In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the operation of connected sum is a geometric modification on
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s. 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
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
s, called the knot sum or composition of knots.
Connected sum at a point
A connected sum of two ''m''-dimensional
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s is a manifold formed by deleting a
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
inside each manifold and
gluing together the resulting boundary
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
s.
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
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
. One can also make this operation work in the
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
, and then the result is unique up to
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
. 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
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of al ...
homeomorphic but not diffeomorphic to a 7-sphere.
However, there is a canonical way to choose the gluing of
and
which gives a unique well defined connected sum. Choose embeddings
and
so that
preserves orientation and
reverses orientation. Now obtain
from the disjoint sum
:
by identifying
with
for each unit vector
and each
. Choose the orientation for
which is compatible with
and
. The fact that this construction is well-defined depends crucially on the
disc theorem, which is not at all obvious. For further details, see
[Kosinski, Differential Manifolds, Academic Press Inc (1992).]
The operation of connected sum is denoted by
; for example
denotes the connected sum of
and
.
The operation of connected sum has the sphere
as an
identity
Identity may refer to:
* Identity document
* Identity (philosophy)
* Identity (social science)
* Identity (mathematics)
Arts and entertainment Film and television
* ''Identity'' (1987 film), an Iranian film
* ''Identity'' (2003 film), ...
; that is,
is homeomorphic (or diffeomorphic) to
.
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
of
tori and some number
of
real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has bas ...
s.
Connected sum along a submanifold
Let
and
be two smooth, oriented manifolds of equal dimension and
a smooth, closed, oriented manifold, embedded as a submanifold into both
and
Suppose furthermore that there exists an isomorphism of
normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Definition
Riemannian manifold
Let (M,g) be a Riemannian m ...
s
:
that reverses the orientation on each fiber. Then
induces an orientation-preserving diffeomorphism
:
where each normal bundle
is diffeomorphically identified with a neighborhood
of
in
, and the map
:
is the orientation-reversing diffeomorphic involution
:
on
normal vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at ...
s. The connected sum of
and
along
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
and on the choice of
.
Loosely speaking, each normal fiber of the submanifold
contains a single point of
, and the connected sum along
is simply the connected sum as described in the preceding section, performed along each fiber. For this reason, the connected sum along
is often called the fiber sum.
The special case of
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
is two less than that of the
. Then the isomorphism
of normal bundles exists whenever their
Euler class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of ...
es are opposite:
:
Furthermore, in this case the
structure group of the normal bundles is the
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
; it follows that the choice of embeddings can be canonically identified with the group of
homotopy
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
classes of maps from
to the circle, which in turn equals the first integral
cohomology
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewe ...
group
. So the diffeomorphism type of the sum depends on the choice of
and a choice of element from
.
A connected sum along a codimension-two
can also be carried out in the category of
symplectic manifold
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form. The study of symplectic manifolds is called sympl ...
s; this elaboration is called the
symplectic sum In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold ...
.
Local operation
The connected sum is a local operation on manifolds, meaning that it alters the summands only in a
neighborhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...
of
. This implies, for example, that the sum can be carried out on a single manifold
containing two
disjoint copies of
, with the effect of gluing
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
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
into the
ambient space
An ambient space or ambient configuration space is the space surrounding an object.
While the ambient space and hodological space are both considered ways of perceiving penetrable space, the former perceives space as ''navigable'', while the latt ...
. 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:
# Consider a planar projection of each knot and suppose these projections are disjoint.
# Find a rectangle in the plane where one pair of sides are arcs along each knot but is otherwise disjoint from the knots and so that the arcs of the knots on the sides of the rectangle are oriented around the boundary of the rectangle in the same direction.
# 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 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
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
with unique
prime factorization, which allows us to define what is meant by a
prime knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
. 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 knot
In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be co ...
s. Higher-dimensional knots can be added by splicing the
-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:
* ''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 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.
See also
*
Band sum
*
Prime decomposition (3-manifold)
In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique ( up to homeomorphism) finite collection of prime 3-manifolds.
A manifold is ''prime'' if it canno ...
*
Manifold decomposition
In topology, a branch of mathematics, a manifold ''M'' may be decomposed or split by writing ''M'' as a combination of smaller pieces. When doing so, one must specify both what those pieces are and how they are put together to form ''M''.
Manifo ...
*
Satellite knot
In the mathematical theory of knots, a satellite knot is a knot that contains an incompressible, non boundary-parallel torus in its complement. Every knot is either hyperbolic, a torus, or a satellite knot. The class of satellite knots include ...
Further reading
*
Robert Gompf
The name Robert is an ancient Germanic given name, from Proto-Germanic "fame" and "bright" (''Hrōþiberhtaz''). Compare Old Dutch ''Robrecht'' and Old High German ''Hrodebert'' (a compound of '' Hruod'' ( non, Hróðr) "fame, glory, honou ...
: A new construction of symplectic manifolds, ''Annals of Mathematics'' 142 (1995), 527–595
*
William S. Massey
William Schumacher Massey (August 23, 1920 – June 17, 2017) was an American mathematician, known for his work in algebraic topology. The Massey product is named for him. He worked also on the formulation of spectral sequences by means of exact ...
, ''A Basic Course in Algebraic Topology'', Springer-Verlag, 1991. .
References
{{Reflist
Differential topology
Geometric topology
Knot theory
Operations on structures