Wedge product of two circles
With the wedge product (after wedge English wedge; also called one-point union or bouquet ) of two dotted topological spaces and one designates their disjoint union, which is glued at one point (the base point). Formally, the definition is as follows:
X
∨
Y
{\ displaystyle X \ vee Y}
X
{\ displaystyle X}
Y
{\ displaystyle Y}
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∨
Y
=
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∐
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/
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p
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{\ displaystyle X \ vee Y = (X \ coprod Y) / (pt \ coprod pt)}
Here denotes the respective base point.
p
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{\ displaystyle pt}
The construction can also be generalized to any set of spaces:
⋁
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∈
I.
X
i
=
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∐
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∈
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i
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/
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∐
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{\ displaystyle \ bigvee _ {i \ in I} X_ {i} = (\ coprod _ {i \ in I} X_ {i}) / (\ coprod _ {i \ in I} pt_ {i})}
In a more abstract way, the wedge product can be understood as the co -product in the category of dotted topological spaces.
Role in algebraic topology
The wedge product behaves well with respect to some functors in algebraic topology . For example, for the fundamental group for locally contractible spaces
X
i
{\ displaystyle X_ {i}}
π
1
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⋁
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∈
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X
i
)
=
∗
i
∈
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π
1
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,
{\ displaystyle \ pi _ {1} (\ bigvee _ {i \ in I} X_ {i}) = * _ {i \ in I} \ pi _ {1} (X_ {i}),}
where denotes the free product of the groups.
∗
{\ displaystyle *}
In the singular homology :
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n
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=
⨁
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∈
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H
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{\ displaystyle H_ {n} (\ bigvee _ {i \ in I} X_ {i}, pt) = \ bigoplus _ {i \ in I} H_ {n} (X_ {i}, pt)}
One can wedge sum in an obvious way into the product embed the quotient
X
∨
Y
{\ displaystyle X \ vee Y}
X
×
Y
{\ displaystyle X \ times Y}
X
∧
Y
: =
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×
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/
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∨
Y
{\ displaystyle X \ wedge Y: = X \ times Y / X \ vee Y}
is the Smash product .
In particular, the reduced suspension is of importance in the stable homotopy theory.
Σ
X
: =
S.
1
∧
X
{\ displaystyle \ Sigma X: = S ^ {1} \ wedge X}
The wedge product is also used in the definition of the link in the homotopy groups .
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