Inmathematics, specifically intopology, the operation ofconnected sum is a geometric modification onmanifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in theclassification of closed surfaces.

More generally, one can also join manifolds together along identical submanifolds; this generalization is often called thefiber sum.

There is also a closely related notion of aconnected sum onknots, called theknot sum orcomposition of knots.

Aconnected sum of twom-dimensionalmanifolds is a manifold formed by deleting aball inside each manifold andgluing together the resulting boundaryspheres.

If both manifolds areoriented, 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 uniqueup tohomeomorphism. One can also make this operation work in thesmoothcategory, and then the result is unique up todiffeomorphism. 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 anexotic sphere homeomorphic but not diffeomorphic to a 7-sphere.

However, there is a canonical way to choose the gluing of${\displaystyle M_1}$ and${\displaystyle M_2}$ which gives a unique well-defined connected sum.^[1] Choose embeddings${\displaystyle i_1:D_n\to M_1}$ and${\displaystyle i_2:D_n\to M_2}$ so that${\displaystyle i_1}$ preserves orientation and${\displaystyle i_2}$ reverses orientation. Now obtain${\displaystyle M_1\mathrm{\#}{M}_2}$ from the disjoint sum

${\displaystyle (M_1-i_1(0))\bigsqcup (M_2-i_2(0))}$

by identifying${\displaystyle i_1(tu)}$ with${\displaystyle i_2((1-t)u)}$ for eachunit vector${\displaystyle u\in S^{n-1}}$ and each. Choose the orientation for${\displaystyle M_1\mathrm{\#}{M}_2}$ which is compatible with${\displaystyle M_1}$ and${\displaystyle M_2}$. The fact that this construction is well-defined depends crucially on thedisc theorem, which is not at all obvious. For further details, see Kosinski,Differential Manifolds.^[2]

The operation of connected sum is denoted by${\displaystyle \mathrm{\#}}$.

The operation of connected sum has the sphere${\displaystyle S^m}$ as anidentity; that is,${\displaystyle M\mathrm{\#}{S}^m}$ is homeomorphic (or diffeomorphic) to${\displaystyle M}$.

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${\displaystyle g}$ oftori and some number${\displaystyle k}$ ofreal projective planes.

The connected sum can be defined along a submanifold.^[3]: §1

Let${\displaystyle M_1}$ and${\displaystyle M_2}$ be two smooth, oriented manifolds of equal dimension and${\displaystyle V}$ a smooth, closed, oriented manifold, embedded as a submanifold into both${\displaystyle M_1}$ and${\displaystyle M_2.}$ Suppose furthermore that there exists anisomorphism ofnormal bundles

${\displaystyle \psi :N_{M_1}V\to N_{M_2}V}$

that reverses the orientation on each fiber. Then${\displaystyle \psi }$ induces an orientation-preserving diffeomorphism

${\displaystyle N_1\setminus V\cong N_{M_1}V\setminus V\to N_{M_2}V\setminus V\cong N_2\setminus V,}$

where each normal bundle${\displaystyle N_{M_i}V}$ is diffeomorphically identified with aneighborhood${\displaystyle N_i}$ of${\displaystyle V}$ in${\displaystyle M_i}$, and the map

${\displaystyle N_{M_1}V\setminus V\to N_{M_2}V\setminus V}$

is the orientation-reversing diffeomorphicinvolution

${\displaystyle v\mapsto v/|v{|}^2}$

onnormal vectors. Theconnected sum of${\displaystyle M_1}$ and${\displaystyle M_2}$ along${\displaystyle V}$ is then the space

${\displaystyle (M_1\setminus V)\bigcup _{N_1\setminus V=N_2\setminus V}(M_2\setminus V)}$

obtained by gluing the deleted neighborhoods together by the orientation-preserving diffeomorphism. The sum is often denoted

${\displaystyle (M_1,V)\mathrm{\#}(M_2,V).}$

Its diffeomorphism type depends on the choice of the two embeddings of${\displaystyle V}$ and on the choice of${\displaystyle \psi }$.

Loosely speaking, each normal fiber of the submanifold${\displaystyle V}$ contains a single point of${\displaystyle V}$, and the connected sum along${\displaystyle V}$ is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along${\displaystyle V}$ is often called thefiber sum.

The special case of${\displaystyle V}$ a point recovers the connected sum of the preceding section.

Another important special case occurs when the dimension of${\displaystyle V}$ is two less than that of the${\displaystyle M_i}$. Then the isomorphism${\displaystyle \psi }$ of normal bundles exists whenever theirEuler classes are opposite:

${\displaystyle e\left(N_{M_1}V\right)=-e\left(N_{M_2}V\right).}$

Furthermore, in this case thestructure group of the normal bundles is thecircle group${\displaystyle SO(2)}$; it follows that the choice of embeddings can be canonically identified with thegroup ofhomotopy classes of maps from${\displaystyle V}$ to thecircle, which in turn equals the first integralcohomology group${\displaystyle H^1(V)}$. So the diffeomorphism type of the sum depends on the choice of${\displaystyle \psi }$ and a choice of element from${\displaystyle H^1(V)}$.

A connected sum along a codimension-two${\displaystyle V}$ can also be carried out in the category ofsymplectic manifolds; this elaboration is called thesymplectic sum.

The connected sum is a local operation on manifolds, meaning that it alters the summands only in aneighborhood of${\displaystyle V}$. This implies, for example, that the sum can be carried out on a single manifold${\displaystyle M}$ containing twodisjoint copies of${\displaystyle V}$, with the effect of gluing${\displaystyle M}$ to itself. 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. 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 itsembedding into theambient 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, aconnected sum (orknot sum, orcomposition) of the original knots. For the connected sum of knots to be well defined, one has to consideroriented knots in 3-space. To define the connected sum for two oriented knots:

1. Consider a planar projection of each knot and suppose these projections are disjoint.
2. Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knotsand so that the arcs of the knots on the sides of the rectangle are oriented around the boundary of the rectangle in thesame direction.
3. Now join the two knots together by deleting these arcs from the knots and adding the arcs that form the other pair of sides of the rectangle.

The resulting connected sum knot inherits an orientation consistent with the orientations of the two original knots, and the orientedambient 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 acommutative monoid with uniqueprime factorization, which allows us to define what is meant by aprime 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. Theunknot is the unit. The twotrefoil knots are the simplest prime knots. Higher-dimensional knots can be added by splicing the${\displaystyle n}$-spheres.

In three dimensions, the unknot cannot be written as the sum of two non-trivial knots. This fact follows from additivity ofknot genus; another proof relies on an infinite construction sometimes called theMazur swindle. In higher dimensions (with codimension at least three), it is possible to get an unknot by adding two nontrivial knots.

If one doesnot 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 knotsK, L which are not equivalent (as unoriented knots); for example take the twopretzel knotsK =P(3, 5, 7) andL =P(3, 5, 9). LetK₊ andK₋ beK with its two inequivalent orientations, and letL₊ andL₋ beL with its two inequivalent orientations. There are four oriented connected sums we may form:

• A =K₊ #L₊
• B =K₋ #L₋
• C =K₊ #L₋
• D =K₋ #L₊

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 aretwo distinct equivalence classes: {A ~B} and {C ~D}. To see thatA andB 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 thatC andD may be constructed from the same pair of disjoint knot projections.

• Band sum
• Prime decomposition (3-manifold)
• Manifold decomposition
• Satellite knot

• Robert Gompf: A new construction of symplectic manifolds,Annals of Mathematics 142 (1995), 527–595
• William S. Massey,A Basic Course in Algebraic Topology, Springer-Verlag, 1991.ISBN 0-387-97430-X.

• Differential topology
• Geometric topology
• Knot theory
• Operations on structures

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