Wedge sum
Encyclopedia
In topology
, the wedge sum is a "one-point union" of a family of topological space
s. Specifically, if X and Y are pointed space
s (i.e. topological spaces with distinguished basepoints x0 and y0) the wedge sum of X and Y is the quotient
of the disjoint union
of X and Y by the identification x0 ∼ y0:
where ∼ is the equivalence closure of the relation {(x0,y0)}.
More generally, suppose (Xi)i∈I is a family of pointed spaces with basepoints {pi}. The wedge sum of the family is given by:
where ∼ is the equivalence relation {(pi, pj) | i,j ∈ I}.
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 {pi}, unless the spaces {Xi} are homogeneous
.
The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to isomorphism).
Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, which is also often called the wedge product.
, while a wedge product of arbitrary spheres is often called a bouquet of spheres.
A common construction in homotopy
is to identify all of the points along the equator of an n-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 n-dimensional homotopy group
of a space X at the distinguished point 
can be understood as the composition of 
and 
with 
:
Here,
and 
are understood to be maps, 
and similarly for 
, which take a distinguished point 
to a point 
. Note that the above defined the wedge sum of two functions, which was possible because 
, which was the point that is equivalenced in the wedge sum of the underlying spaces.
in the category of pointed spaces. Alternatively, the wedge sum can be seen as the pushout
of the diagram X ← {•} → Y in the category of topological spaces
(where {•} is any one point space).
spaces, such as CW complex
es) under which the fundamental group
of the wedge sum of two spaces X and Y is the free product
of the fundamental groups of X and Y.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, the wedge sum is a "one-point union" of a family of topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s. Specifically, if X and Y are pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s (i.e. topological spaces with distinguished basepoints x0 and y0) the wedge sum of X and Y is the quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the disjoint union
Disjoint union (topology)
In general topology and related areas of mathematics, the disjoint union of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology...
of X and Y by the identification x0 ∼ y0:
where ∼ is the equivalence closure of the relation {(x0,y0)}.
More generally, suppose (Xi)i∈I is a family of pointed spaces with basepoints {pi}. The wedge sum of the family is given by:
where ∼ is the equivalence relation {(pi, pj) | i,j ∈ I}.
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 {pi}, unless the spaces {Xi} are homogeneous
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
.
The wedge sum is again a pointed space, and the binary operation is associative and commutative (up to isomorphism).
Sometimes the wedge sum is called the wedge product, but this is not the same concept as the exterior product, 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 n circles is often called a bouquet of circlesBouquet of circles
In mathematics, a rose 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 topology, where they are closely related to free groups.- Definition :A rose is a wedge sum of circles...
, while a wedge product of arbitrary spheres is often called a bouquet of spheres.
A common construction in homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
is to identify all of the points along the equator of an n-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 n-dimensional homotopy groupHomotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
of a space X at the distinguished point can be understood as the composition of and with :Here,
and are understood to be maps, and similarly for , which take a distinguished point to a point . Note that the above defined the wedge sum of two functions, which was possible because , which was the point that is equivalenced in the wedge sum of the underlying spaces.Categorical description
The wedge sum can be understood as the coproductCoproduct
In category theory, the coproduct, or categorical sum, is the category-theoretic construction which includes the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coproduct of a family of objects is essentially the...
in the category of pointed spaces. Alternatively, the wedge sum can be seen as the pushout
Pushout (category theory)
In category theory, a branch of mathematics, a pushout is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common domain: it is the colimit of the span X \leftarrow Z \rightarrow Y.The pushout is the...
of the diagram X ← {•} → Y in the category of topological spaces
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 continuous...
(where {•} is any one point space).
Properties
Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behavedWell-behaved
Mathematicians very frequently speak of whether a mathematical object — a function, a set, a space of one sort or another — is "well-behaved" or not. The term has no fixed formal definition, and is dependent on mathematical interests, fashion, and taste...
spaces, such as CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
es) under which the fundamental group
Fundamental group
In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...
of the wedge sum of two spaces X and Y is the free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
of the fundamental groups of X and Y.