![Wedge of Two Circles](https://upload.wikimedia.org/wikipedia/commons/6/65/Wedge_of_Two_Circles.png)
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 wedge sum is a "one-point union" of a family of
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s. Specifically, if ''X'' and ''Y'' are
pointed space
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
s (i.e. topological spaces with distinguished basepoints
and
) the wedge sum of ''X'' and ''Y'' is the
quotient space of the
disjoint union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A ( ...
of ''X'' and ''Y'' by the identification
where
is the
equivalence closure
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but ...
of the relation
More generally, suppose
is a
indexed family
In mathematics, a family, or indexed family, is informally a collection of objects, each associated with an index from some index set. For example, a ''family of real numbers, indexed by the set of integers'' is a collection of real numbers, whe ...
of pointed spaces with basepoints
The wedge sum of the family is given by:
where
is the equivalence closure of the relation
In other words, the wedge sum is the joining of several spaces at a single point. This definition is sensitive to the choice of the basepoints
unless the spaces
are
homogeneous.
The wedge sum is again a pointed space, and the binary operation is
associative and
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
(up to homeomorphism).
Sometimes the wedge sum is called the wedge product, but this is not the same concept as the
exterior product
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of th ...
, which is also often called the wedge product.
Examples
The wedge sum of two circles is
homeomorphic to a
figure-eight space. The wedge sum of
circles is often called a ''
bouquet of circles
In mathematics, a rose (also known as a bouquet of ''n'' circles) is a topological space obtained by gluing together a collection of circles along a single point. The circles of the rose are called petals. Roses are important in algebraic topo ...
'', while a wedge product of arbitrary spheres is often called a bouquet of spheres.
A common construction in
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 ...
is to identify all of the points along the equator of an
-sphere
. Doing so results in two copies of the sphere, joined at the point that was the equator:
Let
be the map
that is, of identifying the equator down to a single point. Then addition of two elements
of the
-dimensional
homotopy group of a space
at the distinguished point
can be understood as the composition of
and
with
:
Here,
are maps which take a distinguished point
to the point
Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at
the point common to the wedge sum of the underlying spaces.
Categorical description
The wedge sum can be understood as the
coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
in the
category of pointed spaces
In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
. Alternatively, the wedge sum can be seen as the
pushout
A ''pushout'' is a student who leaves their school before graduation, through the encouragement of the school. A student who leaves of their own accord (e.g., to work or care for a child), rather than through the action of the school, is consider ...
of the diagram
in the
category of topological spaces In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again contin ...
(where
is any one-point space).
Properties
Van Kampen's theorem
A van is a type of road vehicle used for transporting goods or people. Depending on the type of van, it can be bigger or smaller than a pickup truck and SUV, and bigger than a common car. There is some varying in the scope of the word across ...
gives certain conditions (which are usually fulfilled for
well-behaved
In mathematics, when a mathematical phenomenon runs counter to some intuition, then the phenomenon is sometimes called pathological. On the other hand, if a phenomenon does not run counter to intuition,
it is sometimes called well-behaved. Th ...
spaces, such as
CW complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cl ...
es) under which the
fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of the wedge sum of two spaces
and
is the
free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and is ...
of the fundamental groups of
and
See also
*
Smash product
In topology, a branch of mathematics, the smash product of two pointed spaces (i.e. topological spaces with distinguished basepoints) (''X,'' ''x''0) and (''Y'', ''y''0) is the quotient of the product space ''X'' × ''Y'' under the id ...
*
Hawaiian earring
In mathematics, the Hawaiian earring \mathbb is the topological space defined by the union of circles in the Euclidean plane \R^2 with center \left(\tfrac,0\right) and radius \tfrac for n = 1, 2, 3, \ldots endowed with the subspace topology:
...
, a topological space resembling, but not the same as, a wedge sum of countably many circles
References
* Rotman, Joseph. ''An Introduction to Algebraic Topology'', Springer, 2004, p. 153.
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Topology
Operations on structures
Homotopy theory