bouquet of spheres
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In
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, 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 po ...
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 x_0 and y_0) 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 x_0 \sim y_0: X \vee Y = (X \amalg Y)\;/, where \,\sim\, is the equivalence closure of the relation \left\. More generally, suppose \left(X_i\right)_ 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, wher ...
of pointed spaces with basepoints \left(p_i\right)_. The wedge sum of the family is given by: \bigvee_ X_i = \coprod_ X_i\;/, where \,\sim\, is the equivalence closure of the relation \left\. 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 \left(p_i\right)_, unless the spaces \left(X_i\right)_ 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 of ...
(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 n circles is often called a '' bouquet of circles'', 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 defor ...
is to identify all of the points along the equator of an n-sphere S^n. Doing so results in two copies of the sphere, joined at the point that was the equator: S^n/ = S^n \vee S^n. Let \Psi be the map \Psi : S^n \to S^n \vee S^n, that is, of identifying the equator down to a single point. Then addition of two elements f, g \in \pi_n(X,x_0) of the n-dimensional homotopy group \pi_n(X,x_0) of a space X at the distinguished point x_0 \in X can be understood as the composition of f and g with \Psi: f + g = (f \vee g) \circ \Psi. Here, f, g : S^n \to X are maps which take a distinguished point s_0 \in S^n to the point x_0 \in X. Note that the above uses the wedge sum of two functions, which is possible precisely because they agree at s_0, 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 coproduc ...
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 of the diagram X \leftarrow \ \to Y in the category of topological spaces (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 of the wedge sum of two spaces X and Y 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 X and Y.


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. {{DEFAULTSORT:Wedge Sum Topology Operations on structures Homotopy theory