
V\to
| N | |
| M2 |
V
that reverses the orientation on each fiber. Then
\psi
N1\setminusV\cong
| N | |
| M1 |
V\setminusV\to
| N | |
| M2 |
V\setminusV\congN2\setminusV,
where each normal bundle
| N | |
| Mi |
V
Ni
V
Mi
| N | |
| M1 |
V\setminusV\to
| N | |
| M2 |
V\setminusV
is the orientation-reversing diffeomorphic involution
v\mapstov/|v|2
on normal vectors. The connected sum of
M1
M2
V
(M1\setminusV)
| cup | |
| N1\setminusV=N2\setminusV |
(M2\setminusV)
obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted
(M1,V)n{\#}(M2,V).
Its diffeomorphism type depends on the choice of the two embeddings of
V
\psi
Loosely speaking, each normal fiber of the submanifold
V
V
V
V
The special case of
V
Another important special case occurs when the dimension of
V
Mi
\psi
| e\left(N | |
| M1 |
V\right)=
| -e\left(N | |
| M2 |
V\right).
SO(2)
V
H1(V)
\psi
H1(V)
A connected sum along a codimension-two
V
The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of
V
M
V
M
There is a closely related notion of the connected sum of two knots. In fact, if one regards a knot merely as a 1-manifold, then the connected sum of two knots is just their connected sum as a 1-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
n
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: and . 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.