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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 ''x''<sub>0</sub> and ''y''<sub>0</sub>) the wedge sum of ''X'' and ''Y'' is the [[quotient space]] of the [[disjoint union (topology)|disjoint union]] of ''X'' and ''Y'' by the identification ''x''<sub>0</sub> ∼ ''y''<sub>0</sub>: | |||
:<math>X\vee Y = (X\amalg Y)\;/ \sim,\,</math> | |||
where ∼ is the [[equivalence closure]] of the relation {(''x''<sub>0</sub>,''y''<sub>0</sub>)}. | |||
More generally, suppose (''X''<sub>''i''</sub>{{pad|0.1em}})<sub>''i''∈''I''</sub> is a [[indexed family|family]] of pointed spaces with basepoints {''p''<sub>''i''</sub>{{pad|0.1em}}}. The wedge sum of the family is given by: | |||
:<math>\bigvee_i X_i = \coprod_i X_i\;/ \sim,\,</math> | |||
where ∼ is the equivalence relation {(''p<sub>i</sub>''{{pad|0.1em}}, ''p<sub>j</sub>''{{pad|0.1em}}) | ''i,j'' ∈ ''I''{{pad|0.1em}}}. | |||
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 {''p''<sub>''i''</sub>}, unless the spaces {''X''<sub>''i''</sub>{{pad|0.1em}}} are [[homogeneous space|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. | |||
==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]] is to identify all of the points along the equator of an ''n''-sphere <math>S^n</math>. Doing so results in two copies of the sphere, joined at the point that was the equator: | |||
:<math>S^n/{\sim} = S^n \vee S^n </math> | |||
Let <math>\Psi</math> be the map <math>\Psi:S^n\to S^n \vee S^n</math>, that is, of identifying the equator down to a single point. Then addition of two elements <math>f,g\in\pi_n(X,x_0)</math> of the ''n''-dimensional [[homotopy group]] <math>\pi_n(X,x_0)</math> of a space ''X'' at the distinguished point <math>x_0\in X</math> can be understood as the composition of <math>f</math> and <math>g</math> with <math>\Psi</math>: | |||
:<math>f+g = (f \vee g) \circ \Psi</math> | |||
Here, <math>f</math> and <math>g</math> are understood to be maps, <math>f:S^n\to X</math> and similarly for <math>g</math>, which take a distinguished point <math>s_0\in S^n</math> to a point <math>x_0\in X</math>. Note that the above defined the wedge sum of two functions, which was possible because <math>f(s_0)=g(s_0)=x_0</math>, 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 [[coproduct]] in the [[category of pointed spaces]]. Alternatively, the wedge sum can be seen as the [[pushout (category theory)|pushout]] of the diagram ''X'' ← {•} → ''Y'' in the [[category of topological spaces]] (where {•} is any one point space). | |||
==Properties== | |||
[[Van Kampen's theorem]] gives certain conditions (which are usually fulfilled for [[well-behaved]] 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''. | |||
==See also== | |||
*[[Smash product]] | |||
*[[Hawaiian earring]], 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. ISBN 0-387-96678-1 | |||
{{DEFAULTSORT:Wedge Sum}} | |||
[[Category:Topology]] | |||
[[Category:Binary operations]] | |||
[[Category:Homotopy theory]] |
Revision as of 22:05, 8 October 2013
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces (i.e. topological spaces with distinguished basepoints x0 and y0) the wedge sum of X and Y is the quotient space 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 (XiTemplate:Pad)i∈I is a family of pointed spaces with basepoints {piTemplate:Pad}. The wedge sum of the family is given by:
where ∼ is the equivalence relation {(piTemplate:Pad, pjTemplate:Pad) | i,j ∈ ITemplate:Pad}. 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 {XiTemplate:Pad} 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.
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 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.
Categorical description
The wedge sum can be understood as the coproduct 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).
Properties
Van Kampen's theorem gives certain conditions (which are usually fulfilled for well-behaved spaces, such as CW complexes) 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.
See also
- Smash product
- Hawaiian earring, 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. ISBN 0-387-96678-1